CMSA/Tsinghua Math-Science Literature Lecture: Marc Lackenby
2024-09-18 09:00 - 10:30 (America/New_York) | Virtual
CMSA/Tsinghua Math-Science Literature Lecture
Date: Wednesday, September 18, 2024
Time: 9:00 – 10:30 am ET
Location: Via Zoom Webinar
Speaker: Marc Lackenby, University of Oxford
Title: The complexity of knots
Abstract: In his final paper in 1954, Alan Turing wrote `No systematic method is yet known by which one can tell whether two knots are the same.’ Within the next 20 years, Wolfgang Haken and Geoffrey Hemion had discovered such a method. However, the computational complexity of this problem remains unknown. In my talk, I will give a survey on this area, that draws on the work of many low-dimensional topologists and geometers. Unfortunately, the current upper bounds on the computational complexity of the knot equivalence problem remain quite poor. However, there are some recent results indicating that, perhaps, knots are more tractable than they first seem. Specifically, I will explain a theorem that provides, for each knot type K, a polynomial p_K with the property that any two diagrams of K with n_1 and n_2 crossings differ by at most p_K(n_1) + p_K(n_2) Reidemeister moves.

Beginning in Spring 2020, the CMSA began hosting a lecture series on literature in the mathematical sciences, with a focus on significant developments in mathematics that have influenced the discipline, and the lifetime accomplishments of significant scholars.

Machine Learning in Science Education Panel Discussion
2024-09-30 15:30 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Machine Learning in Science Education Panel Discussion
Monday, Sep. 30, 2024
3:30-5:30 pm ET
Machine Learning is rapidly influencing many spheres of human activity. As part of the CMSA Mathematics and Machine Learning Program, this panel discussion will explore current and future uses of Machine Learning in science education. Panelists will make brief presentations, which will be followed by discussion and audience questions.
Gregory Kestin (Harvard University)
AI-Supported Activities: Design Principles and Impact on Student Learning
Logan McCarty (Harvard University)
Surveying the Landscape: Teaching and Learning with AI
Alexis Ross (Massachusetts Institute of Technology)
Adaptive Teaching towards Misconceptions with LLMs
Ilia Sucholutsky (New York University)
Why should machines have human-like representations? Towards student-centric AI tutors

Organizers:

Dan Freed (Harvard University and CMSA)
Michael Douglas (CMSA)

Math Science Lectures in Honor of Raoul Bott: Andrew Neitzke
2024-10-16 16:00 - 17:00, 2024-10-17 16:00 - 17:00 | Harvard Science Center, 1 Oxford Street, Cambridge, MA, 02138
Speaker: Andrew Neitzke, Yale University
Location: Harvard University Science Center Hall D & via Zoom webinar
Dates: October 16 & 17, 2024
Time: 4:00 pm

Wednesday, Oct. 16, 2024
Title: Abelianization in analysis of ODEs
Abstract: I will describe the exact WKB method for asymptotic analysis of families of ODEs in one variable, and its interpretation as a kind of abelianization procedure, which replaces GL(N)-connections over a Riemann surface by GL(1)-connections over an N-fold branched cover. This abelianization procedure connects exact WKB to various subjects in geometry (cluster algebras, moduli of Higgs bundles, enumerative geometry). One application is a conjectural description of Hitchin’s hyperkahler metric on the moduli of Higgs bundles; I will review some recent progress on these conjectures.

Thursday, Oct. 17, 2024
Title: Abelianization in quantum topology
Abstract: I will describe new applications of abelianization to various related subjects: perturbative Chern-Simons invariants, skein algebras, and conformal blocks. The aim is to explain how abelianization gives a unifying perspective on constructions familiar in each of these subjects (e.g. dilogarithmic formulas for Chern-Simons invariants, vertex models for computing quantum invariants of links, and iterated-fusion constructions of conformal blocks for the Virasoro algebra), and also suggests various extensions, which are just beginning to be explored.

Raoul Bott (9/24/1923 – 12/20/2005) is known for the Bott periodicity theorem, the Morse–Bott functions, and the Borel–Bott–Weil theorem.

Mathematics and Machine Learning Closing Workshop
2024-10-28 09:00 - 2024-10-30 17:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Mathematics and Machine Learning Closing Workshop
Dates: October 28 – Oct. 30, 2024
Location: Room G10, CMSA, 20 Garden Street, Cambridge MA
The closing workshop will provide a forum for discussing the most current research in these areas, including work in progress and recent results from program participants. We will devote one day to frontier topics in interactive theorem proving, such as mathematical library development and AI for mathematical search and theorem proving.

Youtube Playlist

Organizers

Francois Charton (Meta AI)
Michael R. Douglas (Harvard CMSA)
Michael Freedman (Harvard CMSA)
Fabian Ruehle (Northeastern)
Geordie Williamson (Univ. of Sydney)

Speakers

Ankit Anand, Google Deepmind Montreal
Jeremy Avigad, Carnegie Mellon University
Angelica Babei
Matej Balog, Deepmind
Gergely Bérczi, Aarhus University
Tristan Buckmaster, New York University
Giorgi Butbaia, University of New Hampshire
Edgar Costa, MIT
Alex Davies, DeepMind
Bin Dong, Beijing International Center for Mathematical Research
Kit Fraser-Taliente, University of Oxford
Javier Gomez-Serrano, Brown University
Jim Halverson, Northeastern University
Thomas Harvey, MIT
Amaury Hayat, Ecole des Ponts Paristech
Yang-Hui He, City University of London
Jürgen Jost, Max Planck Institute for Mathematics in the Sciences
Petros Koumoutsakos, Harvard University
Kyu-Hwan Lee, University of Connecticut
David Lowry-Duda, ICERM
Stephane Mallat, Flatiron/College de France
Abbas Mehrabian, Google Deepmind Montreal
Cengiz Pehlevan, Harvard University
Fabian Ruehle, Northeastern University
Eric Vanden-Eijnden, Courant/NYU
Adam Wagner, Worcester Polytechnic Institute
Melanie Matchett Wood, Harvard University

Schedule (download PDF)
Monday Oct. 28, 2024
9:00–9:30 amMorning refreshments
9:30–9:45 amIntroductions
9:45–10:45 amJürgen Jost, Max Planck Institute for Mathematics in the Sciences
Title: Data visualization with category theory and geometry
Abstract: While data often come in a high-dimensional feature space, they typically exhibit intrinsic constraints and regularities, and they can therefore often be represented on a lower-dimensional, but possibly highly curved Riemannian manifold. Still, for visualization purposes, that dimension still needs to be lowered to 2 or 3. We present the mathematical foundations for such schemes, in particular UMAP, and describe an improved such method.
10:45–11:00 amBreak
11:00 am–12:00 pmAnkit Anand, Google Deepmind Montreal, Abbas Mehrabian, Google Deepmind Montreal
Title: From Theorem Proving to Disproving: Modern machine learning versus classical heuristic search in automated theorem proving and extremal graph theory
Abstract: Machine learning is widely believed to outperform classical methods, but this is not always the case. Firstly, we describe how we adapted the idea of hindsight experience replay from reinforcement learning to the automated theorem proving domain, so as to use the intermediate data generated during unsuccessful proofs. We show that provers trained in this way can outperform previous machine learning approaches and compete with the state-of-the-art heuristic-based theorem prover E in its best configuration, on the popular benchmarks MPTP2078, M2k and Mizar40. The proofs generated by our algorithm are also almost always significantly shorter than E’s proofs. Based on this paper, which was presented at ICML 2022: https://proceedings.mlr.press/v162/aygun22a.html. Secondly, we study a central extremal graph theory problem inspired by a 1975 conjecture of Erdős, which aims to find graphs with a given size (number of nodes) that maximize the number of edges without having 3- or 4-cycles. We formulate this problem as a sequential decision-making problem and compare AlphaZero, a neural network-guided tree search, with tabu search, a heuristic local search method. Using either method, by introducing a curriculum—jump-starting the search for larger graphs using good graphs found at smaller sizes—we improve the state-of-the-art lower bounds for several sizes. Joint work with Tudor Berariu, Joonkyung Lee, Anurag Murty Naredla, Adam Zsolt Wagner, and other colleagues at Google DeepMind. Based on this paper, which was presented at IJCAI 2024: https://arxiv.org/abs/2311.03583.
12:00–1:30 pmLunch Break
1:30–2:30 pmFabian Ruehle, Northeastern University, Giorgi Butbaia, University of New Hampshire
Title: Rigorous results  from ML using RL
Abstract: We explain how to use Reinforcement Learning in Mathematics to obtain provably correct results. After a brief introduction to Reinforcement Learning, I will illustrate the idea using an example from Number Theory, where we solve a Diophantine Equation related to String Theory, and two from Knot Theory. The first knot theory problem is to identify unknots, while the second is concerned with identifying so-called ribbon knots. The latter play an important role in the search for counter-examples to the smooth Poincare conjecture.
2:30–2:45 pmBreak
2:45–3:45 pmCengiz Pehlevan, Harvard University
Title: Solvable Models of Scaling and Emergence in Deep Learning
3:45–4:00 pmBreak
4:00–4:30 pmMatej Balog, Deepmindvia Zoom
Title: FunSearch: Mathematical discoveries from program search with large language models
Abstract: Large language models (LLMs) have demonstrated tremendous capabilities in solving complex tasks, from quantitative reasoning to understanding natural language. However, LLMs sometimes suffer from confabulations (or hallucinations), which can result in them making plausible but incorrect statements. This hinders the use of current large models in scientific discovery. We introduce FunSearch (short for searching in the function space), an evolutionary procedure based on pairing a pretrained LLM with a systematic evaluator. We demonstrate the effectiveness of this approach to surpass the best-known results in important problems, pushing the boundary of existing LLM-based approaches. Applying FunSearch to a central problem in extremal combinatorics—the cap set problem—we discover new constructions of large cap sets going beyond the best-known ones, both in finite dimensional and asymptotic cases. This shows that it is possible to make discoveries for established open problems using LLMs. We showcase the generality of FunSearch by applying it to an algorithmic problem, online bin packing, finding new heuristics that improve on widely used baselines. In contrast to most computer search approaches, FunSearch searches for programs that describe how to solve a problem, rather than what the solution is. Beyond being an effective and scalable strategy, discovered programs tend to be more interpretable than raw solutions, enabling feedback loops between domain experts and FunSearch, and the deployment of such programs in real-world applications.
4:30–5:00 pmEdgar Costa, MIT
Title: Machine learning L-functions
Abstract: We report on multiple experiments related to L-functions data. L-functions are complex functions that encode significant information about number theory and algebraic geometry, playing a crucial part in the Langlands program, a foundational set of conjectures connecting number theory with other mathematical domains. We focused on two L-function datasets. The first includes about 250k rational L-functions of small arithmetic complexity with diverse origins. Multiple dimensionality reduction techniques were used to analyze invariants and behavioral trends, focusing on how differing origins impact the results. The second dataset is composed of L-functions associated with Maass forms. Although these L-functions are non-rational, they also share the low arithmetic complexity of the first dataset. The crux of our investigation here is determining whether this set manifests similar characteristics to the first one. Based on this exploration, we propose a simple heuristic method to deduce their Fricke sign, an unknown invariant for 40% of the data. This is joint work with: Joanna Biere, Giorgi Butbaia, Alyson Deines, Kyu-Hwan Lee, David Lowry-Duda, Tom Oliver, Tamara Veenstra, and Yidi Qi.

Tuesday Oct. 29, 2024
9:15–9:45 amMorning refreshments
9:45–10:45 amYang-Hui He, London Institute for Mathematical Sciences Via Zoom
Title: AI assisted mathematics
Abstract: We argue how AI can assist mathematics in three ways: theorem-proving, conjecture formulation, and language processing. Inspired by initial experiments in geometry and string theory in 2017, we summarize how this emerging field has grown over the past years, and show how various machine-learning algorithms can help with pattern detection across disciplines ranging from algebraic geometry to representation theory, to combinatorics, and to number theory. At the heart of the program is the question how does AI help with theoretical discovery, and the implications for the future of mathematics.
10:45–11:00 amBreak
11:00 –11:30Angelica Babei
Title: Learning Euler factors of elliptic curves with transformers
Abstract: The L-function of an elliptic curve is at the core of the BSD conjecture, and its Euler factors encode important arithmetic information about the curve. For example, understanding these Euler factors using machine learning techniques has recently led to discovering the phenomenon of murmurations. In this talk, we present some results on learning Euler factors based on 1. other nearby factors, and 2. the Weierstrass equation of the curve.
11:30–12:00 pmDavid Lowry-Duda, ICERM
Title: Exploring patterns in number theory with deep learning: a case study with the Möbius and squarefree indicator functionsAbstract: We report on experiments using neural networks and Int2Int, the integer sequence to integer sequence transformer made by François Charton for this CMSA program. We initially study the Möbius function. This function appears as the coefficients of the reciprocal of the Riemann zeta function and is famously hard to understand. Predicting the Möbius function is closely related to predicting the squarefree indicator function, leading us to perform similar experiments there. Finally, we’ll discuss how varying the input representation and model affects the strength of the predictions and allows us to explain most (but not all) of the predictive strength and behavior.
12:00–1:30 pmLunch
1:30–2:30 pmAmaury Hayat, Ecole des Ponts Paristech, Melanie Matchett Wood, Harvard University, Alex Davies, DeepMind, Jeremy Avigad, Carnegie Mellon University
Title: Machine learning and theorem proving
Abstract: Recent successes in machine learning have raised hopes that neural networks could one day assist mathematicians in proving theorems. This raises the question of an appropriate setting to apply machine learning methods to theorem proving. Formal languages, such as Lean, provide automatic verification of mathematical proofs and thus offer a natural environment. Nevertheless, some challenges emerge, particularly because these languages are often designed to verify correctness rather than find a solution, while mathematicians often perform reasoning steps to do both at the same time. This talk will present recent applications of machine learning methods to theorem proving in Lean, highlight current challenges, and explore what these developments might mean for the future of mathematics.

2:30–2:45 pmBreak
2:45–3:45 pmAdam Wagner, Worcester Polytechnic Institute, Kit Fraser-Taliente, University of Oxford, Gergely Bérczi, Aarhus University, Thomas Harvey, MIT
Title: Sparse subgraphs of the d-cube with diameter d
Abstract: Erdos et al studied spanning subgraphs of the $d$-cube which have the same diameter $d$ as the cube itself. They asked the following natural question: what is the maximum number of edges one can delete from the $d$-dimensional hypercube, without increasing its diameter? We will discuss how we can use PatternBoost, a simple machine learning algorithm that alternates local and global optimization steps, to find good constructions for this problem
3:45–4:00 pmBreak
4:00–4:30 pmPetros Koumoutsakos, Harvard University
4:30–5:00 pm
Stéphane Mallat, Flatiron/College de France
Title: Image Generation by Score Diffusion and the Renormalisation Group
Abstract: Score based diffusions generate impressive models of images, sounds and complex physical systems. Are they generalising or memorising? How can deep network estimate high-dimensional scores without curse of dimensionality? This talk shows that generalisation does occur for deep network estimation of scores, with enough training data. The ability to avoid the curse of dimensionality seems to rely on multiscale properties revealed by a renormalisation group decomposition coming from statistical physics. Applications to models of turbulences will be introduced and discussed.

Wednesday Oct. 30, 2024
9:15–9:45 amMorning refreshments
9:45–10:45 amBin Dong, Beijing International Center for Mathematical Research(via Zoom)
Title: AI for Mathematics: From Digitization to Intelligentization
Abstract: This presentation explores the synergistic relationship between AI and mathematics, beginning with a brief historical overview of their mutually beneficial interactions. It then examines notable existing work in AI for mathematics, highlighting their achievements and limitations. Next, I will share preliminary findings from the ongoing AI4M research project at Peking University, including our work on creating high-quality mathematical datasets through formalization (digitization), and our future plans for developing intelligent applications using these datasets. The presentation concludes with a forward-looking perspective on the opportunities and challenges within this exciting interdisciplinary field.
10:45–11:00 am Break
11:00 am–12:00 pm Eric Vanden-Eijnden, Courant/NYUvia Zoom
Title: Generative modeling with flows and diffusions, with applications to scientific computing.
Abstract: Generative models based on dynamical transport have recently led to significant advances in unsupervised learning. At mathematical level, these models are primarily designed around the construction of a map between two probability distributions that transform samples from the first into samples from the second. While these methods were first introduced in the context of image generation, they have found a wide range of applications, including in scientific computing where they offer interesting ways to reconsider complex problems once thought intractable because of the curse of dimensionality. In this talk, I will discuss the mathematical underpinning of generative models based on flows and diffusions, and show how a better understanding of their inner workings can help improve their design. These results indicate how to structure the transport to best reach complex target distributions while maintaining computational efficiency, both at learning and sampling stages. I will also discuss applications of generative AI in scientific computing, in particular in the context of application with models and no data (as opposed to the more standard data andno model), such as Monte Carlo sampling, with applications to the statistical mechanics and Bayesian inference, as well as the numerical integration and interpretation of random dynamical systems driven out of equilibrium.
12:00–1:30 pm Lunch
1:30–2:30 pm Kyu-Hwan Lee, University of Connecticut
Title: Discovering New Mathematical Structures with Machine Learning
Abstract: Can machine learning help discover new mathematical structures? In this talk, I will present two case studies: murmurations in number theory and loadings of partitions related to Kronecker coefficients in representation theory and combinatorics. The focus will be on the paradigm of examining mathematical objects collectively, rather than individually, to create datasets suitable for machine learning experiments and interpretations.
2:30–2:45 pm Break
2:45–3:45 pm James Halverson, Northeastern University
Title: Learning the Topological Invariance of Knots
Abstract: This talk focuses on using machine learning for the defining problem in knot theory, the classification of knots up to ambient space isotopy. We will train transformers and convolutional neural networks to distinguish topologically inequivalent knots, given only representatives of the classes and no a priori knowledge of topological invariants. In this scheme, we find that equivalent knots are well-clustered in the embedding space of the neural network, and a trained decoder maps effectively from the embedding space back to knot space. Preliminary results will be presented on a new approach to resolving the Jones unknot conjecture.
3:45–4:00 pmBreak
4:00–5:00 pm Tristan Buckmaster, New York University, Javier Gomez-Serrano, Brown Universityvia Zoom

Image by Sue Side. https://www.sueside.com/

CMSA/Tsinghua Math-Science Literature Lecture: Bjorn Poonen, MIT
2024-11-21 09:00 - 10:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA/Tsinghua Math-Science Literature Lecture
Date: November 21, 2024
Time: 9:00 – 10:30 am ET
Location: CMSA G10, 20 Garden Street, Cambridge MA & via Zoom
Speaker: Bjorn Poonen, MIT
Title: Ranks of elliptic curves
Abstract: Elliptic curves are simplest varieties whose rational points are not fully understood, and they are the simplest projective varieties with a nontrivial group structure. In 1922 Mordell proved that the group of rational points on an elliptic curve is finitely generated. We will survey what is known and what is believed about this group.

Beginning in Spring 2020, the CMSA began hosting a lecture series on literature in the mathematical sciences, with a focus on significant developments in mathematics that have influenced the discipline, and the lifetime accomplishments of significant scholars.

Workshop on Symmetries and Gravity
2025-01-21 09:00 - 2025-01-24 17:00 | CMSA 20 Garden Street Cambridge, Massachusetts 02138 United States
Workshop on Symmetries and Gravity
Dates: January 21-24, 2025
Location: Harvard CMSA, 20 Garden Street, Cambridge, MA 02138
Organizers: Ibrahima Bah (Johns Hopkins University), Patrick Jefferson (Johns Hopkins University), Yiming Chen (Stanford University)
Description: There is a widespread belief, that has its origins in work from the 70s, that a theory of quantum gravity cannot admit global symmetries. Traditionally, this was seen only as a qualitative statement about ordinary symmetries, but there have since been a number of developments that have both widened its scope and sharpened its implications. Recent work has greatly broadened the definition of global symmetries, and characterizes them in terms of topological operators in quantum systems. Concurrently, insights from quantum gravity have suggested ways to quantify the extent of global symmetry violation. Additionally, advances in the swampland program, along with amplitudes and bootstrap techniques, have shown ways to turn high-energy statements into constraints on low-energy effective field theories. In string theory, there are more concrete statements on charge violation in gravity, with proofs in limited context. In general, however, “no global symmetries in quantum gravity” continues to be an open conjecture with broad implications on the nature of quantum gravity and low-energy effective field theory. The main goal of the meeting is to bring together experts in the various arenas of research above, to reassess and develop new strategies for making progress on this long-standing open problem. Some objectives include understanding the violation of various generalized and categorical symmetries in gravity more cohesively, and putting concrete bounds on global charge-violating amplitudes at low energies.
Partially funded by the Simons Collaboration on Global Categorical Symmetries.

Confirmed Participants

Tom Banks, Rutgers
Federico Bonetti, Durham University
Christian Copetti, Oxford
Hector Parra De Freitas, Harvard
Damian van de Heisteeg, Harvard CMSA
Matilda Delgado, IFT
Michele Del-Zotto, Uppsala University
Muldrow Etheredge, UMass Amherst
Iñaki Garcia-Etxebarria, Durham University
Eduardo Garcia-Valdecasas, SISSA, Trieste
Naomi Gendler, Harvard
Kelian Haring, CERN
Daniel Harlow, MIT
Jonathan Heckman, University of Pennsylvania
Ben Heidenreich, UMass Amherst
Aidan Herderschee, IAS
Max Huebner, Uppsala University
Jesús Huertas, Instituto de Física Teórica
Theo Johnson-Freyd, Dalhousie University
Ho Tat Lam, MIT
Adam Levine, MIT
Yue-Zhou Li, Princeton
Jacob McNamara, Caltech
Ruben Minasian, Institute of Theoretical Physics Saclay
Amineh Mohseni, Harvard
Miguel Montero, IFT
Gregory Moore, Rutgers
Leonardo Rastelli, Stony Brook
Matt Reece, Harvard University
Grant Remmen, New York University
Diego Rodriguez-Gomez, University of Oviedo
Konstantinos Roumpedakis, Johns Hopkins
Tom Rudelius, Durham University
Vivek Saxena, Stony Brook and Rutgers
Edgar Shaghoulian, UC Santa Cruz
Shu-Heng Shao, Stony Brook and MIT
Adar Sharon, Simons Center for Geometry and Physics, Stony Brook
Irene Valenzuela, IFT and CERN
Thomas Waddleton, Johns Hopkins
Hao Xu, University of Göttingen
Xingyang Yu, Virginia Tech

Schedule
Tuesday, Jan. 21, 2025
9:00 – 9:30 am
Breakfast
9:30 – 11:00 am
Review
Leonardo Rastelli, Stony Brook University
Youtube Video
11:00 – 11:15 am
Break
11:15 am– 12:00 pm
Kelian Haring, CERN
Title: S-matrix bootstrap and black hole production
Abstract: I will review the expected effects of black hole production in scattering amplitudes. I will consider both symmetry-breaking and elastic amplitudes. I will argue that, in the elastic case, this input can be computationally useful. Then, I will discuss an example of a symmetry-breaking Wilson coefficient as a concrete target for the bootstrap.
Youtube Video
12:00 – 1:45 pm
Lunch Break
1:45 – 2:30 pm
Ho Tat Lam, MIT
Title: Global Aspects of Exactly Marginal Current-Current Deformations
Abstract: Conformal field theories connected by exactly marginal deformations form conformal manifolds. In two dimensions, a large class of conformal manifolds is generated by bilinears of currents, known as current-current deformations. In this talk, we will revisit these deformations and prove that a dense set of points on the conformal manifolds are related to the seed theory through discrete gauging. This perspective enables us to connect the topology of the conformal manifolds with the anomalies of the currents and to show that enhanced invertible symmetries reorganized into non-invertible symmetries away from the symmetry enhanced points. We will also discuss how current-current deformation can be understood from the recently proposed continuous abelian symmetry topological field theory.
Youtube Video
2:30 – 3:15 pm
Tom Banks, Rutgers University
Title: Symmetries in the Hilbert Bundle Formulation of Quantum Gravity
Abstract: Results of Jacobson, Carlip and Solodukhin from the 1990s, as extended by Banks and Zurek in 2021, point to a solution of Einstein’s equations as a hydrodynamic approximation to a quantum gravitational system, which determines the density matrix assigned to each subsystem corresponding to a hydrodynamic causal diamond in terms of the Virasoro generator of a cut off 1 + 1 dimensional CFT. The full quantum system can be viewed as a Hilbert bundle over the space of timelike geodesics on the hydrodynamic background. Isometries of the background generically map one fiber of the bundle to another and don’t act on a fixed Hilbert space. Time evolution along each geodesic is given by an analog of “one sided modular flow in QFT”, which in this context is a sequence of unitary embeddings of smaller diamond Hilbert spaces into larger ones. A full unitary map on the entire Hilbert space of a fiber requires a “Quantum Principle of Relativity” equating the entanglement spectrum of the density matrix of the largest diamond in the overlap between diamonds on different geodesics. In principle, this implies asymptotic symmetries for spacetimes which have them. For the case of asymptotically AdS space, this can be worked out in a hand waving way by using the Tensor Network Renormalization Group of Evenbly and Vidal. For asymptotically flat space we probably require a better non-perturbative definition of the space of asymptotic states to understand the action of the Poincare group. For de Sitter space there is no sense in which the de Sitter group acts on any set of asymptotic observables. Ironically, there IS an approximate de Sitter invariance of at least low point inflationary correlation functions, but I will not have time to discuss that.
Youtube Video
3:15 – 3:45 pm
Break
3:45 – 4:30 pm
Christian Copetti, Oxford University
Title: Non-Invertible Symmetries, Generalized Gauging and Factorization
Abstract: We analyze a toy model for low dimensional holography, in which the dual theory is an ensemble over 2d RCFTs. This simple model lacks factorization on multi-boundary geometries and at the same time has a (generalized) bulk global symmetry. We show that both problems are solved if the path integral prescription is modified by a generalized gauging operation, which can also be interpreted as the insertion of (topological) EOW branes.
Youtube Video
4:30 – 5:00 pm
Free Discussion
Wednesday, Jan. 22, 2025
9:00 – 9:30 am
Breakfast
9:30 – 11:00 am
Review
Daniel Harlow, MIT
Youtube Video
11:00 – 11:15 am
Break
11:15 am– 12:00 pm
Jacob McNamara, Caltech
Title: Conserved Charges of Closed Universes
Abstract: In quantum gravity, while our standard notions of symmetry operator become hard to define, the notion of conserved charge continues to make sense. After a general discussion of conserved charges in quantum gravity, I will present a new kinematic invariant of a gravitational path integral that refines the cobordism groups of quantum gravity: the (higher) category of closed universe charges. By categorifying an argument of Coleman, Giddings, and Strominger, I will argue that conserved charges in quantum gravity of any form degree arise only due to a categorical version of ensemble holography.
Youtube Video
12:00 – 1:45 pm
Lunch Break
1:45 – 2:30 pm
Federico Bonetti, Durham University
Title: Aspects of Categorical Symmetries for Branes
Youtube Video
2:30 – 3:15 pm
Konstantinos Roumpedakis, Johns Hopkins University
Title: Symmetry Operators and Gravity
Abstract: It is widely believed that there are no conserved charges in a theory of gravity, based on arguments involving black holes. Moreover, the modern approach to study global symmetries is the language of topological operators. In this talk, I will revisit the absence of global symmetries in a theory of gravity from the perspective of topological operators. More specifically, I will argue that topological operators for continuous symmetries written in terms of currents need regularization, which effectively gives them a small but finite width. The regulated operator is a finite tension object which fluctuates. In the zero-width limit these fluctuations freeze, recovering the properties of a topological operator. When gravity is turned on, the zero-width limit becomes ill-defined, thereby prohibiting the existence of topological operators. This talk is based on work in collaboration with Ibrahima Bah, Patrick Jefferson, and Thomas Waddleton.
Youtube Video
3:15 – 3:45 pm
Break
3:45 – 4:30 pm
Iñaki Garcia-Etxebarria, Durham University
Title: Some aspects of symmetry descent
Abstract: SymTFTs allow us to encode the symmetry structure of Quantum Field Theories in a convenient way. For those QFTs that arise in geometric engineering, or holography, we expect to be able to derive the SymTFT from the geometric data of the string background. This talk will describe some recent progress in this direction, together with S. Hosseini and with F. Gagliano.
Youtube Video
4:30 – 5:00 pm
Free Discussion
6:00 pm
Dinner at Changsho Restaurant
Thursday, Jan. 23, 2025
9:00 – 9:30 am
Breakfast
9:30 – 11:00 am
Review
Irene Valenzuela, IFT and CERN
Title: Breaking of Symmetries in Gravity
Abstract: Global symmetries are expected to be broken (or gauged) in quantum gravity. However, we can still learn a lot from understanding the mechanisms by which quantum gravity avoids them and quantifying their breaking. Remarkably, several Swampland constraints can be reinterpreted as consequences of breaking global symmetries. I will first focus on quantifying the minimal symmetry violation of axionic shift symmetries, and show how the bottom-up expectation based on black holes seems to hold in string theory examples. I will then discuss how this symmetry violation changes as we move in the moduli space, implying a drop-off of the quantum gravity cut-off when the symmetry is approximate. Finally, I will discuss the fate of non-invertible symmetries in string theory, and how they are typically broken at loop level. Nevertheless, these approximate non-invertible symmetries are still useful to fill in the gaps in the worldsheet proofs of some Swampland conjectures.
11:00 – 11:15 am
Break
11:15 am– 12:00 pm
Tom Rudelius, Durham University
Title: A Symmetry-Centric Perspective on the Geometry of the Landscape and the Swampland
Abstract: As famously observed by Ooguri and Vafa nearly twenty years ago, scalar field moduli spaces in quantum gravity appear to exhibit various universal features. For instance, they seem to be infinite in diameter, have trivial fundamental group, and feature towers of massive particles that become light in their asymptotic limits. In this talk, I will explain how these features can be reformulated in more modern language using generalized notions of global symmetries. Such symmetries are ubiquitous in non-gravitational quantum field theories, but it is widely believed that they must be either gauged or broken in quantum gravity. We will see that the observations of Ooguri and Vafa can be understood as consequences of such gauging or breaking.
12:00 – 1:45 pm
Lunch Break
1:45 – 2:30 pm
Miguel Montero, IFT
Title: Parity symmetry breaking and the membrane Weak Gravity Conjecture
Abstract: Symmetries are expected to be broken or gauged in any consistent theory of quantum gravity, and this also applies to spacetime symmetries such as parity. I will argue that, in the context of 4d N=1 AdS vacua of string theory, the Weak Gravity Conjecture for membranes case only holds if the vacuum has an exact (i.e. gauged) parity symmetry of Pin+ type. I will give top-down examples of M-theory vacua illustrating this, and show that in the DGKT scenario (a putative massive IIA vacuum with scale separation, whose full consistency is the subject of some debate in the literature) there is no parity symmetry, and the membrane WGC is violated. Thus, there is either a pathology in DGKT, or the membrane WGC is wrong. Both possibilities would have interesting consequences, and I will outline ongoing work to figure out which one is it.
2:30 – 3:15 pm
Matilda Delgado, IFT
Title: Dualities, Defects and Duality Defects
Abstract: I will outline how duality symmetries in quantum gravity theories naturally predict the existence of defects associated with duality transformations. While some of these objects are well-understood and extensively studied, others remain enigmatic; I will discuss this with examples. I will conclude by discussing the potential role of dualities in characterising the UV defects predicted by cobordism conjecture (and more generally by the no global symmetries conjecture). Based on: [2412.03640]
3:15 – 3:45 pm
Break
3:45 – 4:30 pm
Max Hübner, Uppsala University
Title: Metric Isometries, Holography, and Continuous Symmetry Operators
Abstract: In the AdS/CFT correspondence, a topological symmetry operator of the boundary CFT is dual to a dynamical brane in the gravitational AdS bulk. Said differently, this predicts a dynamical brane for every global symmetry of the boundary CFT. We analyze this correspondence for continuous symmetries which arise from a consistent truncation of isometries on the “internal” factor X of AdS × X. We discuss how this perspective can be used to both derive properties of the topological symmetry operators and non-topological properties of their bulk duals.
4:30 – 5:00 pm
Free Discussion

Friday, Jan. 24, 2025
9:00 – 9:30 am
Breakfast
9:30 – 10:15 am
Jonathan Heckman, University of Pennsylvania
Title: Cobordism Utopia
Abstract: On general grounds one expects that global symmetries are absent in quantum gravity. We discuss some aspects of this issue, focusing on the recently proposed Swampland Cobordism Conjecture, and related conjectures connected with completeness of the spectrum of states charged under symmetries. In particular, the U-dualities of M-theory provide an excellent arena both for testing aspects of these conjectures, as well as predicting the existence of new dynamical objects. We also comment on how this approach connects to related top down and holographic approaches to constructing and studying gauging and breaking symmetries in quantum gravity. Based on joint work to appear with Braeger, Debray, Dierigl, and Montero.
10:15 – 11:00 am
Naomi Gendler, Harvard University
11:00 – 11:15 am
Break
11:15 am – 12:00 pm
Diego Rodriguez-Gomez, University of Oviedo
Title: Non-BPS branes as holographic symmetry operators
Abstract: We propose a holographic description of the operators implementing continuous global symmetries that are dual to superstring gauge fields in terms of non-BPS D- branes, and consider some possible further extensions.
12:00 – 12:45 pm
Greg Moore, Rutgers University
Title: Summing Over Bordisms In 2d TQFT: Déjà Vu
Abstract: This is basically a rerun of a talk I gave on zoom for the CMSA on March 16, 2022. I will review the contents of a paper I wrote with Anindya Banerjee 2201.00903, but including a few minor updates. I will describe a construction in Topological Field Theory (TFT) which was motivated by developments in the quantum gravity community. The goal is to provide an interpretation of a model discussed by D. Marolf and H. Maxfield 2002.08950 aimed at fitting their model within the functorial framework of Quantum Field Theory (QFT). Given a TFT one can consider – formally – the sum over all bordisms between fixed ingoing and outgoing spatial slices (with appropriate weight factors for the bordisms) of the amplitudes associated to the bordism by the TFT. This construction leads to convergent sums in dleq 2 dimensions, at least for for generic parameters of the TFT. I will describe a curious splitting property satisfied by the total amplitude. I view the splitting property as an alternative to ensemble-type interpretations. There will be a cameo appearance of a very interesting paper by Daniel Friedan 2306.00019 which purports to give an axiomatic framework for Euclidean Quantum Gravity (EQG) analogous to the functorial formalism of QFT. I will also note, in passing, that these extremely simple, low-dimensional, baby baby baby models of EQG admit global symmetries and continuous parameters.
1:00 pm
Farewell Lunch

CMSA/MATH Welcome Back Gathering
2025-01-30 16:00 - 17:30 | CMSA 20 Garden Street Cambridge, Massachusetts 02138 United States
Thursday, Jan. 30, 2025
4:00 pm
All CMSA and Math affiliates are invited.

2025 Ding Shum Lecture: Irit Dinur, IAS: Expanders from local to global
2025-02-13 16:00 - 17:00 | Harvard Science Center, 1 Oxford Street, Cambridge, MA, 02138
On February 13, 2025 the CMSA hosted the sixth annual Ding Shum Lecture, given by Irit Dinur, Institute for Advanced Study.
Location: Harvard Science Center Hall A & via Zoom Webinar
Speaker: Irit Dinur, Institute for Advanced Study

Title: Expanders from local to global
Abstract: Imagine a network—like a social network, a transportation system, or even a biological system—where every part of the network is robustly connected to the rest. Expander graphs are the mathematical idealization of such networks. They are structures where any small group of points (nodes) has many connections to the rest of the graph, ensuring that no part is isolated and information (or influence) spreads efficiently throughout.
We will begin by surveying expander graphs, their discovery and construction, and some fascinating applications such as error-correcting codes, pseudorandomness, and probabilistically checkable proofs (PCPs), highlighting their role as a foundation for many breakthroughs in theoretical computer science. Then, we will shift focus to an exciting new kind of expanders called high dimensional expanders (HDXs). While expanders are well-understood and widely applied, HDXs remain enigmatic, with potential that we are only starting to uncover. We will talk about a fascinating local to global feature that HDXs have, and some applications.

This event is made possible by the generous funding of Ding Lei and Harry Shum.

Program on Classical, quantum, and probabilistic integrable systems - novel interactions and applications
2025-03-24 09:00 - 2025-05-24 17:00 | CMSA 20 Garden Street Cambridge, Massachusetts 02138 United States
Program on Classical, quantum, and probabilistic integrable systems – novel interactions and applications
Dates: March 24–May 24, 2025
Location: CMSA, 20 Garden Street, Cambridge MA 02138
Exactly solvable models have played pivotal roles in mathematics and physics throughout their history. The program is dedicated to exploring and developing a more recent wave of their influence in stochastic models together with accompanying combinatorial, classical, and quantum integrable systems. Topics will include:

Colored and uncolored interacting particle systems with associated vertex models and line ensembles
Yang-Baxter integrability and its applications in algebraic combinatorics, quantum systems, and conformal field theory
Quantum stochastic models, quantum exclusion processes, and free probability
Emerging new aspects of classical and quantum integrable systems – hydrodynamics, large deviations of stochastic models, and random surface models

Organizers:

Amol Aggarwal, Columbia University & Clay Mathematics Institute
Guillaume Barraquand, École normale supérieure, Paris
Alexei Borodin, MIT
Ivan Corwin, Columbia University
Pierre Le Doussal, École normale supérieure, Paris
Michael Wheeler, University of Melbourne

Participants

Denis Bernard, Ecole Normale Supérieure Paris
Alexey Bufetov, University of Leipzig
Pasquale Calabrese, SISSA Trieste
Sylvie Corteel, UC Berkeley
Cesar Cuenca, Ohio State University
Jan De Gier, University of Melbourne
Andrea De Luca, CNRS, Cergy Paris University
Benjamin Doyon, King’s College London
Patrik Ferrari, University of Bonn
Vadim Gorin, UC Berkeley
Tamara Grava, SISSA
Jimmy He, Ohio State University
Jiaoyang Huang, University of Pennsylvania
Kurt Johansson, KTH Stockholm
Richard Kenyon, Yale
Alexandre Krajenbrink, Cambridge Quantum Computing & Quantinuum
Atsuo Kuniba, University of Tokyo
Matteo Mucciconi, National University of Singapore
Greta Panova, University of Southern California
Leonid Petrov, University of Virginia
Sylvain Prolhac, Université Paul Sabatier, Toulouse
Tomaž Prosen, University of Ljubljana
Tomohiro Sasamoto, Tokyo Institute of Technology
Herbert Spohn, Technical University of Munich
Li-Cheng Tsai, University of Utah

Schedule
Week 1
Monday, March 24th
11:00am – 12:00pm Room G-10, Lecture 1 of 4: Denis Bernard, École normale supérieure de Paris: Quantum Exclusion Processes for (and by) Amateurs
12:00 – 2:00pm Common Room: Program Lunch
4:00 – 4:30pm Common Room: CMSA colloquium tea
4:30 – 5:30pm Common Room, CMSA colloquium: Amol Aggarwal, Columbia University: The Toda Lattice as a Soliton Gas

Tuesday, March 25th
3:30 – 4:00pm Common Room: Program tea
4:00 – 5:00pm Room G-10, Seminar: Patrik Ferrari, Universität Bonn: Decoupling and decay of two-point functions in a two-species TASEP

Wednesday, March 26th
11:00am – 12:00pm Room G-10, Lecture 1 of 3: Atsuo Kuniba, University of Tokyo: Multispecies ASEP and t-PushTASEP on a ring and a strange five vertex model
3:00 – 4:00pm Room G-10, Lecture 2 of 4: Denis Bernard, École normale supérieure de Paris: Quantum Exclusion Processes for (and by) Amateurs
4:30 – 5:30pm Common Room: Program wine and cheese reception

Thursday, March 27th
11:00am – 12:00pm Room G-10, Lecture 1 of 2: Benjamin Doyon, King’s College London: The equations of generalised hydrodynamics, and their unusual diffusve corrections
Abstract: I will discuss the hydrodynamics of one-dimensional many-body integrable models. At the Euler scale, this is given by “generalised hydrodynamics”, whose equations only depend on the asymptotic state content and the two-body scattering shift of the model. I will explain how these equations arise, and mention some of their properties: Hamiltonian structure, exact solutions, absence of shocks. At the diffusive scale, generic one-dimensional models with state-dependent currents display super-diffusion. However, integrable models are in a special class of “linearly degenerate systems”, where there is no superdiffusion, and one might expect a standard derivative expansion. I will explain how the diffusive corrections to the Euler equations are not given by a derivative expansion, but instead governed by long-range correlations coming from an Euler-scale fluctuation theory. I will give the general ideas behind this fluctuation theory, where initial fluctuations are deterministically transported by the Euler equation. I will finally provide arguments for the conjecture that, once long-range correlations are accounted for, there is no emergent stochasticity at all scales of hydrodynamics in integrable systems.
3:30 – 4:00pm Common Room: Program tea
4:00 – 5:00pm Room G-10, Seminar: Sylvie Corteel, University of California at Berkeley: Crystal Skeletons

Friday, March 28th
12:00 – 1:00 pm Common Room: Lunch with CMSA Member Seminar
2:00 – 3:00pm Room G-10, Lecture 3 of 4 : Denis Bernard, École normale supérieure de Paris: Quantum Exclusion Processes for (and by) Amateurs
3:30 – 4:00 pm Common Room: Program tea

Week 2
Monday, March 31
11:00am – 12:00pm Room G-10, Lecture 2 of 2: Benjamin Doyon, King’s College London: The equations of generalised hydrodynamics, and their unusual diffusve corrections
Abstract: I will discuss the hydrodynamics of one-dimensional many-body integrable models. At the Euler scale, this is given by “generalised hydrodynamics”, whose equations only depend on the asymptotic state content and the two-body scattering shift of the model. I will explain how these equations arise, and mention some of their properties: Hamiltonian structure, exact solutions, absence of shocks. At the diffusive scale, generic one-dimensional models with state-dependent currents display super-diffusion. However, integrable models are in a special class of “linearly degenerate systems”, where there is no superdiffusion, and one might expect a standard derivative expansion. I will explain how the diffusive corrections to the Euler equations are not given by a derivative expansion, but instead governed by long-range correlations coming from an Euler-scale fluctuation theory. I will give the general ideas behind this fluctuation theory, where initial fluctuations are deterministically transported by the Euler equation. I will finally provide arguments for the conjecture that, once long-range correlations are accounted for, there is no emergent stochasticity at all scales of hydrodynamics in integrable systems.
12:00 – 2:00pm Common Room: Program Lunch
2:00 – 3:00pm Room G-10, Lecture 2 of 3: Atsuo Kuniba, University of Tokyo: Solutions of tetrahedron and 3D reflection equations from quantum cluster algebras

Abstract: Tetrahedron and 3D equations are three-dimensional generalizations of the Yang-Baxter and the reflection equations. I will explain how quantum cluster algebras lead to solutions that generalize and unify many known solutions.

3:30 – 4:00pm Program tea

Tuesday, April 1
11:00am – 12:00pm Room G-10, Lecture 1 of 2: Kurt Johansson, KTH Stockholm: Extremal particles in uniform random Gelfand-Tsetlin patterns
Abstract: I will report on joint work with Elnur Emrah on edge fluctuations in uniform random interlacing patterns with fixed top configuration. The goal is to describe all possible limit processes that can occur, and the conditions under which they occur.
3:30pm – 4:00pm, Common Room: Program tea

Wednesday, April 2
11:00am – 12:00pm Room G-10, Lecture 4 of 4: Denis Bernard, École normale supérieure de Paris: Quantum Exclusion Processes for (and by) Amateurs
3:00 – 4:00pm Room G-10, Lecture 3 of 3: Atsuo Kuniba, University of Tokyo: Box-ball systems
Abstract: Box-ball systems are one-dimensional integrable cellular automata introduced in 1990. This talk surveys major developments that have unfolded consistently over the decades, enriching the scope of the theory. Topics include ultradiscretization, crystal theory in quantum groups, the combinatorial and thermodynamic Bethe ansatz, as well as generalized hydrodynamics.
4:30 – 5:30pm Common Room: Program wine and cheese reception

Thursday, April 3
11:00am – 12:00pm Room G-10, Lecture 2 of 2: Kurt Johansson, KTH Stockholm: Extremal particles in uniform random Gelfand-Tsetlin patterns
Abstract: I will report on joint work with Elnur Emrah on edge fluctuations in uniform random interlacing patterns with fixed top configuration. The goal is to describe all possible limit processes that can occur, and the conditions under which they occur.
3:30pm – 4:00pm Common Room: Program tea

Friday, April 4
12:00 – 1:00pm Common Room: CMSA Member Seminar and Lunch
3:30 – 4:00pm Common Room: Program tea

Week 3
Monday, April 7
12:00 – 2:00pm Common Room: Program lunch
4:00 – 4:30pm Tea with CMSA colloquium
4:30 – 5:30pm CMSA Colloquium: Ben Webster, University of Waterloo and Perimeter Institute: 3-D Mirror Symmetry

Tuesday, April 8
11:00am – 2:00pm Room G-10, Pierre Le Doussal, École normale supérieure de Paris: Exact results for the macroscopic fluctuation theory of the 1D weakly asymmetric exclusion process.
3:30 – 4:00pm Common Room: Program tea

Wednesday, April 9
12:00 – 1:00pm Common Room, CMSA Q&A Seminar and lunch: Eric Maskin, Harvard Economics: The Mathematics of Voting
4:30 – 5:30pm Common Room: Program wine and cheese reception

Thursday, April 10
3:30 – 4:00pm Common Room: Program tea

Friday, April 11
12:00 – 1:00pm Common Room: CMSA member seminar and lunch
3:30 – 4:00pm Common Room: Program tea

Week 4
Monday, April 14
12:00 – 2:00pm Common Room: Program lunch
4:00 – 4:30pm Tea with CMSA colloquium
4:30 –5:30pm CMSA colloquium: Andrey Smirnov, University of North Carolina at Chapel Hill: Quantum K-theory at roots of unity

Tuesday, April 15
11:00 am – 12:00pm Room G-10, Ivan Corwin, Columbia University: How Yang-Baxter unravels Kardar-Parisi-Zhang
Abstract: Over the past few decades, physicists and then mathematicians have sought to uncover the (conjecturally) universal long time and large space scaling limit for the so-called Kardar-Parisi-Zhang (KPZ) class of stochastically growing interfaces in (1+1)-dimensions. Progress has been marked by several breakthroughs, starting with the identification of a few free-fermionic integrable models in this class and their single-point limiting distributions, widening the field to include non-free-fermionic integrable representatives, evaluating their asymptotics distributions at various levels of generality, constructing the conjectural full space-time scaling limit, known as the directed landscape, and checking convergence to it for a few of the free-fermion representatives.
In this talk, I will describe a method that should prove convergence for all known integrable representatives of the KPZ class to this universal scaling limit. The method has been fully realized for the Asymmetric Simple Exclusion Process and the Stochastic Six Vertex Model. It relies on the Yang-Baxter equation as its only input and unravels the rich complexity of the KPZ class and its asymptotics from first principles. This is based on a few works involving Amol Aggarwal, Alexei Borodin, Milind Hegde, Jiaoyang Huang and me.
3:30 – 4:00pm Common Room: Program tea

Wednesday, April 16
11:00am – 12:00pm Room G-10, Tamara Grava, University of Bristol: Random solitons and soliton gas
Abstract: A soliton is a localised travelling wave solution of a nonlinear dispersive equation. When the equation is integrable the interaction of many solitons is elastic. We study the behaviour of a set of N solitons for the Korteweg de Vries equation in the limit N goes to infinity (soliton gas) and the interaction of the soliton gas with a distinct soliton that we call a tracer soliton. We show that the average velocity of the tracer soliton satisfies the Zakharov-El kinetic equations. We then consider a set of random N soliton solution q_N(x,t) and its limiting soliton gas q(x,t). We prove a central limit theorem for the difference q_N(x,t)-q(x,t) for values of x and t that are bounded by log(N).
12:00 – 1:00pm Common Room: CMSA Q&A seminar and lunch: Noah Golowich, MIT: What is length generalization in large language models?
4:30 – 5:30pm Common Room: Program wine and cheese reception

Thursday, April 17
11:00am – 12:00pm Room G-10, Guillaume Barraquand, École normale supérieure de Paris: Large time cumulants of the open KPZ equation
12:00 – 1:00pm Common Room: lunch with featured Yip Lecture speaker Scott Aaronson and CMSA residents
3:30pm Common Room: Program tea
4:00 – 5:00pm Science Center Hall A: Fifth Annual Yip Lecture, Scott Aaronson: How Much Math is Knowable?
5:00 – 6:00pm Math Department Common Room at the Harvard Science Center: Yip Lecture reception

Friday, April 18
12:00 – 1:00pm Common Room: CMSA Member Seminar and lunch: Han Shao, Harvard CMSA, Topic TBD
3:30 – 4:00pm Common Room: Program tea

Week 5

Monday, April 21
11:00am – 12:00pm Room G-10, Tomaz Prosen, University of Ljubljana, Lecture 1 of 3: On Integrable Quantum and Classical Circuits (with Stochastic Boundaries)
Abstract: I will introduce Yang-Baxter integrable brickwork quantum circuit models and discuss their integrability structure, specifically, the transfer matrix, conservation laws etc. A paradigmatic example, XXZ or unitary 6-vertex circuits exhibit an unusual link to KPZ scaling at the isotropic (SU(2) symmetric) point. I will establish the link to corresponding classical integrable Landau-Lifshitz circuits and discuss some aspects of transport and full counting statistics.
12:00 – 2:00pm Common Room: Program Lunch
4:00 – 4:30pm Common Room: CMSA colloquium tea
4:30 – 5:30pm Common Room, CMSA colloquium: Ila Fiete, MIT: Modeling the emergence of complex cortical structure from simple precursors in the brain: maps, hierarchies, and modules

Tuesday, April 22
11:00am – 12:00pm Room G-10, Tomohiro Sasamoto, Tokyo Institute of Technology: Large deviation of symmetric models through classical integrable systems
3:30pm Common Room: Program tea

Wednesday, April 23
11:00am – 12:00pm Room G-10, Tomaz Prosen, University of Ljubljana: On Integrable Quantum and Classical Circuits (with Stochastic Boundaries)
Abstract: I will discuss explicit matrix product solutions for quantum many-body Markov chains, defined for a Yang-Baxter integrable quantum circuit with specific stochastic Kraus processes at its boundaries. In the continuous time limit, these solutions correspond to steady states of boundary driven Lindbladian dynamics and often yield non-trivial quasi-local conservation laws of integrable spin chains. The specific case of XXZ and Hubbard chain will be discussed.
12:00 – 1:00pm Common Room: CMSA Q&A seminar and lunch: Alexei Borodin, MIT: Connections between physics and probability
4:30 – 5:30pm Common Room: Program wine and cheese reception

Thursday, April 24
11:00am – 12:00pm Room G-10, Sylvain Prolhac, Université Paul Sabatier, Toulouse: Approach to stationarity for KPZ fluctuations in finite volume
Abstract: At late times $t$, the cumulants of the height for the KPZ fixed point in finite volume behave as affine functions of time $c_k(t) = a_k t+b_k$, up to exponentially small corrections. The constant term $b_k$ is the last remaining information about the initial state observable at long enough times. Two approaches allow us to compute this constant from the totally asymmetric exclusion process, a discrete version of the KPZ fixed point. First, an iterated version of the matrix product representation for the stationary state leads, for arbitrary initial conditions, to expressions involving extreme value statistics of Brownian paths. On the other hand, Bethe ansatz leads to rather explicit expressions for simple initial conditions. Comparison between the two approaches then provides conjectures for some generating functions of Brownian paths.
3:30pm Common Room: Program tea

Friday, April 25
11:00am – 12:00pm Room G-10, Tomaz Prosen, University of Ljubljana, Lecture 3 of 3: On Integrable Quantum and Classical Circuits (with Stochastic Boundaries)
Abstract: In the last lecture I will discuss the possibility of quantum integrability of many-body quantum Markov chain generators, such as Lindbladians with bulk or boundary dissipation, and the corresponding circuit (Kraus) counterparts. The paradigmatic example is the XX model with dephasing noise which can be mapped to a Hubbard model with imaginary interaction, both in the Hamiltonian and circuit formulation.
3:30 – 4:00pm Common Room: Program tea

Week 6

Monday, April 28
11:00am – 12:00pm Room G-10, Herbert Spohn, Technische Universitaet Muenchen, Lecture 1 of 3: Integral many-body systems and GHD
12:00 – 2:00pm Common Room: Program Lunch
2:00 – 3:00 pm Room G-10, Tomohiro Sasamoto, Tokyo Institute of Technology, Exact density profile and current fluctuations in a tight-binding chain with dephasing noise
Abstract: We consider a tight-binding chain with dephasing noise, whose time evolution is described by the quantum master equation called the Gorini-Kossakowski-Sudarhan-Lindblad (GKSL) equation. By using a connection of this model to the Hubbard model with imaginary coupling [1], we study the density profile [2] and the variance of the current [3] exactly for the model on the infinite line by writing down contour integral formulas using Bethe ansatz. The talk is based on collaborations with Taiki Ishiyama and Kazuya Fujimoto.
4:00 – 4:30pm Common Room: CMSA colloquium tea
4:30 –5:30pm Room G-10, CMSA colloquium: Peter Sarnak, IAS and Princeton University, Bass-Note Spectra of locally uniform geometries

Tuesday, April 29
11:00 am – 12:00pm Room G-10, Pasquale Calabrese, SISSA Trieste, Lecture 1 of 3: Quantum Mpemba effect
2:00 – 3:00 pm Room G-10, Greta Panova, University of Southern California, Grothendieck shenanigans: permutons from pipe dreams via integrable probability
Abstract: Pipe dreams are tiling models originally introduced to study objects related to the Schubert calculus and K-theory of the Grassmannian. They can also be viewed as ensembles of random lattice walks with various interaction constraints. In our quest to understand what the maximal and typical algebraic objects look like, we revealed some interesting permutons. The proofs use the theory of the Totally Asymmetric Simple Exclusion Process (TASEP). Deeper connections with domino tilings of the Aztec diamond and its frozen boundary allow us to describe the extreme cases of the original algebraic problem. This is based on joint work with A. H. Morales, L. Petrov, D. Yeliussizov.
3:30 – 4:00pm Common Room: Program tea

Wednesday, April 30
11:00am – 12:00pm Herbert Spohn, Technische Universitaet Muenchen, Lecture 2 of 3: Integral many-body systems and GHD
12:00 – 1:00pm (tentative) Common Room: CMSA Q&A seminar and lunch
3:00 – 4pm Room G-10, Pasquale Calabrese, SISSA Trieste, Entanglement evolution and quasiparticle picture 1
4:30 – 5:30pm Common Room: Program wine and cheese reception

Thursday, May 1
11:00am – 12:00pm Room G-10, Herbert Spohn, Technische Universitaet Muenchen, Lecture 3 of 3: Integral many-body systems and GHD
2:00 – 3:00 pm Room G-10, Li-Cheng Tsai, University of Utah, Stochastic heat flow by moments
Abstract: The Stochastic Heat Flow (SHF) is the scaling limit of the directed polymers in random environments and the noise-mollified Stochastic Heat Equation (SHE), at the critical dimension of two and near the critical temperature. The finite-dimensional distributions of the SHF was obtained by Caravenna, Sun, and Zygouras (2023) by proving that the discrete polymers converge to a universal (model-independent) limit. In this talk, I will report a new approach based on axioms. We formulate the SHF as a two-parameter continuous measure-valued process that satisfies a set of axioms, and prove the uniqueness in law under these axioms. The key feature of the axioms concerns the matching of the first four moments. As an application, we prove the convergence of the noise-mollified SHE to the SHF, which only requires moment estimates.
3:30pm Common Room: Program tea

Friday, May 2
11:00am – 12:00pm Room G-10, Pasquale Calabrese, SISSA Trieste, Lecture 3 of 3: Entanglement evolution and quasiparticle picture 2
12:00 – 1:00pm Common Room, CMSA Member seminar and lunch
2:00 – 3:00 pm Room G-10, Leonid Petrov, University of Virginia: Random Fibonacci Words
Abstract: Fibonacci words are words of 1’s and 2’s, graded by the total sum of the digits. They form a differential poset YF which is an estranged cousin of the Young lattice powering irreducible representations of the symmetric group. We introduce families of “coherent” measures on YF depending on many parameters, which come from the theory of clone Schur functions (Okada 1994). We characterize parameter sequences ensuring positivity of the measures, and we describe the large-scale behavior of some ensembles of random Fibonacci words. The subject has connections to total positivity of tridiagonal matrices, Stieltjes moment sequences, orthogonal polynomials from the (q-)Askey scheme, and residual allocation (stick-breaking) models. Based on a joint work with Jeanne Scott.
3:30 – 4:00pm Common Room: Program tea

Week 7

Monday, May 5
11:00am – 12:00pm Room G-10, Jan De Gier, University of Melbourne, Lecture 1 of 3: Pfaffian point process for TASEP on the half line
12:00 – 2:00pm Common Room: Program Lunch
2:00 – 3:00 pm Jiaoyang Huang, University of Pennsylvania: Ramanujan Property and Edge Universality of Random Regular Graphs
Abstract: Extremal eigenvalues of graphs are of particular interest in theoretical computer science and combinatorics. Specifically, the spectral gap—the difference between the largest and second-largest eigenvalues—measures the expansion properties of a graph. In this talk, I will focus on random d-regular graphs.I will begin by providing background on the eigenvalues of random d-regular graphs and their connections to random matrix theory. In the second part of the talk, I will discuss our recent results on eigenvalue rigidity and edge universality for these graphs. Eigenvalue rigidity asserts that, with high probability, each eigenvalue concentrates around its classical location as predicted by the Kesten-McKay distribution. Edge universality states that the second-largest eigenvalue and the smallest eigenvalue of random d-regular graphs converge to the Tracy-Widom distribution from the Gaussian Orthogonal Ensemble. Consequently, approximately 69% of d-regular graphs are Ramanujan graphs. This work is based on joint work with Theo McKenzie and Horng-Tzer Yau.

4:00 – 4:30pm Common Room: CMSA colloquium tea
4:30 –5:30pm Common Room, CMSA colloquium: Ariel Procaccia, Harvard University, Thinking Outside the Ballot Box

Tuesday, May 6
11:00 am – 12:00pm Room G-10, Jan De Gier, University of Melbourne, Lecture 2 of 3: Pfaffian point process for TASEP on the half line
2:00 – 3:00 Richard Kenyon, Yale University, Multinomial dimers and 3d limit shapes
Abstract: The “multinomial dimer model” on a graph G is the dimer model on the N-fold blow up of G (the graph obtained by replacing each vertex with N vertices and each edge with a complete bipartite graph K_{N,N}). In the large N limit this model is tractable for general graphs: we find formulas for the partition function and limit shapes in some natural settings, including a three-dimensional version of the Aztec Diamond. This is joint work with Catherine Wolfram (Yale).
3:30 – 4:00pm Common Room: Program tea

Wednesday, May 7
3:00 – 4pm Room G-10, Jan De Gier, University of Melbourne, Lecture 3 of 3: Pfaffian point process for TASEP on the half line
4:30 – 5:30pm Common Room: Program wine and cheese reception

Thursday, May 8:
2:00 – 3:00 pm Room G-10, Andrea De Luca, CNRS Cergy Paris University, Monitored quantum systems, product of random matrices and permutations
3:30pm Common Room: Program tea

Friday, May 9:
12:00 – 1:00pm Common Room: CMSA Member Seminar and lunch, Sergiy Verstyuk, Harvard CMSA, Title TBD
2:00 – 3:00 pm Room G-10, Cesar Cuenca, Ohio State University, Random partitions at high temperature
Abstract: By using Fourier transforms based on Jack symmetric polynomials, we study discrete particle ensembles x_1>x_2>…>x_N with the inverse temperature beta in the regime where beta tends to zero, as the number of particles tends to infinity. We prove the LLN and characterize the limiting measure in terms of a moment problem. For fixed-time distributions of special Markov chains, the limiting measures can be expressed in terms of the eigenvalues of certain Jacobi operators.
3:30 – 4:00pm Common Room: Program tea

Week 8

Monday, May 12
11:00am – 12:00pm Room G-10, Jimmy He, Ohio State University, Symmetries of periodic and free boundary measures on partitions
Abstract: The periodic and free boundary q-Whittaker measures are probability measures on partitions defined in terms of q-Whittaker functions and an additional parameter $u$ controlling the behavior of the system at the boundary. I will explain a hidden distributional symmetry of this model which exchanges the $u$ and $q$ parameters, as well as related results on Hall-Littlewood measures. As a special case, we recover identities of Imamura–Mucciconi–Sasamoto. This is joint work with Michael Wheeler.
12:00 – 2:00pm Common Room: Program Lunch
4:00 – 4:30pm Common Room: CMSA colloquium tea
4:30 – 5:30pm Common Room, CMSA colloquium: Anna Seigal, Harvard University, Factorizations for data analysis

Tuesday, May 13
3:30pm Common Room: Program tea

Wednesday, May 14
12:00 – 1:00pm Common Room: CMSA Conference Reports seminar and lunch: Hugo Cui, Harvard CMSA, reporting on the Perimeter Institute Theory+AI Workshop
3:00 – 4:00pm Room G-10, Alexandre Krajenbrink, Cambridge Quantum Computing and Quantinuum, Unveiling the classical integrable structure of the weak noise theory of the KPZ class: example of the matrix Log–Gamma polymer and the q-TASEP
4:30 – 5:30pm Common Room: Program wine and cheese reception

Thursday, May 15
11:00am – 12:00pm Room G-10: Roger Van Peski, Columbia University, Integrability in discrete random matrix theory

Abstract: Integrable structure has been well-used in classical random matrix theory, and recently is also enjoying application in the parallel world of discrete random matrices (over integers, p-adic integers, and finite fields). In this talk I will try to cover—at least briefly—the following:

Some favorite probabilistic results (convergence of discrete random matrix local limits to a new integrable interacting particle system, the ‘reflecting Poisson sea’),
Some exact formulas with Hall-Littlewood polynomials that make these results possible, and
Some intriguing newer formulas (joint with Jiahe Shen) for Hermitian and antisymmetric p-adic matrices, which naturally feature ‘formal’ Hall-Littlewood processes with negative t parameter.

2:00 – 3:00 pm Room G-10, Matteo Mucciconi, National University Singapore, Orthogonality of spin q-Whittaker polynomials
Abstract: Spin q-Whittaker polynomials are a family of symmetric polynomials that can be defined as partition functions of a solvable lattice model. Their study reveals that they possess mysterious properties such as additional “unorthodox” symmetries, eigenrelations with respect to difference operators and a self orthogonality that I will prove in the talk. A particular case of these results include a novel orthogonality for the Grothendieck polynomials from K-theory of Grassmannian. I will also discuss applications to exact solutions of directed random polymer models with Beta weights.
3:30pm Common Room: Program tea

Friday, May 16
12:00 – 1:00pm Common Room: CMSA Member Seminar and lunch: Samy Jelassi, Echo Chamber: RL Post-training Amplifies Behaviors Learned in Pretraining
3:30 – 4:00pm Common Room: Program tea

Videos are available on the Youtube Playlist.

Current Developments in Mathematics 2025
2025-04-04 09:00 - 2025-04-05 17:00 | Harvard Science Center, 1 Oxford Street, Cambridge, MA, 02138
When: April 4, 2025 – April 5, 2025

Where: Science Center Hall C
Address: 1 Oxford Street, Cambridge, MA 02138, United States

Speaker: Michael Chapman – NYU | Pazit Haim-Kislev – Institute for Advanced Study | Jianfeng Lin – Tsinghua University | Laura Monk – University of Bristol | Ramon van Handel – Princeton University

IN-PERSON REGISTRATION
Limited funding to help defray travel expenses is available for graduate students and recent PhDs. If you are a graduate student or postdoc and would like to apply for support, please register and send a letter to cdm@math.harvard.edu.
A letter indicating your name, address, current status, university affiliation, citizenship, and area of study. F1 visa holders are eligible to apply for support. If you are a graduate student, please send a brief letter of recommendation from a faculty member to explain the relevance of the conference to your studies or research.
Detailed schedule of lectures and events coming soon.
Organizers: David Jerison, Paul Seidel, Nike Sun (MIT); Denis Auroux, Mark Kisin, Lauren Williams, Horng-Tzer Yau, Shing-Tung Yau (Harvard).
Sponsored by the National Science Foundation (pending), Harvard University Mathematics, and the Massachusetts Institute of Technology.
Harvard University is committed to maintaining a safe and healthy educational and work environment in which no member of the University community is, on the basis of sex, sexual orientation, or gender identity, excluded from participation in, denied the benefits of, or subjected to discrimination in any University program or activity. More information can be found here.

Current Developments in Mathematics 2025

CMSA/Tsinghua Math-Science Literature Lecture: Scott Sheffield (MIT): Yang-Mills theory and random surfaces
2025-04-08 09:00 - 10:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA/Tsinghua Math-Science Literature Lecture
Date: April 8, 2025
Time: 9:00 – 10:30 am ET
Location: CMSA G10, 20 Garden Street, Cambridge MA & via Zoom
Speaker: Scott Sheffield (MIT)
Title: Yang-Mills theory and random surfaces
Abstract: The Clay Institute famously offered one million dollars to anyone who could mathematically construct and understand a certain continuum version of “Yang-Mills gauge theory.” This theory is the basis of the standard model of physics, and the heart of the problem is to understand the so-called “Wilson loop expectations.” Following recent work with Sky Cao and Minjae Park, I will explain how the “Wilson loop expectations” in a lattice Yang-Mills model are equivalent to “insertion costs” of loops in a related random-closed-surface-ensemble model. In a sense, these results allow us to convert one famously hard problem into another presumably hard problem. But the new problem is all about random surfaces and random permutations, and it has a lot of relationships with and similarities to other problems we understand (think domino tilings, random planar maps, Young tableaux and symmetric group representation theory, and the Weingarten calculus). It gives us some intuition for *why* certain things should be true like the “area law” or “exponential correlation decay” (what physicists call “quark confinement” or “mass gap”) even if we can’t prove all of them yet.

Beginning in Spring 2020, the CMSA began hosting a lecture series on literature in the mathematical sciences, with a focus on significant developments in mathematics that have influenced the discipline, and the lifetime accomplishments of significant scholars.

Fifth Annual Yip Lecture | Scott Aaronson (UT Austin): How Much Math Is Knowable?
2025-04-17 16:00 - 17:00 | Harvard Science Center, 1 Oxford Street, Cambridge, MA, 02138
Speaker: Scott Aaronson, Department of Computer Science, University of Texas, Austin
Scott Aaronson is the founding director at the Quantum Information Center at the University of Texas at Austin.
Date: April 17, 2025
Time: 4:00-5:00 pm ET (Reception following in the Math Common Room)
Location: Harvard Science Center Hall A

Title: How Much Math Is Knowable?
Abstract: Theoretical computer science has over the years sought more and more refined answers to the question of which mathematical truths are knowable by finite beings like ourselves, bounded in time and space and subject to physical laws. I’ll tell a story that starts with Gödel’s Incompleteness Theorem and Turing’s discovery of uncomputability. I’ll then introduce the spectacular Busy Beaver function, which grows faster than any computable function. Work by me and Yedidia, along with recent improvements by O’Rear and Riebel, has shown that the value of BB(745) is independent of the axioms of set theory; on the other end, an international collaboration proved last year that BB(5) = 47,176,870. I’ll speculate on whether BB(6) will ever be known, by us or our AI successors. I’ll next discuss the P!=NP conjecture and what it does and doesn’t mean for the limits of machine intelligence. As my own specialty is quantum computing, I’ll summarize what we know about how scalable quantum computers, assuming we get them, will expand the boundary of what’s mathematically knowable. I’ll end by talking about hypothetical models even beyond quantum computers, which might expand the boundary of knowability still further, if one is able (for example) to jump into a black hole, create a closed timelike curve, or project oneself onto the holographic boundary of the universe.

The Yip Lecture takes place thanks to the support of Dr. Shing-Yiu Yip.

The previous Yip Lecture featured Josh Tenenbaum (MIT) who spoke on How to grow a mind from a brain: From guessing and betting to thinking and talking

Summer School in Total Positivity and Quantum Field Theory
2025-06-02 09:00 - 2025-06-04 17:00 | CMSA 20 Garden Street Cambridge, Massachusetts 02138 United States
Summer School in Total Positivity and Quantum Field Theory
Dates: June 2–4, 2025
Location: CMSA, 20 Garden Street, Cambridge MA

In the past decade, there has been a great deal of interest and progress in the study of algebro-combinatorial and geometric structures appearing across diverse areas of physics, from particle physics to cosmology. As these research programs expand, there is an ever-growing need for mathematicians and physicists to collaborate effectively and build a shared language. Join us at Harvard University’s Center of Mathematical Sciences and Applications for a week-long summer school dedicated to addressing these interdisciplinary connections. The school welcomes graduate students, postdocs, and early-career researchers drawn to the intersection of mathematics and physics. Whether you are an algebraic combinatorialist looking for a better grasp on the physics, a high energy theorist trying to figure out the math, or a newcomer to both fields, this summer school offers an ideal opportunity for you to learn.

Courses taught by both mathematicians and physicists will connect ideas from total positivity, matroid theory, discrete geometry, and real algebraic geometry with fundamental questions in quantum field theory. Topics will include amplituhedra, cluster algebras, and positive geometry as they relate to scattering amplitudes and cosmological correlators in high-energy physics. Our courses are designed to be accessible to a varied audience; speakers will be mindful of the diverse backgrounds of the participants from both fields.
Amid this exciting period of collaboration between mathematicians and physicists, we look forward to exploring these rich, cutting-edge topics with you.

Courses:

Positive Grassmannian and Cluster Algebras, Lara Bossinger (Instituto de Matemáticas Universidad Nacional Autónoma de México)
slides | exercises

Positive Geometry and Canonical Forms, Simon Telen (MPI Leipzig)
slides

Scattering Amplitudes and Amplituhedra, Marcus Spradlin (Brown)
exercises

Cosmology and Cosmological Polytopes, Nima Arkani-Hamed (IAS)

Organizers: Jonathan Boretsky (McGill University) | Matteo Parisi (Harvard CMSA and IAS Princeton) | Lauren Williams (Harvard University)

Youtube Playlist
Schedule
Monday, June 2, 2025

8:30–9:00 am
Morning Reception

9:00–10:00 am
Lara Bossinger: Positive Grassmannian and Cluster Algebras I

10:00–10:30 am
Coffee Break

10:30–11:10 am
Exercises

11:10 am–12:10 pm
Nima Arkani-Hamed: Cosmology and Cosmological Polytopes I

12:10–2:00 pm
Lunch Break

2:00–3:00 pm
Nima Arkani-Hamed: Cosmology and Cosmological Polytopes II

3:00–3:30 pm
Coffee Break

3:30–4:10 pm
Exercises

4:10–5:10 pm
Nima Arkani-Hamed: Cosmology and Cosmological Polytopes III

Tuesday, June 3, 2025

8:30–9:00 am
Morning Reception

9:00–10:00 am
Lara Bossinger: Positive Grassmannian and Cluster Algebras II

10:00–10:30 am
Coffee Break

10:30–11:30 am
Marcus Spradlin: Scattering Amplitudes and Amplituhedra I

11:30 am–12:10 pm
Exercises

12:10–2:00 pm
Lunch Break

2:00–3:00 pm
Simon Telen: Definitions and first examples of positive geometries

3:00–3:30 pm
Coffee Break

3:30–4:30 pm
Lightning Talks

4:30–5:30 pm
Simon Telen: Positive geometry of polytopes

Wednesday, June 4, 2025

8:30–9:00 am
Morning Reception

9:00–10:00 am
Lara Bossinger: Positive Grassmannian and Cluster Algebras III

10:00–10:30 am
Coffee Break

10:30–11:10 am
Exercises

11:10 am–12:10 pm
Marcus Spradlin: Scattering Amplitudes and Amplituhedra II

12:10–2:00 pm
Lunch Break

2:00–3:00 pm
Marcus Spradlin: Scattering Amplitudes and Amplituhedra III

3:00–3:30 pm
Coffee Break

3:30–4:30 pm
Simon Telen: Positive geometry of polypols

4:30–5:10 pm
Exercises

Image credit: Annabel Ma (Harvard College)

Quantum Field Theory and Topological Phases via Homotopy Theory and Operator Algebras
2025-06-30 09:00 - 2025-07-11 17:00 | Hybrid
Workshop on Quantum Field Theory and Topological Phases via Homotopy Theory and Operator Algebras
Dates: June 30 – July 11, 2025
Location: CMSA, 20 Garden Street, Cambridge MA and Max Planck Institute for Mathematics, Bonn, Germany
This event is a twinned workshop at the CMSA (Harvard) and the Max Planck Institute for Mathematics (Bonn). Lectures will alternate between the two sites, watched simultaneously on both sides, and there will be opportunities for dialogue between the locations. The first week will contain four pedagogical lecture series; lecturers and locations are
Michael Hopkins, Harvard (CMSA)Alexei Kitaev, Caltech (CMSA)Pieter Naaijkens, Cardiff (MPIM)Bruno Nachtergaele, UC Davis (MPIM)
The second week will consist of research talks.
Participants are strongly encouraged to attend at the location that minimizes travel and hence the ecological impact of the conference.
The application deadline was March 16, 2025.
Directions to CMSA
MPIM-Bonn location: https://www.mpim-bonn.mpg.de/qft25

Register for Zoom Webinar

Quantum Field Theory (QFT) and Quantum Statistical Mechanics are central to high energy physics and condensed matter physics; they also raise deep questions in mathematics. The application of operator algebras to these areas of physics is well-known. Recent developments indicate that to understand some aspects QFT properly a further ingredient is needed: homotopy theory and infinity-categories. One such development is the recognition that symmetry in a QFT is better described by a homotopy type rather than a group (so-called generalized symmetries). Another one is the work of Lurie and others on extended Topological Field Theory (TFT) and the Baez-Dolan cobordism hypothesis. Finally, there is a conjecture of Kitaev that invertible phases of matter are classified by homotopy groups of an Omega-spectrum. This workshop will bring together researchers and students approaching this physics using different mathematical techniques: operator algebras, homotopy theory, higher category theory, etc. The goal is to catalyze new interactions between different communities. At the workshop recent developments will be reviewed and hopefully progress can be made on two outstanding problems: the Kitaev conjecture as well as the long-standing goal of finding a proper mathematical formulation for QFT.
Organizers:

Dan Freed, Harvard University CMSA & Math
Dennis Gaitsgory, MPIM Bonn
Owen Gwilliam, UMass Amherst
Anton Kapustin, Caltech
Catherine Meusburger, University of Erlangen-Nürnberg

Talks are recorded and available on the CMSA Youtube Playlist.

BACKGROUND READING
Participants are encouraged to have some basic familiarity with the definition of a C*-algebra and quantum spin system. Some knowledge of quantum channels (completely positive trace-preserving maps) and quantum circuits will be useful. Some knowledge of Clifford algebras will also be helpful.
Possible references include:
1) arXiv:1311.2717 (Sections 2.1, 2.2, 2.4, and 2.5 up to Theorem 2.5.3)
2) Lectures by Daniel Spiegel on “C*-Algebraic Foundations of Quantum Spin Systems”, at the Summer School on C*-Algebraic Quantum Mechanics and Topological Phases of Matter, University of Colorado Boulder, July 29 to August 2, 2024. (lecture notes and video recordings: https://sites.google.com/colorado.edu/caqm).
3) https://nextcloud.tfk.ph.tum.de/etn/wp-content/uploads/2022/09/JvN_lecture_notes_S2016_abcde-1.pdf
4) https://en.wikipedia.org/wiki/Classification_of_Clifford_algebras
5) Karoubi, K-theory, section III.3
6.) Alexei Kitaev: A norm bound for 1D local matrices (pdf)

Schedule Times are Eastern Time
download schedule pdf
Workshop on Quantum Field Theory and Topological Phases via Homotopy Theory and Operator Algebras
June 30 – July 11, 2025

Monday, June 30

8:00–9:00 am

MPIM

Bruno Nachtergaele, UC Davis
Title: Ground states of quantum lattice systems: Quantum Lattice Systems: observables, dynamics, ground states, GNS representation, ground state gap, examples

9:00–9:30 am

Breakfast break

9:30–10:30 am

CMSA

Michael Hopkins, Harvard
Title: Lattice models and topological quantum field theories I
Abstract: This series will cover the relationship between gapped Hamiltonian lattice models and topological quantum field theories, with an emphasis on a conjecture of Kitaev.

10:30–10:45 am

break

10:45–11:45 am

MPIM

Pieter Naajkens, Cardiff
Title: Introduction to superselection sector theory: Motivation and introduction of basic setting
Abstract: (week 1 lectures) In this series of lectures, I will give an introduction to the operator-algebraic (Doplicher-Haag-Roberts) approach to study the superselection sectors of a (2D) gapped quantum spin system. The sectors have a rich mathematical structure of a braided monoidal category. This category describes all the algebraic properties of the ‘anyons’ or ‘charges’ such quantum spin systems can have. The aim of these lectures is to build up this theory from first principles, using simple examples of topologically ordered models to illustrate the main ideas. If time permits, I will elaborate on how this fits into the larger programme of the classification of gapped phases of matter, and long-range entangled states in particular. No prior knowledge of operator algebras or tensor categories is assumed.
SLIDES (pdf)

11:45 am –12:00 pm

break

12:00–1:00 pm

CMSA

Alexei Kitaev, Caltech
Title: Local definitions of gapped Hamiltonians and topological and invertible states I

Tuesday, July 1

8:00–9:00 am

MPIM

Bruno Nachtergaele, UC Davis
Title: Ground states of quantum lattice systems: Quasilocality: almost local observables and interactions, Lieb-Robinson bounds, quasi-adiabatic evolution, stability I

9:00–9:30 am

Breakfast break

9:30–10:30 am

CMSA

Michael Hopkins, Harvard
Title: Lattice models and topological quantum field theories II

10:30–10:45 am

break

10:45–11:45 am

MPIM

Pieter Naajkens, Cardiff
Title: Introduction to superselection sector theory: Building the braided (fusion) category of superselection sectors I
SLIDES (pdf)

11:45 am –12:00 pm

break

12:00–1:00 pm

CMSA

Alexei Kitaev, Caltech
Title: Local definitions of gapped Hamiltonians and topological and invertible states II

Wednesday, July 2

8:00–9:00 am

MPIM

Bruno Nachtergaele, UC Davis
Title: Ground states of quantum lattice systems: Quantum Entanglement in many-body systems: short-range entangled states, topological entanglement, stability II

9:00–9:30 am

Breakfast break

9:30–10:30 am

CMSA

Michael Hopkins, Harvard
Title: Lattice models and topological quantum field theories III

10:30–10:45 am

break

10:45–11:45 am

MPIM

Pieter Naajkens, Cardiff
Title: Introduction to superselection sector theory: Building the braided (fusion) category of superselection sectors II
SLIDES (pdf)

11:45 am –12:00 pm

break

12:00–1:00 pm

CMSA

Alexei Kitaev, Caltech
Title: Local definitions of gapped Hamiltonians and topological and invertible states III

Thursday, July 3

8:00–9:00 am

MPIM

Bruno Nachtergaele, UC Davis
Title: Ground states of quantum lattice systems: Quantum Phase Diagrams: order parameters, topological invariants, examples

9:00–9:30 am

Breakfast break

9:30–10:30 am

CMSA

Michael Hopkins, Harvard
Title: Lattice models and topological quantum field theories IV

10:30–10:45 am

break

10:45–11:45 am

MPIM

Pieter Naajkens, Cardiff
Title: Introduction to superselection sector theory: Classification of phases and long-range entanglement
SLIDES (pdf)

11:45 am –12:00 pm

break

12:00–1:00 pm

CMSA

Alexei Kitaev, Caltech
Title: Local definitions of gapped Hamiltonians and topological and invertible states IV

No talks Friday July 4

Monday July 7

8:00–9:00 am

MPIM

Jackson van Dyke, TU Munich
Title: Moduli spaces of projective 3d TQFTs
Abstract: A gapped quantum system is well-approximated at low energy by a projective topological field theory. Therefore questions concerning the classification, symmetries, and anomalies of gapped quantum systems can be reinterpreted via the homotopy theory of the moduli space of such theories. I will describe a moduli space of 3-dimensional TQFTs, and the sense in which its homotopy theory informs us about the low energy behavior of gapped systems in 2+1 dimensions. This moduli space depends on the fixed target category: Explicitly, it is built from the classifying spaces of higher groups of automorphisms of ribbon categories. The emphasis will be on target categories which have convenient algebraic features, yet are analytically robust enough to allow for boundary/relative theories defined in terms of unitary representations on topological vector spaces.

9:00–9:30 am

Breakfast break

9:30–10:30 am

CMSA

Constantin Teleman, UC Berkeley
Title: Quantizing homotopy types
Abstract: Kontsevich (90’s) proposed a topological quantization of (sigma-models into) finite homotopy types to top dimensions (d, d+1). Its enhancement to a `fully extended’ TQFT was described later (Freed, Hopkins, Lurie and the speaker) in the target category of iterated algebras. Independently, Chas and Sullivan constructed a (partially defined) 2-dimensional TQFT (d=1) with target compact oriented manifolds. I will briefly review the features of the finite homotopy theory and its boundary conditions, with particular interest in Dirichlet conditions; their analogue in Chas-Sullivan theory (older work by Blumberg, Cohen and the speaker). Finally, I propose a generalization combining these to a higher-dimensional Chas-Sullivan theory.
Slides (link)

10:30–10:45 am

break

10:45–11:45 am

MPIM

Matthias Ludewig, University of Greifswald
Title: Generalized Kitaev Pairings and Higher Berry curvature in coarse geometry
Abstract: In Appendix C of his “Anyons” paper, Kitaev introduced the notion of a “generalized Chern number” for a 2-dimensional system by diving the system in three ordered parts and measuring a signed rotational flux. This construction has since been used by several authors to measure topological non-triviality of a physical system. In recent work with Guo Chuan Thiang, we observe that the recipe provided by Kitaev can be interpreted in coarse geometry as the pairing of a K-theory class with a coarse cohomology class. A corresponding index theorem then provides a proof that the set of values of this “Kitaev pairing” is always quantized, as already argued by Kitaev. In our work, we generalize Kitaev’s definition and the corresponding quantization result to arbitrary dimensions. By replacing a single Hamiltonian with a whole family of Hamiltonians (parametrized by a space X), we recover and extend the construction of “Higher Berry curvatures” by Kapustin and Spodyneiko. Given a coarse cohomology class, we obtain a characteristic class on the parameter space X, which is integral whenever integrated against a cycle in X that lies in the image of the homological Chern character (so, in particular, spheres in X).

11:45 am –12:00 pm

break

12:00–1:00 pm

CMSA

Theo Johnson-Freyd, Perimeter Institute
Title: Some thoughts about the Kapustin–Kitaev cobordism conjecture
Abstract: In 2013, Kitaev explained that, under some reasonable locality hypotheses, gapped invertible phases of bosonic lattice models in different dimensions are naturally organized into an Omega-spectrum. The following year, Kapustin conjectured that this spectrum is dual to a Thom spectrum, specifically (smooth) oriented bordism MSO, and that for fermionic lattice models one sees instead the dual to spin bordism. In 2016, Freed and Hopkins proved Kapustin’s conjecture for invertible phases of continuous unitary QFTs valued in an at-the-time conjectural universal target category. Freed and Hopkins put bordism categories into the statement of the problem, by working from the beginning with continuous QFTs. Kapustin’s conjecture for lattice models remains open.David Reutter and I, in ongoing work in progress, have investigating Kapustin’s conjecture from the perspective of deeper category theory. We have built the universal target category for phases satisfying a finite semisimplicity hypothesis, and we are working on relaxing finite semisimplicity. We can show that any spectrum of invertible finite-semisimple phases will indeed be dual to a Thom spectrum for some topological group G acting on the spectrum of spheres. For example, if one looks just at those bosonic phases which can be topologically condensed from the vacuum, G is almost the (oriented) piecewise linear group, whose Thom spectrum is the bordism spectrum MSPL is the (oriented) *piecewise* smooth manifolds; the difference between MSPL and MSO is only visible in dimensions 7 and above. I say almost because in fact our G is what you would get if you tried to build MSPL, but could only make finitary measurements, which surely is explained by our restriction to condensable semisimple TQFTs. We conjecture that MSPL, rather than MSO, classifies invertible gapped phases of bosonic lattice models.The general relation between MSPL and topological phases is explained by a certain “surgery exact sequence” for topological phases that mirrors the surgery sequence for MSPL. By studying this sequence, we can also answer the question of which invertible phases admit a gapped boundary condition. In particular that only (the trivial phase and) the Arf–Kervaire invariants admit finite-semisimple gapped boundary conditions.
SLIDES (pdf)

Tuesday, July 8

8:00–9:00 am

MPIM

David Reutter, University of Hamburg
Title: On the categorical spectrum of topological quantum field theories
Abstract: As originally suggested by Kitaev, invertible topological quantum field theories of varying dimensions should assemble into a spectrum/generalized homology theory. A candidate for such a spectrum of invertible TQFTs was proposed by Freed and Hopkins, with the defining property that (isomorphism classes of) n-dimensional invertible TQFTs are completely determined by their partition functions on closed n-manifolds. More generally, not-necessarily-invertible TQFTs should assemble into a ‘categorical spectrum’, an analogue of a spectrum with non-invertible cells at each level. In this talk, I will explain that there exists a unique such categorical spectrum satisfying a list of reasonable assumptions on the collection of (compact/very finite & discrete) TQFTs; one of these assumptions being that its invertibles agree with Freed and Hopkins’ suggestion. I will explain these assumptions, sketch how this categorical spectrum looks like in low-dimensions, outline its construction, and how it may be used to learn about gapped boundaries of anomaly theories in high dimensions. This is based on work in progress with Theo Johnson-Freyd.

9:00–9:30 am

Breakfast break

9:30–10:30 am

CMSA

Agnes Beaudry, UC Boulder
Title: An algebraic theory of planon-only fracton orders
Abstract: In this talk, I will describe an algebraic theory for planon-only abelian fracton orders. These are three-dimensional gapped phases with the property that fractional excitations are abelian particles restricted to move in parallel planes. The fusion and statistics data can be identified with a finitely generated module over a Laurent polynomial ring together with a U(1)-valued quadratic form. These systems thus lend themselves to an elegant algebraic theory which we expect will lead to easily computable phase invariants and a classification. As a starting point, we establish a necessary condition for physical realizability, the excitation-detector principle, which I will explain. We conjecture that this criterion is also sufficient for realizability. I will also discuss preliminary classification results.This talk is based on joint with Michael Hermele, Wilbur Shirley and Evan Wickenden.

10:30–10:45 am

break

10:45–11:45 am

MPIM

João Faria Martins, University of Leeds
Title: A categorification of Quinn’s finite total homotopy TQFT with application to TQFTs and once-extended TQFTs derived from discrete higher gauge theory
Abstract: Quinn’s Finite Total Homotopy TQFT is a topological quantum field theory defined for any dimension n of space, depending on the choice of a homotopy finite space B. For instance, B can be the classifying space of a finite group or a finite 2-group.In this talk, I will report on recent joint work with Tim Porter on once-extended versions of Quinn’s Finite Total Homotopy TQFT, taking values in the symmetric monoidal bicategory of groupoids, linear profunctors, and natural transformations between linear profunctors. The construction works in all dimensions, yielding (0,1,2)-, (1,2,3)-, and (2,3,4)-extended TQFTs, given a homotopy finite space B. I will show how to compute these once-extended TQFTs when B is the classifying space of a homotopy 2-type, represented by a crossed module of groups.Reference: Faria Martins J, Porter T: “A categorification of Quinn’s finite total homotopy TQFT with application to TQFTs and once-extended TQFTs derived from strict omega-groupoids.” arXiv:2301.02491 [math.CT]

11:45 am –12:00 pm

break

12:00–1:00 pm

CMSA

Emil Prodan, Yeshiva University
Title: Mapping the landscape of frustration-free models
Abstract: Frustration-free models are of great interest because they are amenable to specialized techniques and their understanding is more complete among the general quantum spin models. In this talk, I will establish an almost bijective relation between frustration-free families of projections and a subclass of hereditary subalgebras defined by an intrinsic property. This relation sets further synergies between frustration-free models and open projections in double duals, and subsets of pure states spaces. These connections enable a better understanding of the class of frustration-free models. For example, the open projections in the double dual derived from frustration-free models is dense in the norm-topology in the space of generic open projections, thus assuring us that, for many purposes, we can choose to work with frustration-free models without losing generality. Furthermore, the Cuntz semigroup, originally designed to classify the positive elements of C*-algebra, has been proven to also classify the open projections. Given the mentioned connections, we now have a new device to investigate the ground states of quantum spin models.
SLIDES (pdf)

Wednesday, July 9

8:00–9:00 am

MPIM

Alexander Schenkel, University of Nottingham
Title: C*-categorical prefactorization algebras for superselection sectors and topological order
Abstract: I will present a geometric framework to encode the algebraic structures on the category of superselection sectors of an algebraic quantum field theory on the n-dimensional lattice Z^n. I will show that, under certain assumptions which are implied by Haag duality, the monoidal C*-categories of localized superselection sectors carry the structure of a locally constant prefactorization algebra over the category of cone-shaped subsets of Z^n. Employing techniques from higher algebra, one extracts from this datum an underlying locally constant prefactorization algebra defined on open disks in the cylinder R^1 x S^{n-1}. While the sphere S^{n-1} arises geometrically as the angular coordinates of cones, the origin of the line R^1 is analytic and rooted in Haag duality. The usual braided (for n=2) or symmetric (for n>2) monoidal C*-categories of superselection sectors are recovered by removing a point of the sphere and using the equivalence between E_n-algebras and locally constant prefactorization algebras defined on open disks in R^n. The non-trivial homotopy groups of spheres induce additional algebraic structures on these E_n-monoidal C*-categories, which in the simplest case of Z^2 is given by a braided monoidal self-equivalence arising geometrically as a kind of ‘holonomy’ around the circle S^1.This talk is based on joint work with Marco Benini, Victor Carmona and Pieter Naaijkens.

9:00–9:30 am

Breakfast break

9:30–10:30 am

CMSA

Lukasz Fidkowski, University of Washington
Title: Non-invertible bosonic chiral symmetry on the lattice
Abstract: We construct a Hamiltonian lattice realization of the non-invertible chiral symmetry that mimics an axial rotation at a rational angle in a U(1) gauge theory with bosonic charged matter. We provide a heuristic argument that this setup allows a symmetric Hamiltonian which flows, at low energies, to a known field theory with this symmetry.

10:30–10:45 am

break

10:45–11:45 am

MPIM

Nils Carqueville, University of Vienna
Title: Gauging categorical symmetries
Abstract: Orbifold data are categorical symmetries that can be gauged in oriented defect topological quantum field theories. We review the general construction and apply it to 2-group symmetries of 3-dimensional TQFTs; upon further specialisation this leads to equivariantisation of G-crossed braided fusion categories. We also describe a proposal, via higher dagger categories, to gauging categorical symmetries in the context of other tangential structures. This is based on separate projects with Benjamin Haake and Tim Lüders.

11:45 am –12:00 pm

break

12:00–1:00 pm

CMSA

Nikita Sopenko, IAS
Title: Reflection positivity and invertible phases of 2d quantum many-body systems
Abstract: Reflection positivity is a property that is usually taken as an assumption in the classification of topological phases of matter via continuous quantum field theories. For general quantum many-body systems, this property does not hold. This raises the question of whether it somehow emerges in the effective theory from the microscopic description, thereby justifying the field-theoretic approach.In this talk, I will discuss reflection positivity in the context of invertible phases of two-dimensional lattice systems. I will explain why every such phase admits a reflection-positive representative, and why inverse phases are represented by complex conjugate states. I will also introduce an index that distinguishes these phases and is conjecturally related to the chiral central charge.

Thursday, July 10

8:00–9:00 am

MPIM

Ilka Brunner, Ludwig-Maximilians University of Munich
Title: Defects as functors between phases of Abelian gauged linear sigma models
Abstract: Defects act naturally on boundary conditions, providing functors between D-brane categories. In the context of gauged linear sigma models, one can use defects to transport branes from one phase to another. In this talk, I will show how to construct such defects explicitly.

9:00–9:30 am

Breakfast break

9:30–10:30 am

CMSA

David Penneys, Ohio State
Title: Holography for bulk-boundary local topological order
Abstract: In previous joint work [arXiv:2307.12552] with C. Jones, Naaijkins and Wallick, we introduced local topological order (LTO) axioms for quantum spin systems which allowed us to define a physical boundary manifested by a net of boundary algebras in one dimension lower. This gives a formal setting for topological holography, where the braided tensor category of DHR bimodules of the physical boundary algebra captures the bulk topological order.In joint work with C. Jones and Naaijkens, we extend the LTO axioms to quantum spin systems equipped with a topological boundary, again producing a physical boundary algebra for the bulk-boundary system, whose category of (topological) boundary DHR bimodules recovers the topological boundary order. We perform this analysis in explicit detail for Levin-Wen and Walker-Wang bulk-boundary systems.Along the way, we introduce a 2D braided categorical net of algebras built from a unitary braided fusion category (UBFC), which arise as boundary algebras of Walker-Wang models. We consider the canonical state on this braided categorical net corresponding to the standard topological boundary for the Walker-Wang model. Interestingly, in this state, the cone von Neumann algebras are type I with finite dimensional centers, in contrast with the type II and III cone von Neumann algebras from the Levin-Wen models studied in [arXiv:2307.12552]. Their superselection sectors recover the underlying unitary category of our UBFC, and we conjecture the superselection category also captures the fusion and braiding.

10:30–10:45 am

break

10:45–11:45 am

MPIM

Christoph Schweigert, University of Hamburg
Title: Tensor network states: a topological field theory perspective.
Abstract: Projected entangled pair states (PEPS) and matrix product operators (MPO) are standard tools in quantum information theory and quantum many-body physics. We explain how to understand them in terms of Turaev-Viro models on manifolds with boundary. We then sketch how a recently developed categorical Morita theory for spherical module categories can be used to find generalizations of the standard PEPS tensors.

11:45 am –12:00 pm

break

12:00–1:00 pm

CMSA

Greg Moore, Rutgers
Title: p-form puzzles
Abstract: It is commonly stated that level k BF theory for a p-form (and a form of complementary dimension) is equivalent to a homotopy sigma model with target space K(A,p) where A is a cyclic group of order k. Some aspects of this standard statement are puzzling me. I’ll explain what they are. (Perhaps someone in the audience can resolve my puzzles.) Then I’ll revisit the (again standard) electromagnetic duality of p-form electrodynamics. The conclusion will be that a slightly modified version of Ray-Singer torsion is the partition function of an invertible topological field theory.

Friday, July 11Note: On Friday, there will be separate schedules for Bonn and CMSA.
To view the Bonn schedule, please visit the program page at: https://www.mpim-bonn.mpg.de/qft25

8:00–9:00 am

CMSA

Markus Pflaum, UC Boulder
Title: A tour d’horizon through homotopical aspects of C*-algebraic quantum spin systems
Abstract: In the talk I report on joint work with Beaudry, Hermele, Moreno, Qi and Spiegel, where a homotopy theoretic framework for studying state spaces of quantum lattice spin systems has been introduced using the language of C*-algebraic quantum mechanics. First some old and new results about the state space of the quasi-local algebra of a quantum lattice spin system when endowed with either the natural metric topology or the weak* topology will be presented. Switching to the algebraic topological side, the homotopy groups of the unitary group of a UHF algebra will then be determined and it will be indicated that the pure state space of any UHF algebra in the weak* topology is weakly contractible. In addition, I will show at the example of non-commutative tori that also in the case of a not commutative C*-algebra, the homotopy type of the state space endowed with the weak* topology can be non-trivial and is neither deformation nor Morita invariant. Finally, I indicate how such tools together with methods from higher homotopy theory such as E_infinity spaces may lead to a framework for constructing Kitaev’s loop-spectrum of bosonic invertible gapped phases of matter.

9:00–9:30 am

Breakfast break

9:30–11:00 am

CMSA

Speed Talks
Ben Gripaios, University of CambridgeTitle: Locality and smoothness of QFTs
Carolyn Zhang, Harvard UniversityTitle: SymTFT approach for (non-)invertible symmetries of mixed states
Roman Geiko, UCLATitle: Omega-spectrum of stabilizer invertible phases

11:00–11:15 am

break

11:15–12:45 pm

CMSA

Speed Talks continued
Eric Roon, Michigan State UniversityTitle: Finitely Correlated States Driven by Topological Dynamics
Dmitri Pavlov, Texas Tech UniversityTitle: The classification of two-dimensional extended conformal field theories
Bowen Shi, University of Illinois Urbana-ChampaignTitle: Mathematical Puzzles from the Entanglement Bootstrap: On Immersions and regular homotopySLIDES (pdf)

Fall CMSA Welcome Event
2025-09-03 16:00 - 17:30 | CMSA 20 Garden Street Cambridge, Massachusetts 02138 United States
Fall CMSA Welcome Event
Date: September 3, 2025
Time: 4:00 pm
Location: CMSA Common Room, 20 Garden Street, Cambridge MA

All CMSA and Math affiliates are invited.

Math and Machine Learning Reunion Workshop
2025-09-08 09:00 - 2025-09-10 17:00 | CMSA 20 Garden Street Cambridge, Massachusetts 02138 United States
Math and Machine Learning Reunion Workshop
Dates: September 8–10, 2025
Location: Harvard CMSA, Room G10, 20 Garden Street, Cambridge MA
Machine learning and AI are increasingly important tools in all fields of research. In the fall of 2024, the CMSA Mathematics and Machine Learning Program hosted 70 mathematicians and machine learning experts, ranging from beginners to established leaders in their field, to explore ML as a research tool for mathematicians, and mathematical approaches to understanding ML. More than 20 papers came out of projects started and developed during the program. The MML Reunion workshop will be an opportunity for the participants to share their results, review subsequent developments, and develop directions for future research.
Invited Speakers

Angelica Babei, Howard University
Gergely Bérczi, Aarhus University
Joanna Bieri, University of Redlands
Giorgi Butbaia, University of New Hampshire
Randy Davila, RelationalAI, Rice University
Alyson Deines, IDA/CCR La Jolla
Sergei Gukov, Caltech
Yang-Hui He, University of Oxford
Mark Hughes, Brigham Young University
Kyu-Hwan Lee, University of Connecticut
Eric Mjolsness, UC Irvine
Maria Prat Colomer, Brown University
Sébastien Racanière, Google DeepMind
Eric Ramos, Stevens Institute of Technology
Tamara Veenstra, IDA-CCR La Jolla

Organizer:Michael Douglas, CMSA

Schedule
Monday Sep. 8, 2025

9:00–9:30 am
Morning refreshments

9:30–9:45 am
Introductions

9:45–10:45 am
Angelica Babei, Howard University
Title: Predicting Euler factors of elliptic curves
Abstract: Two non-isogenous elliptic curves will have distinct traces of Frobenius at a large enough prime, and a finite set of $a_p(E)$ values determines all others. However, even when enough $a_p(E)$ values are provided to uniquely identify the isogeny class, no efficient algorithm is known for determining the remaining $a_p(E)$ values from this finite set. Preliminary results show that ML models can learn to predict the next trace of Frobenius with a surprising degree of accuracy from relatively few nearby entries. We investigate some possible reasons for this performance. Based on joint work with François Charton, Edgar Costa, Xiaoyu Huang, Kyu-Hwan Lee, David Lowry-Duda, Ashvni Narayanan, and Alexey Pozdnyakov.

10:45–11:00 am
Break

11:00 am–12:00 pm
Kyu-Hwan Lee, University of Connecticut
Title: Machine learning mutation-acyclicity of quivers

12:00–1:30 pm
Lunch

1:30–2:30 pm
Gergely Bérczi, Aarhus University
Title: Diffusion Models for Sphere Packings

2:30–2:45 pm
Break

2:45–3:45 pm
Randy Davila, RelationalAI, Rice University
Title: Recent Developments in Automated Conjecturing
Abstract: The dream of a machine capable of generating deep mathematical insight has inspired decades of research—from Fajtlowicz’s Graffiti program in graph theory and chemistry to DeepMind’s neural breakthroughs in knot theory. In this talk, we briefly trace the evolution of automated conjecturing systems and present recent advances that deepen our understanding of what it means for machines to conjecture—a pursuit long embodied by our system, TxGraffiti. Building on this legacy, we introduce a new framework that integrates optimization, enumeration, and convex geometric methods with creative heuristics and symbolic translation. This extended system produces not only conjectured inequalities, but also necessary and sufficient condition statements, which can then be automatically ranked by IRIS (Inequality Ranking and Inference System) model and translated into Lean 4 for formal verification. The result is a flexible architecture capable of generating precise, human-readable, and logically rigorous conjectures with minimal manual intervention.
We showcase results across a range of mathematical areas, including graph theory, polyhedral theory, number theory, and—for the first time—conjectures in string theory, derived from the dataset of complete intersection Calabi–Yau (CICY) threefolds. Together, these developments suggest that with the right blend of structure, strategy, and aesthetic, machines can generate conjectures that not only withstand scrutiny but invite it—offering a glimpse into a future where AI contributes meaningfully to the creative process of mathematics.

3:45–4:00 pm
Break

4:00–5:00 pm
Eric Ramos, Stevens Institute of Technology
Title: An AI approach to a conjecture of Erdos
Abstract: Given a graph G, its independence sequence is the integral sequence a_1,a_2,…,a_n, where a_i is the number of independent sets of vertices of size i. In the 90’s Erdos and coauthors showed that this sequence need not be unimodal for general graphs, but conjectured that it is always unimodal whenever G is a tree. This conjecture was then naturally generalized to claim that the independence sequence of trees should be log concave, in the sense that a_i^2 is always above a_{i-1}a_{i+1}. This stronger version of the conjecture was shown to hold for all trees of at most 25 vertices. In 2023, however, using improved computational power and a considerably more efficient algorithm, Kadrawi, Levit, Yosef, and Mirzrachi proved that there were exactly two trees on 26 vertices whose independence sequence was not log concave. They also showed how these two examples could be generalized to create two families of trees whose members are all not log concave. Finally, in early 2025, Galvin provided a family of trees with the property that for any chosen positive integer k, there is a member T of the family where log concavity breaks at index alpha(T) – k, where alph(T) is the independence number of T. Outside of these three families, not much else was known about what causes log concavity to break.In this presentation, I will discuss joint work of myself and Shiqi Sun, where we used the PatternBoost architecture to train a machine to find counter-examples to the log concavity conjecture. We will discuss the successes of this approach – finding tens of thousands of new counter-examples with vertex set sizes varying from 27 to 101 – and some of its fascinating failures.

Tuesday, Sep. 9, 2025

9:00–9:30 am
Morning refreshments

9:30–10:30 am
Maria Prat Colomer, Brown University
Title: From PINNs to Computer-Assisted Proofs for Fluid Dynamics
Abstract: Physics-Informed Neural Networks (PINNs) have emerged as an alternative to traditional numerical methods for solving partial differential equations (PDEs). We apply PINNs to the study of low regularity problems in fluid dynamics, focusing on the incompressible 2D Euler equations. In particular, we study V-states, which are a class of weak, non-smooth solutions for which the vorticity is the characteristic function of a domain that rotates with constant angular velocity. We have obtained an approximate solution of a limiting V-state using a PINN and we are currently working on a rigourous proof of the existence of a nearby solution through a computer-assisted proof. Our PINN-based numerical approximation significantly improves on traditional methods, a key factor being the integration of prior mathematical knowledge of the problem to effectively explore the solution space.

10:30–11:00 am
Break

11:00 am–12:00 pm
Sebastian Racaniere, Google DeepMind
Title: Generative models and high dimensional symmetries: the case of Lattice QCD
Abstract: Applying normalizing flows, a machine learning technique for mapping distributions, to Lattice QCD offers a promising route to enhance simulations and overcome limitations of traditional methods like Hybrid Monte Carlo. LQCD aims to compute expectation values of observables from an intractable distribution defined over a lattice of fields. Normalizing flows can learn this complex distribution and generate new configurations, improving efficiency and addressing challenges such as critical slowing down and topological freezing. Topological freezing, in particular, traps simulations in local minima and prevents exploration of the full configuration space, affecting accuracy. This approach incorporates the symmetries of LQCD through gauge equivariant flows, leading to successful definitions and good effective sample sizes on smaller lattices. Beyond accelerating configuration generation, normalizing flows also find application in variance reduction for observable calculation and exploring phenomena at different scales within LQCD. While further research is needed to apply these methods at the scale of state-of-the-art LQCD calculations, these advancements hold significant potential to improve the accuracy, efficiency, and reach of future simulations.

12:00–1:30 pm
Lunch break

1:30–2:30 pm
Sergei Gukov, Caltech
Title: On sparse reward problems in mathematics
Abstract: An alternative title for this talk could be “Learning Hardness.” To see why, we will explore some long-standing open problems in mathematics and examine what makes them hard from a computational perspective. We will argue that, in many cases, the difficulty arises from a highly uneven distribution of hardness within families of related problems, where the truly hard cases lie far out in the tail. We will then discuss how recent advances in AI may provide new tools to tackle these challenges. Based in part on the recent work with A.Shehper, A.Medina-Mardones, L.Fagan, B.Lewandowski, A.Gruen, Y.Qiu, P.Kucharski, and Z.Wang.

2:30–2:45 pm
Break

2:45–3:45 pm
Alyson Deines, IDA-CCR La Jolla; Tamara Veenstra, IDA-CCR La Jolla; Joanna Bieri, University of Redlands
Title: Machine learning $L$-functions
Abstract: We study the vanishing order of rational $L$-functions and Maass form $L$-functions from a data scientific perspective. Each $L$-function is represented by finitely many Dirichlet coefficients, the normalization of which depends on the context. We observe murmurations by averaging over these datasets. For rational $L$-functions, we find that PCA clusters rational $L$-functions by their vanishing order and record that LDA and neural networks may accurately predict this quantity. For Maass form $L$-functions, while PCA does not cluster these $L$-functions, we still find that LDA and neural networks may accurately predict this quantity.

3:45–4:00 pm
Break

4:00–5:00 pm
Mark Hughes, Brigham Young University
Title: Modelling the concordance group via contrastive learning
Abstract: The concordance group of knots in 3-space is an abelian group formed by the equivalence classes of knots under the relation of concordance, where two knots are concordant if they are the boundary of a smooth annulus properly embedded in the 4-dimensional product space S^3 x I. Though studied since 1966, properties of the concordance groups (and even the recognition problem of deciding when a knot is null-concordant, or slice) are difficult to study. In this talk I will outline ongoing attempts to model the concordance group using contrastive learning. This is joint work with Onkar Singh Gujral.

Wednesday Sep. 10, 2025

9:00–9:30 am
Morning refreshments

9:30–10:30 am
Yang-Hui He, University of Oxford (Via Zoom)
Title: AI for Mathematics: Bottom-up, Top-Down, Meta-
Abstract: We argue how AI can assist mathematics in three ways: theorem-proving, conjecture formulation, and language processing. Inspired by initial experiments in geometry and string theory in 2017, we summarize how this emerging field has grown over the past years, and show how various machine-learning algorithms can help with pattern detection across disciplines ranging from algebraic geometry to representation theory, to combinatorics, and to number theory. At the heart of the programme is the question how does AI help with theoretical discovery, and the implications for the future of mathematics.

10:30–11:00 am
Break

11:00 am–12:00 pm
Giorgi Butbaia, University of New Hampshire
Title: Computational String Theory using Machine Learning
Abstract: Calabi-Yau compactifications of the $E_8times E_8$ heterotic string provide a promising route to recovering the four-dimensional particle physics described by the Standard Model. While the topology of the Calabi-Yau space determines the overall matter content in the low-energy effective field theory, further details of the compactification geometry are needed to calculate the normalized physical couplings and masses of elementary particles. In this talk, we present novel numerical techniques for computing physically normalized Yukawa couplings in a number of heterotic models in the standard embedding using geometric machine learning and equivariant neural networks. We observe that the results produced using these techniques are in excellent agreement with the expected values for certain special cases, where the answers are known. In the case of the Tian-Yau manifold, which defines a model with three generations and has $h^{2,1}>1$, we provide a first-of-its-kind calculation of the normalized Yukawa couplings. As part of this work, we have developed a Python library called cymyc, which streamlines calculation of the Calabi-Yau metric and the Yukawa couplings on arbitrary Calabi-Yau manifolds that are realized as complete intersections and provides a framework for studying the differential geometric properties, such as the curvature.

12:00–1:30 pm
Lunch break

1:30–2:30 pm
Eric Mjolsness, UC Irvine
Title: Graph operators for science-applied AI/ML
Abstract: Scalable, structured graphs play a central role in mathematical problem definition for scientific applications of artificial intelligence and machine learning. Qualitatively diverse kinds of operators are necessary to bring these graphs to life. Continuous-time processes govern the evolution of spatial graph embeddings and other graph-local differential equation systems, as well as the flow of probability between locally similar graph structures in a probabilistic Fock space, according to rules in a dynamical graph grammar (DGG). Both kinds of dynamics have biophysical application eg. to dynamic cytoskeleton, and both obey graph-centric time-evolution operators in an operator algebra that can be differentiated for learning. On the other hand coarse-scale discrete jumps in graph structure such as global mesh refinement can be modeled with a “graph lineage”: a sequence of sparsely interrelated graphs whose size grows roughly exponentially with level number. Graph lineages permit the definition of substantially more cost-efficient skeletal graph products, as versions of classic binary graph operators such as the Cartesian product and direct product of graphs, with analogous but not identical properties. Application to deep neural networks and to multigrid numerical methods are shown.
These two graph operator frameworks are interrelated. Further graph lineage operators allow the definition of graph frontier spaces, accommodating graph grammars and supporting the definition of skeletal graph-graph function spaces. In return, “confluent” graph grammars e.g. for adaptive mesh generation permit the definition of graph lineages through iteration. I will also sketch the design of compatible AI for Science systems that may exploit DGGs.
Joint work with Cory Scott and Matthew Hur.

2:30–3:00 pm
Break

3:00–5:00 pm
Panel and Discussion Group: Jordan Ellenberg, Tamara Veenstra, Sébastien Racaniere, Kyu-Hwan Lee, Sergei Gukov

Big Data Conference 2025
2025-09-11 09:00 - 2025-09-12 17:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Big Data Conference 2025
Dates: Sep. 11–12, 2025
Location: Harvard University CMSA, 20 Garden Street, Cambridge & via Zoom
The Big Data Conference features speakers from the Harvard community as well as scholars from across the globe, with talks focusing on computer science, statistics, math and physics, and economics.
Invited Speakers

Markus J. Buehler, MIT
Yiling Chen, Harvard
Jordan Ellenberg, UW Madison
Yue M. Lu, Harvard
Pankaj Mehta, BU
Nick Patterson, Harvard
Gautam Reddy, Princeton
Trevor David Rhone, Rensselaer Polytechnic Institute
Tess Smidt, MIT

Organizers:
Michael M. Desai, Harvard OEB | Michael R. Douglas, Harvard CMSA | Yannai A. Gonczarowski, Harvard Economics | Efthimios Kaxiras, Harvard Physics | Melanie Weber, Harvard SEAS

Big Data Youtube Playlist

Schedule
Thursday, Sep. 11, 2025

9:00 am
Refreshments

9:30 am
Introductions

9:45–10:45 am
Gautam Reddy, Princeton
Title: Global epistasis in genotype-phenotype maps

10:45–11:00 am
Break

11:00 am –12:00 pm
Nick Patterson, Harvard
Title: The Origin of the Indo-Europeans
Abstract: Indo-European is the largest family of human languages, with very wide geographical distribution and more than 3 billion native speakers. How did this family arise and spread? This question has been discussed for nearly 250 years but with the advent of the availability of DNA from ancient fossils is now largely understood, at least in broad outlines. We will describe what we now know about the origins.

12:00–1:30 pm
Lunch break

1:30–2:30 pm
Markus Buehler, MIT
Title: Superintelligence for scientific discovery
Abstract: AI is moving beyond prediction to become a partner in invention. While today’s models excel at interpolating within known data, true discovery requires stepping outside existing truths. This talk introduces superintelligent discovery engines built on multi-agent swarms: diverse AI agents that interact, compete, and cooperate to generate structured novelty. Guided by Gödel’s insight that no closed system is complete, these swarms create gradients of difference – much like temperature gradients in thermodynamics – that sustain flow, invention, and surprise. Case studies in protein design and music composition show how swarms escape data biases, invent novel structures, and weave long-range coherence, producing creativity that rivals human processes. By moving from “big data” to “big insight”, these systems point toward a new era of AI that composes knowledge across science, engineering, and the arts.

2:30–2:45 pm
Break

2:45–3:45 pm
Jordan Ellenberg, UW Madison
Title: What does machine learning have to offer mathematics?

3:45–4:00 pm
Break

4:00–5:00 pm
Pankaj Mehta, Boston University
Title: Thinking about high-dimensional biological data in the age of AI
Abstract: The molecular biology revolution has transformed our view of living systems. Scientific explanations of biological phenomena are now synonymous with the identification of the genes and proteins. The preeminence of the molecular paradigm has only become more pronounced as new technologies allow us to make measurements at scale. Combining this wealth of data with new artificial intelligence (AI) techniques is widely viewed as the future of biology. Here, I will discuss the promise and perils of this approach. I will focus on our unpublished work with collaborators on two fronts: (i) transformer-based models for understanding genotype-to-phenotype maps, and (ii) LLM-based ‘foundational models’ for cellular identity, such as TranscriptFormer, which is trained on single-cell RNA sequencing (scRNAseq) data. While LLMs excel at capturing complex evolutionary and demographic structure in DNA sequence data, they are much less adept at elucidating the biology of cellular identity. We show that simple parameter-free models based on linear-algebra outperform TranscriptFormer on downstream tasks related to cellular identity, even though TranscriptFormer has nearly a billion parameters. If time permits, I will conclude by showing how we can combine ideas from linear algebra, bifurcation theory, and statistical physics to classify cell fate transitions using scRNAseq data.

Friday, Sep. 12, 2025

9:00-9:45 am
Refreshments

9:45–10:45 am
Yiling Chen, Harvard
Title: Data Reliability Scoring
Abstract: Imagine you are trying to make a data-driven decision, but the data at hand may be noisy, biased, or even strategically manipulated. Can you assess whether such a dataset is reliable—without access to ground truth?
We initiate the study of reliability scoring for datasets reported by potentially strategic data sources. While the true data remain unobservable, we assume access to auxiliary observations generated by an unknown statistical process that depends on the truth. We introduce the Gram Determinant Score, a reliability measure that evaluates how well the reported data align with the unobserved truth, using only the reported data and the auxiliary observations. The score comes with provable guarantees: it preserves several natural reliability orderings. Experimentally, it effectively captures data quality in settings with synthetic noise and contrastive learning embeddings.
This talk is based on joint work with Shi Feng, Fang-Yi Yu, and Paul Kattuman.

10:45–11:00 am
Break

11:00 am –12:00 pm
Yue M. Lu, Harvard
Title: Nonlinear Random Matrices in High-Dimensional Estimation and Learning
Abstract: In recent years, new classes of structured random matrices have emerged in statistical estimation and machine learning. Understanding their spectral properties has become increasingly important, as these matrices are closely linked to key quantities such as the training and generalization performance of large neural networks and the fundamental limits of high-dimensional signal recovery. Unlike classical random matrix ensembles, these new matrices often involve nonlinear transformations, introducing additional structural dependencies that pose challenges for traditional analysis techniques.
In this talk, I will present a set of equivalence principles that establish asymptotic connections between various nonlinear random matrix ensembles and simpler linear models that are more tractable for analysis. I will then demonstrate how these principles can be applied to characterize the performance of kernel methods and random feature models across different scaling regimes and to provide insights into the in-context learning capabilities of attention-based Transformer networks.

12:00–1:30 pm
Lunch break

1:30–2:30 pm
Trevor David Rhone, Rensselaer Polytechnic Institute
Title: Accelerating the discovery of van der Waals quantum materials using AI
Abstract: van der Waals (vdW) materials are exciting platforms for studying emergent quantum phenomena, ranging from long-range magnetic order to topological order. A conservative estimate for the number of candidate vdW materials exceeds ~106 for monolayers and ~1012 for heterostructures. How can we accelerate the exploration of this entire space of materials? Can we design quantum materials with desirable properties, thereby advancing innovation in science and technology? A recent study showed that artificial intelligence (AI) can be harnessed to discover new vdW Heisenberg ferromagnets based on Cr2Ge2Te6 [1], [2] and magnetic vdW topological insulators based on MnBi2Te4 [3]. In this talk, we will harness AI to efficiently explore the large chemical space of vdW materials and to guide the discovery of vdW materials with desirable spin and charge properties. We will focus on crystal structures based on monolayer Cr2I6 of the form A2X6, which are studied using density functional theory (DFT) calculations and AI. Magnetic properties, such as the magnetic moment are determined. The formation energy is also calculated and used as a proxy for the chemical stability. We also investigate monolayers based on MnBi2Te4 of the form AB2X4 to identify novel topological materials. Further to this, we study heterostructures based on MnBi2Te4/Sb2Te3 stacks. We show that AI, combined with DFT, can provide a computationally efficient means to predict the thermodynamic and magnetic properties of vdW materials [4],[5]. This study paves the way for the rapid discovery of chemically stable vdW quantum materials with applications in spintronics, magnetic memory and novel quantum computing architectures.
[1] T. D. Rhone et al., “Data-driven studies of magnetic two-dimensional materials,” Sci. Rep., vol. 10, no. 1, p. 15795, 2020.
[2] Y. Xie, G. Tritsaris, O. Granas, and T. Rhone, “Data-Driven Studies of the Magnetic Anisotropy of Two-Dimensional Magnetic Materials,” J. Phys. Chem. Lett., vol. 12, no. 50, pp. 12048–12054.
[3] R. Bhattarai, P. Minch, and T. D. Rhone, “Investigating magnetic van der Waals materials using data-driven approaches,” J. Mater. Chem. C, vol. 11, p. 5601, 2023.
[4] T. D. Rhone et al., “Artificial Intelligence Guided Studies of van der Waals Magnets,” Adv. Theory Simulations, vol. 6, no. 6, p. 2300019, 2023.
[5] P. Minch, R. Bhattarai, K. Choudhary, and T. D. Rhone, “Predicting magnetic properties of van der Waals magnets using graph neural networks,” Phys. Rev. Mater., vol. 8, no. 11, p. 114002, Nov. 2024.
This work used the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation Grant No. ACI-1548562. This research used resources of the Argonne Leadership Computing Facility, which is a DOE Office of Science User Facility supported under Contract No. DE-AC02-06CH11357. This material is based on work supported by the National Science Foundation CAREER award under Grant No. 2044842.

2:30–2:45 pm
Break

2:45–3:45 pm
Tess Smidt, MIT
Title: Applications of Euclidean neural networks to understand and design atomistic systems
Abstract: Atomic systems (molecules, crystals, proteins, etc.) are naturally represented by a set of coordinates in 3D space labeled by atom type. This poses a challenge for machine learning due to the sensitivity of coordinates to 3D rotations, translations, and inversions (the symmetries of 3D Euclidean space). Euclidean symmetry-equivariant Neural Networks (E(3)NNs) are specifically designed to address this issue. They faithfully capture the symmetries of physical systems, handle 3D geometry, and operate on the scalar, vector, and tensor fields that characterize these systems.
E(3)NNs have achieved state-of-the-art results across atomistic benchmarks, including small-molecule property prediction, protein-ligand binding, force prediciton for crystals, molecules, and heterogeneous catalysis. By merging neural network design with group representation theory, they provide a principled way to embed physical symmetries directly into learning. In this talk, I will survey recent applications of E(3)NNs to materials design and highlight ongoing debates in the AI for atomistic sciences community: how to balance the incorporation of physical knowledge with the drive for engineering efficiency.

The Geometry of Machine Learning
2025-09-15 09:00 - 2025-09-18 17:00 | CMSA 20 Garden Street Cambridge, Massachusetts 02138 United States
The Geometry of Machine Learning
Dates: September 15–18, 2025
Location: Harvard CMSA, Room G10, 20 Garden Street, Cambridge MA 02138
Despite the extraordinary progress in large language models, mathematicians suspect that other dimensions of intelligence must be defined and simulated to complete the picture. Geometric and symbolic reasoning are among these. In fact, there seems to be much to learn about existing ML by considering it from a geometric perspective, e.g. what is happening to the data manifold as it moves through a NN? How can geometric and symbolic tools be interfaced with LLMs? A more distant goal, one that seems only approachable through AIs, would be to gain some insight into the large-scale structure of mathematics as a whole: the geometry of math, rather than geometry as a subject within math. This conference is intended to begin a discussion on these topics.
Speakers

Maissam Barkeshli, University of Maryland
Eve Bodnia, Logical Intelligence
Adam Brown, Stanford
Bennett Chow, USCD & IAS
Michael Freedman, Harvard CMSA
Elliot Glazer, Epoch AI
James Halverson, Northeastern
Jesse Han, Math Inc.
Junehyuk Jung, Brown University
Alex Kontorovich, Rutgers University
Yann Lecun, New York University & META*
Jared Duker Lichtman, Stanford & Math Inc.
Brice Ménard, Johns Hopkins
Michael Mulligan, UCR & Logical Intelligence
Patrick Shafto, DARPA & Rutgers University

Organizers: Michael R. Douglas (CMSA) and Mike Freedman (CMSA)

Geometry of Machine Learning Youtube Playlist

Schedule
Monday, Sep. 15, 2025

8:30–9:00 am
Morning refreshments

9:00–10:00 am
James Halverson, Northeastern
Title: Sparsity and Symbols with Kolmogorov-Arnold Networks
Abstract: In this talk I’ll review Kolmogorov-Arnold nets, as well as new theory and applications related to sparsity and symbolic regression, respectively. I’ll review essential results regarding KANs, show how sparsity masks relate deep nets and KANs, and how KANs can be utilized alongside multimodal language models for symbolic regression. Empirical results will necessitate a few slides, but the bulk will be chalk.

10:00–10:30 am
Break

10:30–11:30 am
Maissam Barkeshli, University of Maryland
Title: Transformers and random walks: from language to random graphs
Abstract: The stunning capabilities of large language models give rise to many questions about how they work and how much more capable they can possibly get. One way to gain additional insight is via synthetic models of data with tunable complexity, which can capture the basic relevant structures of real data. In recent work we have focused on sequences obtained from random walks on graphs, hypergraphs, and hierarchical graphical structures. I will present some recent empirical results for work in progress regarding how transformers learn sequences arising from random walks on graphs. The focus will be on neural scaling laws, unexpected temperature-dependent effects, and sample complexity.

11:30 am–12:00 pm
Break

12:00–1:00 pm
Adam Brown, Stanford
Title: LLMs, Reasoning, and the Future of Mathematical Sciences
Abstract: Over the last half decade, the mathematical capabilities of large language models (LLMs) have leapt from preschooler to undergraduate and now beyond. This talk reviews recent progress, and speculates as to what it will mean for the future of mathematical sciences if these trends continue.

Tuesday, Sep. 16, 2025

8:30–9:00 am
Morning refreshments

9:00–10:00 am
Junehyuk Jung, Brown University
Title: AlphaGeometry: a step toward automated math reasoning
Abstract: Last summer, Google DeepMind’s AI systems made headlines by achieving Silver Medal level performance on the notoriously challenging International Mathematical Olympiad (IMO) problems. For instance, AlphaGeometry 2, one of these remarkable systems, solved the geometry problem in a mere 19 seconds!
In this talk, we will delve into the inner workings of AlphaGeometry, exploring the innovative techniques that enable it to tackle intricate geometric puzzles. We will uncover how this AI system combines the power of neural networks with symbolic reasoning to discover elegant solutions.

10:00–10:30 am
Break

10:30–11:30 am
Bennett Chow, USCD and IAS
Title: Ricci flow as a test for AI

11:30 am–12:00 pm
Break

12:00–1:00 pm
Jared Duker Lichtman, Stanford & Math Inc. and Jesse Han, Math Inc.
Title: Gauss – towards autoformalization for the working mathematician
Abstract: In this talk we’ll highlight some recent formalization progress using a new agent – Gauss. We’ll outline a recent Lean proof of the Prime Number Theorem in strong form, completing a challenge set in January 2024 by Alex Kontorovich and Terry Tao. We hope Gauss will help assist working mathematicians, especially those who do not write formal code themselves.

5:00–6:00 pm
Special Lecture: Yann LeCun, Science Center Hall C

Wednesday, Sep. 17, 2025

8:30–9:00 am
Refreshments

9:00–10:00 am
Michael Mulligan, UCR and Logical Intelligence
Title: Spontaneous Kolmogorov-Arnold Geometry in Vanilla Fully-Connected Neural Networks
Abstract: The Kolmogorov-Arnold (KA) representation theorem constructs universal, but highly non-smooth inner functions (the first layer map) in a single (non-linear) hidden layer neural network. Such universal functions have a distinctive local geometry, a “texture,” which can be characterized by the inner function’s Jacobian, $J(mathbf{x})$, as $mathbf{x}$ varies over the data. It is natural to ask if this distinctive KA geometry emerges through conventional neural network optimization. We find that indeed KA geometry often does emerge through the process of training vanilla single hidden layer fully-connected neural networks (MLPs). We quantify KA geometry through the statistical properties of the exterior powers of $J(mathbf{x})$: number of zero rows and various observables for the minor statistics of $J(mathbf{x})$, which measure the scale and axis alignment of $J(mathbf{x})$. This leads to a rough phase diagram in the space of function complexity and model hyperparameters where KA geometry occurs. The motivation is first to understand how neural networks organically learn to prepare input data for later downstream processing and, second, to learn enough about the emergence of KA geometry to accelerate learning through a timely intervention in network hyperparameters. This research is the “flip side” of KA-Networks (KANs). We do not engineer KA into the neural network, but rather watch KA emerge in shallow MLPs.

10:00–10:30 am
Break

10:30–11:30 am
Eve Bodnia, Logical Intelligence
Title:
Abstract: We introduce a method of topological analysis on spiking correlation networks in neurological systems. This method explores the neural manifold as in the manifold hypothesis, which posits that information is often represented by a lower-dimensional manifold embedded in a higher-dimensional space. After collecting neuron activity from human and mouse organoids using a micro-electrode array, we extract connectivity using pairwise spike-timing time correlations, which are optimized for time delays introduced by synaptic delays. We then look at network topology to identify emergent structures and compare the results to two randomized models – constrained randomization and bootstrapping across datasets. In histograms of the persistence of topological features, we see that the features from the original dataset consistently exceed the variability of the null distributions, suggesting that the observed topological features reflect significant correlation patterns in the data rather than random fluctuations. In a study of network resiliency, we found that random removal of 10 % of nodes still yielded a network with a lesser but still significant number of topological features in the homology group H1 (counts 2-dimensional voids in the dataset) above the variability of our constrained randomization model; however, targeted removal of nodes in H1 features resulted in rapid topological collapse, indicating that the H1 cycles in these brain organoid networks are fragile and highly sensitive to perturbations. By applying topological analysis to neural data, we offer a new complementary framework to standard methods for understanding information processing across a variety of complex neural systems.

11:30 am–12:00 pm
Break

12:00–1:00 pm
Alex Kontorovich, Rutgers University
Title: The Shape of Math to Come
Abstract: We will discuss some ongoing experiments that may have meaningful impact on what working in research mathematics might look like in a decade (if not sooner).

5:00–6:00 pm
Mike Freedman Millennium Lecture: The Poincaré Conjecture and Mathematical Discovery (Science Center Hall D)

Thursday, Sep. 18, 2025

8:30–9:00 am
Morning refreshments

9:00–10:00 am
Elliott Glazer, Epoch AI
Title: FrontierMath to Infinity
Abstract: I will discuss FrontierMath, a mathematical problem solving benchmark I developed over the past year, including its design philosophy and what we’ve learned about AI’s trajectory from it. I will then look much further out, speculate about what a “perfectly efficient” mathematical intelligence should be capable of, and discuss how high-ceiling math capability metrics can illuminate the path towards that ideal.

10:00–10:30 am
Break

10:30–11:30 am
Brice Ménard, Johns Hopkins
Title:Demystifying the over-parametrization of neural networks
Abstract: I will show how to estimate the dimensionality of neural encodings (learned weight structures) to assess how many parameters are effectively used by a neural network. I will then show how their scaling properties provide us with fundamental exponents on the learning process of a given task. I will comment on connections to thermodynamics.

11:30 am–12:00 pm
Break

12:00–12:30 pm
Patrick Shafto, Rutgers
Title: Math for AI and AI for Math
Abstract: I will briefly discuss two DARPA programs aiming to deepen connections between mathematics and AI, specifically through geometric and symbolic perspectives. The first aims for mathematical foundations for understanding the behavior and performance of modern AI systems such as Large Language Models and Diffusion models. The second aims to develop AI for pure mathematics through an understanding of abstraction, decomposition, and formalization. I will close with some thoughts on the coming convergence between AI and math.

12:30–12:45 pm
Break

12:45–2:00 pm
Mike Freedman, Harvard CMSA
Title: How to think about the shape of mathematics
Followed by group discussion

Support provided by Logical Intelligence.

Geometry of Machine Learning Special Lecture: Yann LeCun
2025-09-16 17:00 - 18:00 | Harvard Science Center, 1 Oxford Street, Cambridge, MA, 02138
Geometry of Machine Learning Special Lecture: Yann LeCun
Title: Self-Supervised Learning, JEPA, World Models, and the future of AI
Date: Tuesday, Sep. 16, 2025
Time: 5:00 pm ET
Location: Harvard Science Center, Hall C & via Zoom Webinar

Millennium Prize Problems Lecture - Michael Freedman: The Poincaré Conjecture and Mathematical Discovery
2025-09-17 17:00 - 18:00 | Harvard Science Center Hall D, 1 Oxford Street, Cambridge, MA, 02138, United States
Millennium Prize Problems Lecture
Date: September 17, 2025
Location: Harvard Science Center Hall D & via Zoom Webinar
Time: 5:00–6:00 pm
Speaker: Michael Freedman, Harvard CMSA and Logical Intelligence
Title: The Poincaré Conjecture and Mathematical Discovery
Abstract: The AI age requires us to re-examine what mathematics is about. The Seven Millenium Problems provide an ideal lens for doing so. Five of the seven are core mathematical questions, two are meta-mathematical – asking about the scope of mathematics. The Poincare conjecture represents one of the core subjects, manifold topology. I’ll explain what it is about, its broader context, and why people cared so much about finding a solution, which ultimately arrived through the work of R. Hamilton and G. Perelman. Although stated in manifold topology, the proof requires vast developments in the theory of parabolic partial differential equations, some of which I will sketch. Like most powerful techniques, the methods survive their original objectives and are now deployed widely in both three- and four-dimensional manifold topology.

Read more about the Poincaré Conjecture at the Clay Math website.
Organizers: Martin Bridson, Clay Mathematics Institute | Dan Freed, Harvard University and CMSA | Mike Hopkins, Harvard University

Millennium Prize Problems Lecture Series

Mathematical foundations of AI
2025-10-06 09:00 - 2025-10-10 17:00 | CMSA 20 Garden Street Cambridge, Massachusetts 02138 United States
Mathematical foundations of AI
Date: October 6–10, 2025
Location: Harvard CMSA, Room G10, 20 Garden Street, Cambridge MA & via Zoom
Artificial intelligence (AI) has achieved unprecedented advances, yet our theoretical understanding lags significantly behind. This gap poses a significant obstacle to improving AI’s safety and reliability. Since the classical tools of learning theory have proven insufficient for understanding AI, researchers are now drawing insights from a vast array of fields—including functional analysis, probability theory, optimal transport, optimization, PDEs, information theory, geometry, statistics, electrical engineering, and ergodic theory. Those interdisciplinary efforts are gradually shedding light on the underlying principles governing modern AI. This workshop centers around these mathematical and interdisciplinary developments. It will feature a series of talks from people in various subfields. Open problem and small-group sessions will help foster new connections and new research avenues.

Speakers

Jason Altschuler, University of Pennsylvania
Guy Bresler, MIT
Sinho Chewi, Yale University
Lenaic Chizat, EPFL
Nabarun Deb, University of Chicago
Edgar Dobriban, University of Pennsylvania
Ahmed El Alaoui, Cornell University
Zhou Fan, Yale University
Boris Hanin, Princeton University
Jason Klusowski, Princeton University
Tengyu Ma, Stanford University
Alexander Rakhlin, MIT
Yuting Wei, University of Pennsylvania
Tijana Zrnic, Stanford University

Organizer: Morgane Austern, Harvard Statistics

Schedule
Monday, Oct. 6, 2025

8:30–9:00 am
Morning refreshments

9:00–10:00 am
Yuting Wei, U Penn
To Intrinsic Dimension and Beyond: Efficient Sampling in Diffusion Models
The denoising diffusion probabilistic model (DDPM) has become a cornerstone of generative AI. While sharp convergence guarantees have been established for DDPM, the iteration complexity typically scales with the ambient data dimension of target distributions, leading to overly conservative theory that fails to explain its practical efficiency. This has sparked recent efforts to understand how DDPM can achieve sampling speed-ups through automatic exploitation of intrinsic low dimensionality of data. This talk explores two key scenarios: (1) For a broad class of data distributions with intrinsic dimension k, we prove that the iteration complexity of the DDPM scales nearly linearly with k, which is optimal under the KL divergence metric; (2) For mixtures of Gaussian distributions with k components, we show that DDPM learns the distribution with iteration complexity that grows only logarithmically in k. These results provide theoretical justification for the practical efficiency of diffusion models.

10:00–10:30 am
Break

10:30–11:30 am
Jason Klusowski, Princeton
The Value of Side Information in Unlabeled Data
Practitioners often work in settings with limited labeled data and abundant unlabeled data. During training, they may even have access to extra side information (some labeled, some not) that won’t be available once the model is deployed. When can this side information actually improve performance? I’ll present a simple framework where a rich-view model that sees the extra features generates pseudo-labels on the large unlabeled data, and a deployment model that only sees the standard features is trained on both real and pseudo-labels. The two are trained iteratively: each deployment model update calibrates the next round of pseudo-labels, and those refined pseudo-labels in turn guide the deployment model. Our theory shows that side information helps precisely when the rich-view and deployment models make different kinds of errors. We formalize this with a decorrelation score that quantifies how independent those errors are; the more independent, the greater the performance gains.

11:3 0am–12:00 pm
Break

12:00–1:00 pm
Guy Bresler, MIT
Global Minimizers of Sigmoid Contrastive Loss
The meta-task of obtaining and aligning representations through contrastive pre-training is steadily gaining importance since its introduction in CLIP and ALIGN. In this paper we theoretically explain the advantages of synchronizing with trainable inverse temperature and bias under the sigmoid loss, as implemented in the recent SigLIP models of Google DeepMind. Temperature and bias can drive the loss function to zero for a rich class of configurations that we call (m,b)-Constellations. (m,b)-Constellations are a novel combinatorial object related to spherical codes and are parametrized by a margin m and relative bias b. We use our characterization of constellations to theoretically justify the success of SigLIP on retrieval, to explain the modality gap present in SigLIP, and to identify the necessary dimension for producing high-quality representations. We also propose a reparameterization of the sigmoid loss with explicit relative bias, which appears to improve training dynamics. Joint work with Kiril Bangachev, Iliyas Noman, and Yury Polyanskiy.

Tuesday, Oct. 7, 2025

8:30–9:00 am
Morning refreshments

9:00–10:00 am
Lénaïc Chizat, EPFL
The Hidden Width of Deep ResNets
We present a mathematical framework to analyze the training dynamics of deep ResNets that rigorously captures practical architectures (including Transformers) trained from standard random initializations. Our approach combines stochastic approximation of ODEs with propagation-of-chaos arguments. It yields three main insights:
– Depth begets width: infinite-depth ResNets of any hidden width behave throughout training as if they were infinitely wide;
– Unified phase diagram: the phase diagram of Transformers mirrors that of two-layer perceptrons, once the appropriate substitutions are made;
– Optimal shape scaling: for a given parameter budget P, a Transformer with optimal shape converges to its limiting dynamics at rate P^{-1/6}.
This is based on https://arxiv.org/abs/2509.10167

10:00–10:30 am
Break

10:30–11:30 am
Boris Hanin, Princeton
Kernel Learning on Manifolds
This talk concerns the L_2 risk of minimum norm interpolation with n samples in the RKHS of a kernel K. Unlike most prior work in this space our kernels will be defined on any close d-dimensional Riemannian manifold, and we require only that the kernels are trace class and elliptic. With these assumptions we get nearly sharp L_2 risk bounds with high probability over the data. Like prior work on round spheres our results essentially say that the number of samples n, the dimension of the manifold, and some details of the kernel determine a natural spectral cutoff lambda(n,d,K) and that minimal norm interpolation essentially learns exactly the projection of the data generating process onto the eigenfunctions of the Laplacian with frequency at most lambda(n,d,K). Joint work with Mengxuan Yang.

11:30–12:00
Break

12:00–1:00
Zhou Fan, Yale
Dynamical mean-field analysis of adaptive Langevin diffusions
In many applications of statistical estimation via sampling, one may wish to sample from a high-dimensional target distribution that is adaptively evolving to the samples already seen. We study an example of such dynamics, given by a Langevin diffusion for posterior sampling in a Bayesian linear regression model with i.i.d. regression design, whose prior continuously adapts to the Langevin trajectory via a maximum marginal-likelihood scheme. Using techniques of dynamical mean-field theory (DMFT), we provide a precise characterization of a high-dimensional asymptotic limit for the joint evolution of the prior parameter and law of the Langevin sample. We then carry out an analysis of the equations that describe this DMFT limit, under conditions of approximate time-translation-invariance which include, in particular, settings where the posterior law satisfies a log-Sobolev inequality. In such settings, we show that this adaptive Langevin trajectory converges on a dimension-independent time horizon to an equilibrium state that is characterized by a system of replica-symmetric fixed-point equations, and the associated prior parameter converges to a critical point of a replica-symmetric limit for the model free energy. We explore the nature of the free energy landscape and its critical points in a few simple examples, where such critical points may or may not be unique.

Wednesday, Oct. 8, 2025

8:30–9:00 am
Morning refreshments

9:00–10:00 am
Jason Altschuler, U Penn
Negative Stepsizes Make Gradient-Descent-Ascent Converge
Solving min-max problems is a central question in optimization, games, learning, and controls. Arguably the most natural algorithm is Gradient-Descent-Ascent (GDA), however since the 1970s, conventional wisdom has argued that it fails to converge even on simple problems. This failure spurred the extensive literature on modifying GDA with extragradients, optimism, momentum, anchoring, etc. In contrast, we show that GDA converges in its original form by simply using a judicious choice of stepsizes. The key innovation is the proposal of unconventional stepsize schedules that are time-varying, asymmetric, and (most surprisingly) periodically negative. We show that all three properties are necessary for convergence, and that altogether this enables GDA to converge on the classical counterexamples (e.g., unconstrained convex-concave problems). The core intuition is that although negative stepsizes make backward progress, they de-synchronize the min/max variables (overcoming the cycling issue of GDA) and lead to a slingshot phenomenon in which the forward progress in the other iterations is overwhelmingly larger. This results in fast overall convergence. Geometrically, the slingshot dynamics leverage the non-reversibility of gradient flow: positive/negative steps cancel to first order, yielding a second-order net movement in a new direction that leads to convergence and is otherwise impossible for GDA to move in. Joint work with Henry Shugart.

10:00–10:30 am
Break

10:30–11:30 am
Nabarun Deb, U Chicago
Generative Modeling via Parabolic Monge-Ampère PDEs
We introduce a novel generative modeling framework based on a discretized parabolic Monge-Ampère PDE, which emerges as a continuous limit of the Sinkhorn algorithm commonly used in optimal transport. Our method performs iterative refinement in the space of Brenier maps using a mirror gradient descent step. We establish theoretical guarantees for generative modeling through the lens of no-regret analysis, demonstrating that the iterates converge to the optimal Brenier map under a variety of step-size schedules. As a technical contribution, we derive a new Evolution Variational Inequality tailored to the parabolic Monge-Ampère PDE, connecting geometry, transportation cost, and regret. Our framework accommodates non-log-concave target distributions, constructs an optimal sampling process via the Brenier map, and integrates favorable learning techniques from generative adversarial networks and score-based diffusion models.

11:30–12:00
Break

12:00–1:00
Sinho Chewi, Yale
Discretization and distribution learning in diffusion models
First, I will review some literature on discretization of diffusion models, focusing on the use of randomized midpoints for deterministic vs. stochastic samplers. Then, I will argue that such sampling guarantees reduce distribution learning, in the form of learning to generate a sample, to score matching. To complement this result, we reduce other forms of distribution learning (parameter estimation and density estimation) to score matching as well. This leads to new consequences for diffusion models, such as asymptotic efficiency of a DDPM-based parameter estimator and algorithms for Gaussian mixture density estimation, as well as to a general approach for establishing cryptographic hardness results for score estimation.

Thursday, Oct. 9, 2025

8:30–9:00 am
Morning refreshments

9:00–10:00 am
Ahmed El Alaoui, Cornell
How abundant are good interpolators?
We consider classifying labelled data in the interpolation regime where there exist linear classifiers (with possibly negative margin) correctly classifying all points in the dataset. Under the logistic model with gaussian features, we derive the large deviation rate function of the event that an interpolator chosen uniformly at random achieves a given generalization error. This describes the proportion of interpolators having any desired performance. We remark that in a wide regime of parameters, the vast majority of interpolators have inferior performance than the one found via a simple linear programming procedure, showing that the latter algorithm produces an atypically good classifier.
This is based on joint work with August Chen.

10:00–10:30 am
break

10:30–11:30 am
Tengyu Ma, Stanford
Self-play LLM Theorem Provers with Iterative Conjecturing and Proving
I will discuss some works on using RL for theorem proving, especially in the possible future regime where we ran out of high-quality training data. To keep improving the models with limited data, we draw inspiration from mathematicians, who continuously develop new results, partly by proposing novel conjectures or exercises (which are often variants of known results) and attempting to solve them. We design the Self-play Theorem Prover (STP) that simultaneously takes on two roles, conjecturer and prover, each providing training signals to the other. The model achieves state-of-the-art performance among whole-proof generation methods on miniF2F-test (65.0%, pass@3200), Proofnet-test (23.9%, pass@3200) and PutnamBench (8/644, pass@3200).

11:30–12:00
break

12:00–1:00
Edgar Dobriban, U Penn
Leveraging synthetic data in statistical inference
The rapid proliferation of high-quality synthetic data — generated by advanced AI models or collected as auxiliary data from related tasks — presents both opportunities and challenges for statistical inference. This paper introduces a GEneral Synthetic-Powered Inference (GESPI) framework that wraps around any statistical inference procedure to safely enhance sample efficiency by combining synthetic and real data. Our framework leverages high-quality synthetic data to boost statistical power, yet adaptively defaults to the standard inference method using only real data when synthetic data is of low quality. The error of our method remains below a user-specified bound without any distributional assumptions on the synthetic data, and decreases as the quality of the synthetic data improves. This flexibility enables seamless integration with conformal prediction, risk control, hypothesis testing, and multiple testing procedures, all without modifying the base inference method. We demonstrate the benefits of our method on challenging tasks with limited labeled data, including AlphaFold protein structure prediction, and comparing large reasoning models on complex math problems.

Friday, Oct. 10, 2025

8:30–9:00 am
Morning refreshments

9:00–10:00 am
Tijana Zrnic, Stanford
Probably Approximately Correct Labels
Obtaining high-quality labeled datasets is often costly, requiring either extensive human annotation or expensive experiments. We propose a method that supplements such “expert” labels with AI predictions from pre-trained models to construct labeled datasets more cost-effectively. Our approach results in probably approximately correct labels: with high probability, the overall labeling error is small. This solution enables rigorous yet efficient dataset curation using modern AI models. We demonstrate the benefits of the methodology through text annotation with large language models, image labeling with pre-trained vision models, and protein folding analysis with AlphaFold. This is joint work with Emmanuel Candes and Andrew Ilyas.

10:00–10:30 am
Break

10:30–11:30 am
Alexander Rakhlin, MIT
Elements of Interactive Decision Making
Machine learning methods are increasingly deployed in interactive environments, ranging from dynamic treatment strategies in medicine to fine-tuning of LLMs using reinforcement learning. In these settings, the learning agent interacts with the environment to collect data and necessarily faces an exploration-exploitation dilemma. We present a general framework for interactive decision making that subsumes multi-armed bandits, contextual bandits, structured bandits, and reinforcement learning. We focus on both the statistical aspect of learning—aiming to develop a tight characterization of sample complexity in terms of properties of the class of models—and on the basic algorithmic primitives.

Millennium Prize Problems Lecture - Sourav Chatterjee: Yang-Mills and the foundations of quantum field theory
2025-10-15 17:00 - 18:00 | Harvard Science Center Hall D, 1 Oxford Street, Cambridge, MA, 02138, United States
Millennium Prize Problems Lecture
Date: October 15, 2025
Time: 5:00–6:00 pm
Location: Harvard Science Center Hall D, 1 Oxford St., Cambridge MA
Speaker: Sourav Chatterjee, Stanford University
Title: Yang-Mills and the foundations of quantum field theory
Abstract: Yang-Mills theories are the building blocks of the Standard Model of quantum mechanics, which is the best available model for our universe at the quantum scale. Yet, these theories do not have a rigorous mathematical foundation. Physical calculations are based on perturbation theory, but there are various phenomena that are believed to be out of the reach of perturbative arguments. Building a mathematical foundation is, therefore, important even from the physics point of view. A program with this objective, known as “constructive field theory”, was initiated in the 1960s. In spite of many successes, the program has not reached its original goal. Completing this program is the Clay Millennium Prize problem of Yang-Mills existence and mass gap. I will give a general introduction to the main questions, and an overview of exciting recent progress that has rejuvenated the quest for a solution in the last ten years.
Read more about the Yang-Mills Existence and Mass Gap at the Clay Math website.
Organizers: Martin Bridson, Clay Mathematics Institute | Dan Freed, Harvard University and CMSA | Mike Hopkins, Harvard University

Millennium Prize Problems Lecture Series

Math Science Lectures in Honor of Raoul Bott | Dennis Gaitsgory, MPIM | Function-theoretic implications of geometric Langlands
2025-10-20 16:30 - 17:30 | Harvard Science Center, 1 Oxford Street, Cambridge, MA, 02138
Two talks on Function-theoretic implications of geometric Langlands
Dates: October 20 & 21, 2025
Time: 4:30–5:30 pm
Location: Science Center Lecture Hall A and via Webinar

Speaker: Dennis Gaitsgory, Max Planck Institute for Mathematics
Abstract: The recently established geometric Langlands equivalence implies an explicit description of the space of (unramified) automorphic functions in terms of Langlands parameters. In these lectures, we will derive these description and explain how far we can go with it in order to deduce some expected properties of automorphic functions, e.g., Ramanujan and Arthur multiplicity conjectures. This is joint work with Vincent Lafforgue and Sam Raskin.

Lecture 1: Monday, October 20, 2025
From geometric to classical Langlands

Lecture 2: Tuesday, October 21, 2025
Analytic properties of automorphic functions as seen from algebraic geometry

Harvard Mathematics Professor Raoul Bott (1923 – 2005), was a Hungarian-American mathematician known for numerous foundational contributions to geometry in its broad sense. He is best known for his Bott periodicity theorem, the Morse–Bott functions which he used in this context, and the Borel–Bott–Weil theorem.

Math Science Lectures in Honor of Raoul Bott | Dennis Gaitsgory, MPIM | Function-theoretic implications of geometric Langlands
2025-10-21 16:30 - 17:30 | Harvard Science Center, 1 Oxford Street, Cambridge, MA, 02138
Two talks on Function-theoretic implications of geometric Langlands
Dates: October 20 & 21, 2025
Time: 4:30–5:30 pm
Location: Science Center Lecture Hall A and via Webinar
Speaker: Dennis Gaitsgory, Max Planck Institute for Mathematics
Abstract: The recently established geometric Langlands equivalence implies an explicit description of the space of (unramified) automorphic functions in terms of Langlands parameters. In these lectures, we will derive these description and explain how far we can go with it in order to deduce some expected properties of automorphic functions, e.g., Ramanujan and Arthur multiplicity conjectures. This is joint work with Vincent Lafforgue and Sam Raskin.

Lecture 1: Monday, October 20, 2025
Function-theoretic implications of geometric Langlands: From geometric to classical Langlands

Lecture 2: Tuesday, October 21, 2025
Function-theoretic implications of geometric Langlands: Analytic properties of automorphic functions as seen from algebraic geometry

Harvard Mathematics Professor Raoul Bott (1923 – 2005), was a Hungarian-American mathematician known for numerous foundational contributions to geometry in its broad sense. He is best known for his Bott periodicity theorem, the Morse–Bott functions which he used in this context, and the Borel–Bott–Weil theorem.

Geometry Meets Physics: Finiteness, Tameness, and Complexity
2025-11-12 09:00 - 2025-11-14 17:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Geometry Meets Physics: Finiteness, Tameness, and Complexity
Dates: November 12–14, 2025
Location: CMSA G10, 20 Garden Street, Cambridge MA 02138
(note: this event is in-person only)
Finiteness is a fundamental property in consistent physical theories. From the earliest days of quantum field theory and string theory, the drive to eliminate unphysical infinities has been a guiding principle. More recently, finiteness has emerged as a key criterion for constraining effective theories that can be embedded in quantum gravity. Formulating and testing these constraints remains a central challenge in current research.
In parallel, mathematics has made remarkable advanced in addressing finiteness questions using tame geometry. Built on the framework of o-minimal structures, tame geometry offers a precise language for describing objects of finite geometric complexity. Recent developments, such as sharp o-minimality, go further by introducing a quantitative notion of complexity, opening new directions for analyzing finiteness in mathematics and physics alike.
This workshop brings together mathematicians and physicists to exchange ideas, explore new perspectives, and spark collaborations at the interface of geometry, logic, and fundamental physics.
Invited Speakers

Vijay Balasubramanian (UPenn)
Gregorio Baldi (CNRS, IMJ-PRG & IAS)
Gal Binyamini (Weizmann Institute & IAS)
Raf Cluckers (Lille, France)
Matilda Delgado (Max Planck Institute Munich)
Bruno Klingler (Humboldt University, Berlin & IAS)
Adele Padgett (Vienna)
David Prieto (Utrecht)
Washington Taylor (MIT)
David Urbanik (IHES, France & IAS)
Cumrun Vafa (Harvard)
Mick van Vliet (Utrecht)
Benny Zak (Weizmann Institute & IAS)

Organizers: Thomas Grimm, Harvard CMSA & Utrecht University | Gal Binyamini, Weizmann Institute & IAS | Bruno Klingler, Humboldt University, Berlin & IAS

Schedule
(download pdf)
Wednesday Nov. 12, 2025
8:30–8:55 am
Morning refreshments (Common Room)
8:55–9:00 am
Introductions
9:00–10:30 am
Lecture
Speaker: Gal Binyamini, Weizmann Institute & IAS
Title: O-minimality: finiteness and complexity
Abstract: O-minimality is a mathematical formalism of “tame geometry”: a geometry where every set has finite geometric complexity. I will give an introduction to o-minimality in general, and to quantitative variants where one measures the complexity of sets in terms of some natural parameters. I’ll try to focus on the main examples that potentially come up in the interaction with physics, and describe the state of the art and some conjectures.
10:30–11:00 am
Break
11:00 am–12:00 pm
Speaker: Benny Zak, Weizmann Institute & IAS
Title: Analytic tameness – complex cells
Abstract: Complex cells are a complex anayltic version of cells from o-minimality, invented by Binyamini and Novikov. We aim to introduce complex cells, and demonstrate their usefullness in quantifying the analytic information present in a complex set. If time permits, we will discuss applications of this theory.
12:00–1:00 pm
Catered Lunch (Common Room)
1:00–2:30 pm
Lecture
Speakers: David Prieto and Mick van Vliet, Utrecht
Title: Tameness and Complexity in Physical Theories
Abstract: We give an introductory overview of recent applications of o-minimality to physics, focusing on quantum field theories and quantum gravity. In the first part of the lecture we explain how o-minimality makes a first appearance in physical theories when considering amplitudes in quantum field theory. In the second part, we concentrate on a class of theories where finiteness principles seem to be essential, namely the quantum field theories which are consistent with quantum gravity. We review some of these finiteness principles and interpret them through the lens of the o-minimal framework. Along the way, we highlight recent progress in this direction, as well as open questions to explore in the future.
2:30–3:00 pm
Break with refreshments (Common Room)
3:00–4:00 pm
Speaker: Matilda Delgado, Max Planck Institute Munich
Title: Dualities and the Compactifiability of Moduli Space
Abstract: After introducing (self-)dualities in string theory and their action on the field content & spectrum of the theory, I will present the notion of compactifiability for the moduli space of massless fields as the condition that its volume is finite or grows no faster than Euclidean space. I will argue that compactifiability generically implies the existence of non-trivial dualities by providing evidence from string theory. Moreover, I will explain how one can connect compactifiability to the condition that the spectrum of objects charged under the duality group transform in a semisimple representation. Finally, I will provide a bottom-up argument for compactifiability, and argue that it (at least in supersymmetric cases) can be explained by the finiteness of the number of massless states upon compactification to 1D. Based on arXiv:2412.03640.
5:00 PM
Millennium Lecture and Reception: Pierre Deligne (IAS) (Science Center Hall D)
Title: What is the Hodge conjecture?

Thursday, Nov. 13, 2025
8:30–9:00 am
Morning refreshments (Common Room)
9:00–10:30 am
Lecture
Speaker: Bruno Klingler, Humboldt University, Berlin & IAS
Title: Tame geometry and Hodge theory
Abstract: I will give an introduction to applications of o-minimality in complex geometry, in particular in Hodge theory.
10:30–11:00 am
Break
11:00 am–12:00 pm
Speaker: Cumrun Vafa, Harvard
Title: The Swampland Program
12:00–1:30 pm
Catered Lunch (Common Room)
1:30–2:30 pm
Speaker: Gregorio Baldi, CNRS, IMJ-PRG & IAS
Title: The Hodge locus
Abstract: We will survey various recent results around the distribution of the Hodge locus of a (mixed) variation of Hodge structures. Various concrete applications to moduli spaces will also be presented.
2:30–3:00 pm
Break with refreshments (Common Room)
3:00–4:00 pm
Speaker: Vijay Balasubramanian, U Penn
Title: Chaos and complexity in quantum dynamics
4:30–5:30
Discussion/Q&A session
6:30 PM
Dinner: Changsho Restaurant, 1712 Massachusetts Ave., Cambridge, MA 02138

Friday Nov. 14, 2025
8:30–9:00 am
Morning refreshments (Common Room)
9:00–10:00 am
Speaker: Washington Taylor, MIT
Title: Finiteness, connectivity, and the power of fibrations in the Calabi-Yau landscape
10:00–10:30am
Break
10:30–11:30 am
Speaker: Adele Padgett, Vienna
Title: Tameness of multisummable series
Abstract: There are sophisticated theories of summability that map divergent series solutions of differential or functional equations to solutions that are holomorphic in sector-like domains. Van den Dries and Speissegger proved that functions obtained from real multisummable power series have tame geometric behavior when restricted to the real numbers. It would be desirable to know that these functions are also tame on their whole sector-like domains, but recently Speissegger and I proved that these functions are in general only tame on part of their domains. I will present this result and discuss the domains on which some examples are tame, including the Stirling series which appears in the asymptotic expansion of the Gamma function. In this talk, “tame” means definable in an o-minimal structure.
11:30 am–1:00 pm
Catered Lunch (Common Room)
1:00–2:00 pm
Speaker: Raf Cluckers, Lille, France
Title: Finiteness and tameness in (non-archimedean) geometry
Abstract: Non-archimedean geometry work with orders of magnitude rather than with precise measurements. The former works for example with orders of vanishing of functions, and the latter typically works with real or complex numbers. I will discuss recent progress on non-archimedean tame geometry. I will present analogues of o-minimality, of Pila-Wilkie’s o-minimal counting results, and of other finiteness results, in non-archimedean settings.
2:00–2:30 pm
Break with refreshments (Common Room)
2:30–3:30 pm
Speaker: David Urbanik, IHES, France & IAS
Title: Degrees of Hodge Loci

Millennium Prize Problems Lecture - Pierre Deligne: What is the Hodge conjecture?
2025-11-12 17:00 - 18:00 | Harvard Science Center Hall D, 1 Oxford Street, Cambridge, MA, 02138, United States
Date: November 12, 2025
Time: 5:00–6:00 pm
Location: Harvard Science Center Hall D, 1 Oxford St., Cambridge MA
Speaker: Pierre Deligne, Institute for Advanced Study
Title: What is the Hodge conjecture?
Abstract: The Hodge conjecture is about projective non-singular complex algebraic varieties. It characterizes the cohomology classes coming from algebraic cycles. I will explain these terms, tell why the conjecture is so hard to attack, and why we care.

Series Pamphlet (pdf)
Read more about the Hodge Conjecture at the Clay Math website.
Organizers: Martin Bridson, Clay Mathematics Institute | Dan Freed, Harvard University and CMSA | Mike Hopkins, Harvard University

Millennium Prize Problems Lecture Series

Conference on Geometry and Statistics
2025-11-17 09:00 - 2025-11-19 17:00 | CMSA 20 Garden Street Cambridge, Massachusetts 02138 United States
Conference on Geometry and Statistics
Dates: November 17–19, 2025
Location: CMSA G10, 20 Garden Street, Cambridge MA & via Zoom

Speakers

Charles Fefferman, Princeton University
Stephan Huckemann, Georg-August Universität Göttingen
Sungkyu Jung, Seoul National University
Kei Kobayashi, Keio University
Clément Levrard, Université de Rennes
Ker-Chau Li, University of California, Los Angeles
Rong Ma, Harvard University
Steve Marron, University of North Carolina
Ezra Miller, Duke University
Hans-Georg Müller, University of California, Davis
Wilderich Tuschmann, Karlsruhe Institute of Technology
Melanie Weber, Harvard University
Andrew Wood, Australian National University
Horng-Tzer Yau, Harvard University

Organizer: Zhigang Yao, National University of Singapore

Youtube Playlist

SCHEDULE
download pdf
Monday, Nov. 17, 2025
9:00–9:25 am
Morning refreshments
9:25–9:30 am
Introductions
9:30–10:30 am
Speaker: Stephan Huckemann, Georg-August Universität Göttingen
Title: The Probability of the Cut Locus of a Fréchet Mean
Abstract: We show that the cut locus of a Fréchet mean of a random variable on a connected and complete Riemanian manifold has zero probability, a result known previously in special cases (Le and Barden, 2014) and conjectured in general. The proof is based on first order and second order considerations, where the latter are based on a recent result by Générau (2020) on “Laplacians in the barrier sense”. This generalizes to Fréchet p-means for p > 2. The former allow also to rule out stickiness on Riemannian manifolds, and for generalization to 1 <= p < 2, with a conjecture. We close with discussing and conjecturing extensions to noncomplete manifolds and more general metric spaces. This is joint work with Alexander Lytchak.

Générau, F. (2020). Laplacian of the distance function on the cut locus on a Riemannian manifold. Nonlinearity 33(8), 3928.
Le, H. and D. Barden (2014). On the measure of the cut locus of a Fréchet mean. Bulletin of the London Mathematical Society 46(4), 698–708.
Lytchak, A. and S. F. Huckemann (2025). Zero mass at the cut locus of a Fréchet mean on a Riemannian manifold. arXiv preprint arXiv:2508.00747.

10:30–10:45 am
break
10:45 am–11:45 am
Speaker: Hans-Georg Müller, University of California, Davis
Title: Conformal Inference for Random Objects
Abstract: The underlying probability measure of random objects, i.e., metric-space-valued random variables, can be probed by distance profiles. These are one-dimensional distributions of probability mass falling into balls of increasing radius. In a regression setting with Euclidean covariates X and responses Y that are random objects, one can consider conditional Fréchet means that can be implemented with Fréchet regression and also conditional distance profiles, conditioning on X. Conditional distance profiles can then be leveraged to obtain conditional average transport costs, the expected cost for transporting a fixed conditional distance profile to a randomly selected conditional distance profile. The conditional average transport costs can then be utilized to obtain conditional conformity scores. In conjunction with the split conformal algorithm these scores lead to conditional prediction sets located in the object space with asymptotic conditional validity and attractive finite sample behavior. Based on joint work Hang Zhou (UNC).
11:45 am–1:15 pm
Lunch (Catered)
1:15–2:15 pm
Speaker: Horng-Tzer Yau, Harvard
Title: Ramanujan property of random regular graphs and delocalization of random band matrices
Abstract: In this lecture, we review recent works on random matrices. The first result is about the normalized adjacency matrix of a random $d$-regular graph on $N$ vertices with any fixed degree $dgeq 3$ and denote its eigenvalues as $lambda_1=d/sqrt{d-1}geq lambda_2geqlambda_3cdotsgeq lambda_N$. We establish the edge universality for random $d$-regular graphs, namely, the distributions of $lambda_2$ and $-lambda_N$ converge to the Tracy-Widom$_1$ distribution associated with the Gaussian Orthogonal Ensemble. As a consequence, for sufficiently large $N$, approximately $69%$ of $d$-regular graphs on $N$ vertices.
are Ramanujan, meaning $max{lambda_2,|lambda_N|}leq 2$. This resolves a conjecture by Sarnak and Miller-Novikoff-Sabelli
The second result concerns $ N times N$ Hermitian $d$-dimensional random band matrices with band width $W$. In the bulk of the spectrum and in the large $ N $ limit, we prove that all $ L^2 $- normalized eigenvectors are delocalized in all dimensions under suitable conditions on $W$ and $N$. In addition, we proved that the eigenvalue statistics are given by those of the Gaussian unitary ensemble.
2:15–2:45 pm
break with refreshments
2:45–3:45 pm
Speaker: Clément Levrard, Université de Rennes
Title: Optimal reach estimation
Abstract: The reach of an embedded submanifold, a notion that dates back to the famous work Curvature measures of H. Federer, may be understood as a scale under which the submanifold is flat enough so that traditional Euclidean techniques in statistics locally apply, up to some approximation. I will expose several ways to estimate the reach from sample (on the submanifold), some of them being optimal from the point of view of minimax estimation theory. Along the way, intermediate estimation problems of local and global quantities will arise (curvature estimation, weak feature size estimation, distance estimation, etc.), for which various phenomenons can occur from a statistical point of view (different convergence rates, inconsistency). This will be an opportunity to provide a selective overview of the state of the art on these issues.
4:30–5:30 pm
CMSA Colloquium
Speaker: Zhigang Yao (National University of Singapore)
Title: Interaction of Statistics and Geometry: A New Landscape for Data Science
Abstract: Classical statistics views data as real numbers or vectors in Euclidean space, but modern challenges increasingly involve data with intrinsic geometric structures. A central problem in this direction is manifold fitting, with origins in H. Whitney’s work of the 1930s. The Geometric Whitney Problems ask: given a set, when can we construct a smooth 𝑑-dimensional manifold that approximates it, and how accurately can we estimate it?
In this talk, I will discuss recent progress on manifold fitting and its role in bridging geometry and data science. While many existing methods rely on restrictive assumptions, the manifold hypothesis—that data often lie near non-Euclidean structures—remains fundamental in modern statistical learning. I will highlight both theoretical insights and algorithmic challenges, drawing on recent works with, as well as ongoing research.
Youtube video

Tuesday, Nov. 18, 2025
9:00–9:30 am
Morning refreshments
9:30–10:30 am
Speaker: Charles Fefferman, Princeton University (via Zoom)
Title: Extrinsic and intrinsic manifold learning, old and new
Abstract: The talk will include an exposition of the old paper “Testing the manifold hypothesis”, joint work with S. Mitter and H. Narayanan, on extrinsic manifold learning (the manifold to be learned is assumed to be embedded in a high-dimensional Euclidean space). The talk will also include a new result on intrinsic manifold learning (the manifold to be learned is not assumed to be embedded, and the data consist of intrinsic distances corrupted by noise), provided the result is proven by the time of the conference.
10:30–10:45 am
break
10:45 am–11:45 am
Speaker: Steve Marron, University of North Carolina
Title: Data Integration Via Analysis of Manifolds (DIVAM)
Abstract: A major challenge in the age of Big Data is the integration of disparate data types into a single data analysis. That was tackled by Data Integration Via Analysis of Subspaces (DIVAS) in the context of data blocks measured on a common set of experimental cases. Joint variation was defined in terms of modes of variation having identical scores across data blocks. DIVAS allowed mathematically rigorous formulation of individual variation within each data block in terms of individual modes. The goal of DIVAM is to intrinsically extend the DIVAS approach to data objects lying in manifolds, such as shape data.
11:45 am–1:15 pm
Lunch Break
1:15–2:15 pm
Speaker: Ker-Chau Li, University of California, Los Angeles
Title: Investigation of Data clouds: From Galton’s Ellipses to Explainable AI (XAI), modeling or molding?
Abstract: Francis Galton’s seminal 1886 visualization of regression toward the mean in trait inheritance is arguably the first and most influential example of geometric thinking applied to statistical modeling. The pioneering geometric insight driving Galton’s use of elliptical contours to discover the bivariate normal distribution laid down the foundation for classic multivariate analysis (e.g., PCA, canonical correlation) and profoundly impacts modern methods like diffusion models.
Statistical models, particularly those based on parsimony, are effective for characterizing data distribution and facilitating scientific rule induction. However, the rise of unstructured big data (like images) has challenged these parsimonious approaches, necessitating the use of deep learning models. These models, containing billions of parameters, sacrifice transparency to excel in prediction. Seeking solutions to this “black-box” dilemma is now the heart of Explainable AI (XAI).
Leveraging the simplicity of elementary geometric concepts, this talk will present a new path toward interpretable and parsimonious XAI. Unstructured big data is highly plastic. Our approach moves beyond the standard data modeling perspective—which answers what the data is—and introduces a novel data molding perspective. This shift is key to unlocking the full potential of data’s plasticity, allowing us to effectively answer the crucial question: what the data can be used for.
I will first discuss a connection between manifold learning and my earlier works, helical confounding and liquid association. I will then turn to the data molding perspective and present two novel notions: mold-compliance and artificial-trait configurative-generation (ATCG). These notions guide our recent efforts in formulating novel algorithms for image data investigation, addressing issues like prediction validity and within-class heterogeneity. Data molding entails a dramatically different feature space extraction, which consequently shifts the subsequent investigation on the data clouds from out-of-distribution (OOD) to mold-violation, and from UMAP clustering to ATCG-induced hierarchical clustering.
2:15–2:45 pm
break with refreshments
2:45–3:45 pm
Speaker: Andrew Wood, Australian National University
Title: Empirical likelihood methods for Fréchet means on open books
Abstract: The open book is a simple example of a stratified space that captures some (but not all) of the properties of stratified spaces. Central limit theory for open books plus relevant background is given by Hotz et al. (2013, Annals of Applied Probability). In this talk I will describe some basic inference procedures for Fréchet means in open books based on empirical likelihood (Owen, book, 2001). Empirical likelihood (EL) is a type of nonparametric likelihood that can be useful for many types of data, including manifold-valued data and data from stratified spaces. An EL approach to basic inference for Fréchet means will be described. In particular, it will be shown how the non-regularity in the geometry of open books can result in non-regular behaviour in Wilks’s theorem (i.e. the large sample likelihood ratio test). The talk will also discuss difficulties in extending the EL inference theory from open books to more general stratified spaces, where the difference in dimension of adjacent strata can be 2 or more. For discussion of more general stratified spaces than open books, see the orthant spaces discussed in Barden and Le (2018, Proc of London Math Society) and the general stratified space setting considered by Mattingly et al. (2023, arxiv).
3:45–4:00 pm
break
4:00–5:00 pm
Speaker: Wilderich Tuschmann, Karlsruhe Institute of Technology
Title: A Spectator’s Perspective on the Manifold Hypothesis
Abstract: At its core, the Manifold Hypothesis asserts that real-world, high-dimensional data is not uniformly or randomly distributed throughout its high-dimensional “ambient” space, but concentrated on or near a low-dimensional manifold (or a collection of manifolds) embedded within that high-dimensional ambient space.
In my talk, I will discuss reasons and facts that speak for as well as against this hypothesis and also address geometric alternatives.

Wednesday, Nov. 19, 2025
9:00–9:30 am
Morning refreshments
9:30–10:30 am
Speaker: Melanie Weber, Harvard University
Title: Ricci Curvature, Ricci Flow, and the Geometry of Learning
Abstract: Geometric structure in data plays a crucial role in machine learning. In this talk, we study this observation through the lens of Ricci curvature and its associated Ricci flow. We start by reviewing a discrete notion of Ricci curvature introduced by Ollivier and the geometric flow that it induces. We further discuss the relationship between discrete Ricci curvature and its continuous counterpart via discrete-to-continuum consistency results, which imply that discrete Ricci curvature can provably characterize the geometry of a data manifold based on a finite sample. This provides a theoretical foundation for several applications of discrete Ricci curvature in machine learning, two of which we discuss in the remainder of this talk. First, we analyze learned feature representations in deep neural networks and show that they transform during training in ways that closely resemble a discrete Ricci flow. Our analysis reveals that nonlinear activations shape class separability and suggests geometry-informed training principles such as early stopping and depth selection. Second, we turn to deep learning on graphs, where we address representational limitations of state of the art graph neural networks through curvature-based data augmentations. We show that augmenting input graphs with geometric information provably increases the representational power of such models and yields performance gains in practice.
10:30–10:45 am
break
10:45 am–11:45 am
Speaker: Ezra Miller, Duke University
Title: Extracting bar lengths from multiparameter persistent homology
Abstract: Persistent homology in one parameter can be summarized using bar codes or persistence diagrams, which are elementary gadgets with many features amenable to vectorization and hence statistical analysis. For example, early work with Bendich, Marron, Pieloch, and Skwerer showed how to extract meaningful statistics from the top 100 bar lengths in persistent homology summaries of brain arteries. The story for persistent homology with multiple parameters, on the other hand, is still developing. Although it has the potential to be much more flexible and informative, multipersistence has structural issues that present fundamental mathematical challenges. There is no consensus on what might be meant by a “bar”, let alone “the top 100 bar lengths”. This talk recalls the basics of single and multiparameter persistent homology and discusses some of the mathematical issues, including obstacles and potential routes forward.
11:45 am–1:15 pm
Lunch Break
1:15–2:15 pm
Speaker: Kei Kobayashi, Keio University
Title: Metric Transformations of Data Spaces: Curvature Control and Related Developments
Abstract: We present our proposed method of increasing the accuracy of data analysis by means of two transformations of the metric of the data space. The first transformation is based on the curve length defined by the integral of the power of the density function, which can be computed approximately using an empirical graph; the second transformation can be interpreted as the extrinsic distance when the data space is embedded in a metric cone. The advantage of both distance transformations is that the hyperparameters allow the curvature to be monotonically transformed in a specific sense. Some statistical applications of these transformations and theoretical justifications are presented. Detailed analyses of the geodesics obtained by this method for several simple probability distributions will also be presented. The main part of this work is based on joint works with Henry P. Wynn.
2:15–2:45 pm
break with refreshments
2:45–3:45 pm
Speaker: Sungkyu Jung, Seoul National University
Title: Generalized Frechet means with random minimizing domains and its strong consistency
Abstract: In this talk, I will discuss a novel extension of Frechet means, referred to as generalized Frechet means, as a comprehensive framework for describing the characteristics of random elements. The generalized Frechet mean is defined as the minimizer of a cost function, and the framework encompasses various extensions of Frechet means that have appeared in the literature. The most distinctive feature of the proposed framework is that it allows the domain of minimization for the empirical generalized Frechet means to be random and different from that of its population counterpart. This flexibility broadens the applicability of the Frechet mean framework to various statistical scenarios, including sequential dimension reduction for non-Euclidean data. We establish a strong consistency theorem for generalized Frechet means. Applications such as verifying the consistency of principal geodesic analysis on the hypersphere, compositional principal component analysis on the composition space, and k-medoids clustering for data on a metric space will be discussed.
3:45–4:00 pm
break
4:00–5:00 pm
Speaker: Rong Ma, Harvard University
Title: Modern Nonlinear Embedding Methods Unpacked
Abstract: Learning and representing low-dimensional structures from noisy, high-dimensional data is a cornerstone of modern data science. Stochastic neighbor embedding algorithms, a family of nonlinear dimensionality reduction and data visualization methods, with t-SNE and UMAP as two leading examples, have become very popular in recent years. Yet despite their wide applications, these methods remain subject to points of debate, including limited theoretical understanding, ambiguous interpretations, and sensitivity to tuning parameters. In this talk, I will present our recent efforts to decipher and improve these nonlinear embedding approaches. Our key results include a rigorous theoretical framework that uncovers the intrinsic mechanisms, large-sample limits, and fundamental principles underlying these algorithms; a set of theory-informed practical guidelines for their principled use in trustworthy biological discovery; and a collection of new algorithms that address current limitations and improve performance in areas such as bias reduction and stability. Throughout the talk, I will highlight how these advances not only deepen our theoretical understanding but also open new avenues for scientific discovery.

Millennium Prize Problems Lecture - Madhu Sudan: P vs NP Problem
2025-12-03 17:00 - 18:00 | Harvard Science Center Hall D, 1 Oxford Street, Cambridge, MA, 02138, United States
Pamphlet (pdf)
Slides (pdf)
Date: December 3, 2025
Time: 5:00–6:00 pm
Location: Harvard Science Center Hall D, 1 Oxford St., Cambridge MA
Speaker: Madhu Sudan, Harvard University
Title: The P vs. NP problem: An Existential Question for Mathematics
At the beginning of the twentieth century, in response to questions raised by Hilbert, illustrious mathematicians such as Godel, Church and Turing formalized the notion of theorems and proofs. Proofs were automatically verifiable while theorems are logical propositions for which proofs exist. The formal definition of a computer, a definition that had strong influence on the later development of the technology, was a by-product of the effort to define the phrase “automatically verifiable”!
While the resulting theory had major implications already, one notion was however missing in the early definitions. Proofs were meant to be easily verifiable, while determining the truth of a proposition/conjecture (arguably a core task of mathematics) was not necessarily so. But what is “easiness” and how is it to be defined? While this was already hinted at by Godel in the 50s, the notion was finally formalized in seminal works of Cook, Levin and Karp in the early 70s. Central notions here included the adoption of the notion that polynomial time algorithms are (the only) tractable ones, and the realization that algorithms seeking to remove the existential quantifier in the definition of a “theorem” lead naively to exponential time algorithms. But are there no sophisticated algorithms to search for proofs? This is the profound “Is P = NP?” question.
In this talk we will introduce the question and explain implications of resolutions of this question to the modern computing infrastructure, to mathematics and other sciences. We will briefly describe the state of progress on this question and recent progress on weaker forms of this question. Finally we will also aim to connect this question, and why one may believe that P != NP (proof search can not be automated) even in the face of accumulating evidence on the ability of computers to solve more and more complex mathematical problems, which seem to implement brute force search in less than polynomial time.

Read more about the P vs NP Problem at the Clay Math website.

Organizers: Martin Bridson, Clay Mathematics Institute | Dan Freed, Harvard University and CMSA | Mike Hopkins, Harvard University

Millennium Prize Problems Lecture Series

Millennium Prize Problems Lecture - Barry Mazur: About the Birch and Swinnerton–Dyer Conjecture
2026-02-04 17:00 - 18:00 | Harvard Science Center Hall D, 1 Oxford Street, Cambridge, MA, 02138, United States
Date: February 4, 2026
Time: 5:00–6:00 pm
Location: Harvard Science Center Hall C, 1 Oxford St., Cambridge MA
Speaker: Barry Mazur, Harvard University
Title: About the Birch and Swinnerton–Dyer Conjecture
Abstract:
In the 1950s Bryan Birch and Peter Swinnerton–Dyer made computations that suggested a striking connection between a basic global invariant of an elliptic curve E over the field of rational numbers (namely, the rank of its group of rational points) and certain asymptotics of its local arithmetic invariants (i.e., the number of its rational points over finite fields).
This initial observation has evolved into their conjecture. My lecture will be an introduction to the general ideas behind its ever-expanding development.
Read more about the Birch and Swinnerton–Dyer Conjecture at the Clay Math website.

Organizers: Martin Bridson, Clay Mathematics Institute | Dan Freed, Harvard University and CMSA | Mike Hopkins, Harvard University
Barry Mazur joined the Harvard University faculty in 1959 as a Junior Fellow in the Society of Fellows and advanced through the ranks to become the Gerhard Gade University Professor of Mathematics, a position he has held since 1998. During his tenure at Harvard, he has mentored 60 doctoral students and served as a pivotal figure in bridging topology and number theory, notably through his classification of the possible torsion subgroups of elliptic curves over the rational numbers (Mazur’s torsion theorem), which identifies exactly 15 possible finite groups. This theorem, detailed in his 1977 paper “Modular curves and the Eisenstein ideal,” provided crucial insights into the Taniyama-Shimura conjecture and laid groundwork for Andrew Wiles’s 1994 proof of Fermat’s Last Theorem.
His broader research includes seminal works on étale homotopy theory (co-authored with Michael Artin in 1969), the arithmetic moduli of elliptic curves (with Nicholas M. Katz in 1985), and the Iwasawa main conjecture (proved with Andrew Wiles in 1984), as well as advancements in p-adic L-functions and the formulation of the Fontaine-Mazur conjecture on Galois representations. Mazur’s influence extends to public communication of mathematics; he has authored books like Imagining Numbers (2003), exploring historical perspectives on complex numbers.
Among his numerous honors, Mazur received the Cole Prize in Number Theory from the American Mathematical Society in 1982, the Chauvenet Prize in 1994 for expository writing, the Leroy P. Steele Prize for Lifetime Achievement in 2000, and election to the National Academy of Sciences in 1982. In 2011 (presented in 2013), he was awarded the National Medal of Science by President Barack Obama for his pioneering work in these fields.Most recently, in 2022, he received the Chern Medal from the International Mathematical Union, recognizing his profound discoveries and mentorship.

Millennium Prize Problems Lecture Series
