Formality Theorem and Webs
2024-10-25 12:00 - 13:00 (America/New_York) | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Ahsan Khan
Title: Formality Theorem and Webs
Abstract: The “formality theorem” of Kontsevich was a key result that implies that every Poisson manifold admits a deformation quantization. I will review the ideas behind the formality theorem and discuss a potentially novel viewpoint on it involving webs and twisted masses.

Freedman CMSA Seminar
2024-10-25 14:30 - 17:30 | Virtual
Freedman CMSA Seminar
*Note: via Zoom only*
2:00-3:30 pm ET
Speaker: Matt Hastings, Microsoft Quantum Program
Title: Invertible Phases of Matter and Quantum Cellular Automata: Dimensions One to Three
Abstract: A Quantum Cellular Automaton (QCA) is a *-automorphism of the algebra of local operators. While local quantum circuits provide one example of QCA, we are most interested in nontrivial QCA which are those which cannot be written as conjugation by a local quantum circuit. For systems in one and two spatial dimensions, all nontrivial QCA are shifts (i.e., translations by some amount), up to conjugation by a quantum circuit, but in three and higher dimensions, other examples are known. I’ll explain the relation between QCA and a certain “boundary algebra” of operators in one lower spatial dimension, and also the relation to invertible phases of matter on the boundary, and use this to explain and motivate some of these results in dimensions one through three.

3:30-4:00 pm ET
Break/Discussion

4:00-5:30 pm ET
Speaker: Lukasz Fidkowski, U Washington, Physics
Title: Invertible Phases of Matter and Quantum Cellular Automata: Higher dimensions
Abstract: We discuss the explicit construction of a non-trivial QCA in 3 dimensions, one which takes the form of multiplication by a discrete Chern-Simons functional in an appropriate basis for the Hilbert space. We relate the non-trivialness of the QCA to the fact that the Chern-Simons action is not the integral of a gauge invariant local quantity. One property of this QCA is that it creates a specific non-trivial time reversal symmetry protected topological (SPT) phase when acting on a non-trivial tensor product state. Motivated by this, we construct a general class of QCA in arbitrary dimensions based on time reversal protected SPTs, and conjecture a general correspondence between unoriented cobordism (which classifies such SPTs) and QCA.

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/

Foundation Seminar: Singularity Theorems, Part II
2024-10-28 09:30 - 10:30 | 20 Garden Street, Cambridge, MA 02138, MA, MA, 02138, United States
Foundation Seminar (Joint Seminar with BHI)
Location: BHI
Title: Singularity Theorems, Part II
Journal Club Discussion

Formation of Trapped Surfaces in Geodesic Foliation
2024-10-29 11:00 - 12:00 | Virtual
General Relativity Seminar
Speaker: Xuantao Chen, Johns Hopkins University
Title: Formation of Trapped Surfaces in Geodesic Foliation
Abstract: We reprove the formation of trapped surfaces for the Einstein vacuum equation using the incoming geodesic foliation. All previous results, starting with the seminal work of Christodoulou, make use of the double null foliation. In the new proof, all Ricci coefficients are estimated by transport equations in the incoming null direction, and the incoming initial data is only used in the curvature (energy) estimates. The result is based on a version of the non-integrable PT frame, which was introduced in the proof of Kerr stability with small angular momentum.

Boundaries and duality for 3d gauge theories
2024-10-29 16:15 - 18:15 | Science Center Hall E, 1 Oxford Street, Cambridge, MA, 02138, United States
Geometry and Quantum Theory Seminar
Speaker: Ben Gammage, Harvard University
Title: Boundaries and duality for 3d gauge theories
Abstract: 3d N=4 supersymmetric gauge theory has a pair of topological twists, the A-model and B-model, the latter of which is also known as Rozansky-Witten theory. Conjecturally, boundary conditions for these TFTs ought to admit descriptions in terms of (microlocal) perverse or coherent sheaves of categories, respectively. Unfortunately, neither of these admits a general mathematical definition; nevertheless, in some cases these are well-defined 2-categories. We will survey these situations and the duality, known as 3d mirror symmetry, which relates the A- and B-models of different theories, together with its relation to the relative Langlands duality of Ben-Zvi–Sakellaridis-Venkatesh.

CMSA Q&A Seminar: Michael Freedman
2024-10-30 12:00 - 13:00 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA Q&A Seminar
Speaker: Michael Freedman, Harvard CMSA
Topic: Broad perspective on manifolds: all dimensions, all structures, classification and dynamics

Mirror Construction for Nakajima Quiver Varieties
2024-10-31 10:00 - 11:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Mathematical Physics and Algebraic Geometry Seminar
Speaker: Ju Tan (Boston University)
Title: Mirror Construction for Nakajima Quiver Varieties
Abstract: Quiver possesses a rich representation theory. On the one hand, it exhibits a deep connection with instantons and coherent sheaves as illuminated by the ADHM construction and the works of many others. On the other hand, quivers also capture the formal deformation space of a Lagrangian submanifold. In this talk, we will discuss these relations more explicitly from the perspective of SYZ mirror symmetry. In particular, we will introduce the notion of framed Lagrangian immersions, the Maurer-Cartan deformation spaces of which are Nakajima quiver varieties/ stacks. Besides, we will realize the ADHM construction as a mirror symmetry phenomenon. This is based on the joint work with Jiawei Hu and Siu-Cheong Lau.

Foundation Seminar
2024-11-04 09:30 - 10:30 | Black Hole Initiative, 20 Garden Street, Cambridge MA, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Foundation Seminar (Joint Seminar with BHI)
Location: BHI

Speaker: Christoph Kehle (MIT)

Title: On the cosmic censorship conjectures
Abstract: I will present the modern formulations of the weak and strong cosmic censorship conjectures and discuss some recent developments in the context of gravitational collapse.

The mathematics of evolution
2024-11-04 16:30 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Colloquium
Speaker: Martin Nowak (Harvard)
Title: The mathematics of evolution
Abstract: All living systems are guided by evolutionary dynamics. Evolution is a search process which occurs in populations of reproducing individuals. The three fundamental forces of evolution are mutation, selection and cooperation. I will present basic ideas in the mathematical description of evolutionary dynamics, including quasi-species theory, evolutionary game theory, and evolutionary graph theory. I will discuss specific problems such as origin of life, emergence of complexity, mechanisms of cooperation, evolution of cancer and how to overcome resistance to targeted therapy.

Introduction to Factorization algebras
2024-11-05 16:15 - 18:15 | Science Center Hall E, 1 Oxford Street, Cambridge, MA, 02138, United States
Geometry and Quantum Theory Seminar
Speaker: Dan Freed, Harvard University
Title: Introduction to Factorization algebras

CMSA Q&A Seminar: Michael Douglas
2024-11-06 12:00 - 13:00 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA Q&A Seminar
Speaker: Michael Douglas, Harvard CMSA
Topic: What is Argyres-Douglas theory?

Is Behavior Cloning All You Need? Understanding Horizon in Imitation Learning
2024-11-06 14:00 - 15:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
New Technologies in Mathematics Seminar
Speaker: Dylan Foster, Microsoft Research
Title: Is Behavior Cloning All You Need? Understanding Horizon in Imitation Learning
Abstract: Imitation learning (IL) aims to mimic the behavior of an expert in a sequential decision making task by learning from demonstrations, and has been widely applied to robotics, autonomous driving, and autoregressive language generation. The simplest approach to IL, behavior cloning (BC), is thought to incur sample complexity with unfavorable quadratic dependence on the problem horizon, motivating a variety of different online algorithms that attain improved linear horizon dependence under stronger assumptions on the data and the learner’s access to the expert.In this talk, we revisit the apparent gap between offline and online IL from a learning-theoretic perspective, with a focus on general policy classes up to and including deep neural networks. Through a new analysis of behavior cloning with the logarithmic loss, we will show that it is possible to achieve horizon-independent sample complexity in offline IL whenever (i) the range of the cumulative payoffs is controlled, and (ii) an appropriate notion of supervised learning complexity for the policy class is controlled. When specialized to stationary policies, this implies that the gap between offline and online IL is smaller than previously thought. We will then discuss implications of this result and investigate the extent to which it bears out empirically.
Bio: Dylan Foster is a principal researcher at Microsoft Research, New York. Previously, he was a postdoctoral fellow at MIT, and received his PhD in computer science from Cornell University, advised by Karthik Sridharan. His research focuses on problems at the intersection of machine learning, AI, interactive decision making. He has received several awards for his work, including the best paper award at COLT (2019) and best student paper award at COLT (2018, 2019).

Bounds and Dualities of Type II Little String Theories
2024-11-07 10:00 - 11:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Mathematical Physics and Algebraic Geometry Seminar
Speaker: Fabian Ruehle (Northeastern University)
Title: Bounds and Dualities of Type II Little String Theories
Abstract: The goal of this seminar is to introduce Type II Little String Theories (LSTs), which are six-dimensional supersymmetric QFTs. We explore how to geometrically engineer these theories within the context of M-/F-theory (top-down) as well as consistent QFT realizations (bottom-up). After that, we turn to the worldsheet theory of LSTs, which are two-dimensional N=(0,4) SCFTs. Using anomaly inflow and unitarity, we derive strong constraints on the rank of their global symmetry algebras.

Representations of minimal W-algebras: unitarity and modular invariance
2024-11-08 10:00 - 11:30 | Virtual
Quantum Field Theory and Physical Mathematics Seminar
Speaker: Victor Kac (MIT)
Title: Representations of minimal W-algebras: unitarity and modular invariance
Abstract: The minimal W-algebras, obtained by quantum Hamiltonian reduction from affina vertex algebras, form the most interesting class of vertex algebras, which includes all superconformal algebras: Virasoro, Neveu-Scharz, N=2, 3, 4, and big N=4. I will explain a unified classification of their unitary representations, and their character formulas. For N=0, 1, and 2 these vertex algebras are modular invariant (meaning that tr q^L_0-c/24 is a modular function). However for all other minimal W-algebra modular invariance fails, and one needs the “modification” of characters to restore modular invariance. Unfortunately the representation-theoretical or physical meaning of the modification is not known (at least to me).

ADHM spaces and their quantizations
2024-11-08 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Vasily Krylov, CMSA
Title: ADHM spaces and their quantizations
Abstract: In their paper “Construction of Instantons,” Atiyah, Drinfeld, Hitchin, and Manin introduced an algebraic construction of the moduli space of instantons on R^4, now also known as the “ADHM space.” This is a Poisson complex variety; it has been actively studied by both mathematicians and physicists. In this talk, I will review the ADHM construction, present examples, and discuss various geometric and algebraic properties of ADHM spaces. I will also describe natural quantizations of these Poisson varieties. I will explain a joint result with Etingof, Losev, and Simental, providing explicit formulas for the dimensions and characters of all finite-dimensional representations of these quantizations. Time permitting, I will illustrate some predictions of the 3D mirror symmetry in the example of ADHM spaces, following our joint paper with Shlykov.

pp Waves: Quasinormal Modes & Hidden Symmetries of Black Holes
2024-11-12 11:00 - 12:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
General Relativity Seminar
Speaker: Ahmed Seta, Harvard University
Title: pp Waves: Quasinormal Modes & Hidden Symmetries of Black Holes
Abstract: The spectrum of quasinormal modes of 4D flat space black holes is not analytically tractable, but there are two asymptotic limits where the QNM spectrum is under control: weak damping and strong damping. In this talk, I will explain how these asymptotic QNMs are controlled by dynamical symmetries of the wave equation in certain kinematic limits. These two asymptotic limits are, in turn, captured by the two classes of bound null geodesics in the black hole geometry: the photon ring and the horizon. I will also discuss the Penrose limit: a scaling limit into the geometry experienced by these geodesics, which results in a plane-wave spacetime where the dynamical symmetries get enhanced into isometries.

Frontier of Formal Theorem Proving with Large Language Models: Insights from the DeepSeek-Prover Series
2024-11-13 10:00 - 23:00 | Virtual
New Technologies in Mathematics Seminar
Speaker: Huajian Xin, DeepSeek
Title: Frontier of Formal Theorem Proving with Large Language Models: Insights from the DeepSeek-Prover Series
Abstract: Recent advances in large language models have markedly influenced mathematical reasoning and automated theorem proving within artificial intelligence. Yet, despite their success in natural language tasks, these models face notable obstacles in formal theorem proving environments such as Lean and Isabelle, where exacting derivations must adhere to strict formal specifications. Even state-of-the-art models encounter difficulty generating accurate and complex formal proofs, revealing the unique blend of mathematical rigor required in this domain. In the DeepSeek-Prover series (V1 and V1.5), we have explored specialized methodologies aimed at addressing these challenges. This talk will delve into three foundational areas: the synthesis of training data through autoformalization, reinforcement learning that utilizes feedback from proof assistants, and test-time optimization using Monte Carlo tree search. I will also provide insights into current model capabilities, persistent challenges, and the future potential of large language models in automated theorem proving.

(Un)likely intersections
2024-11-14 10:00 - 11:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Mathematical Physics and Algebraic Geometry Seminar
Speaker: Tom Scanlon, UC Berkeley
Title: (Un)likely intersections

Abstract: The Zilber-Pink conjectures predicts that for an ambient special variety (such as an abelian variety or a Shimura variety), if is an irreducible algebraic subvariety which is not contained a proper special subvariety of (e.g. a proper algebraic subgroup in the abelian variety case or a variety of Hodge type in the case of Shimura varieties), then the union of the unlikely intersections as ranges over the special subvarieties of with is not Zariski dense in . While various instances of this conjecture have been proven, it remains open in most cases of interest. In this lecture, I will describe some of my work with Jonathan Pila in which we prove an effective function field version of this conjecture along with a counterpart to the Zilber-Pink conjecture proven with Sebastian Eterović: after accounting for some geometric obstructions, the likely intersections, i.e. the union of the intersections with special and , are dense in the Euclidean topology in . Our techniques for both results come from o-minimal complex analysis and differential algebra.

Quantum Criticality in Black Hole Scattering
2024-11-15 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Uri Kol
Title: Quantum Criticality in Black Hole Scattering
Abstract: Perturbation theory around rotating black holes captures a few important effects in the physics of gravitational waves emitted from binary mergers. Despite a long and rich history, developing a qualitative understanding of the system remains a challenging problem. In this talk I will describe an emergent critical phenomena arising in black hole perturbation theory, which is reminiscent of the structure found in quantum many-body systems. A critical point is identified at zero temperature, giving rise to a wide “quantum” critical region at finite temperatures that is dominated by critical fluctuations. In the critical region, the physics is exclusively described by a set of critical exponents, therefore leading to robust predictions.

Freedman CMSA Seminar
2024-11-15 14:30 - 17:30 | Virtual
Freedman CMSA Seminar
*Note: via Zoom only*

2:00-3:30 pm ET
Speaker: Michael Freedman, Harvard CMSA
Title: Some questions and theorems about closed 3 manifolds embedded in S^4
Abstract: Much is unknown about smooth embeddings of 3-manifolds in S^4; the Schoenflies problem (Is there only one smoothly embedded 3-sphere in S^4 up to isotopy?) is the best-known example. There has long been a hope that 3-manifold reasoning applied to level-sets will be helpful. I’ll mention some successes and failures of this method and revisit a classical theorem of Hantzsche in this light. (Hantzsche: If a 3-manifold embeds in S^4 its linking form is hyperbolic.)

3:30-4:00 pm ET
Break/Discussion

4:00-5:30 pm ET
Speaker: Slava Krushkal, University of Virginia
Title: A higher order torsion linking form for 3-manifolds
Abstract: This talk is based on a joint work with Mike Freedman defining a triple linking form for rational homology spheres, assuming that the classical torsion linking pairing of three classes pairwise vanishes. I will discuss its vanishing for 3-manifolds in S^4, and its relation to the Matsumoto triple intersection form on 4-manifolds.

Emergent Non-Invertible Symmetries —The Adjoint QCD Example
2024-11-18 14:00 - 15:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Quantum Field Theory and Physical Mathematics Seminar
Speaker: Shani Nadir Meynet (Uppsala)
Title: Emergent Non-Invertible Symmetries — The Adjoint QCD Example
Abstract: After reviewing some general properties of generalized symmetries and the renormalization group (RG) flow for quantum field theories (QFT), I’ll describe how the recently discovered non-invertible symmetries can be used to study theories at strong coupling. I’ll illustrate these facts using (3+1)-dimensional adjoint QCD with two flavors as an example. This theory can be obtained by mass deforming a pure N=2 super Yang-Mills theory. Relying on supersymmetric results, dynamical abelianization and monopole condensation, we are able to get to the description of an infrared (IR) phase as an abelian theory flowing to a CP1 sigma model. In this scenario, the IR phase has an emergent non-invertible symmetry, which is matched with the non-invertible symmetry of the IR CP1 phase. This result illustrates how an emergent non-invertible symmetry can be used to provide a bridge connecting gauge theories at strong coupling and their IR via dynamical abelianization.

The Einstein-Euler system with a physical vacuum boundary in spherical symmetry
2024-11-19 11:00 - 12:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
General Relativity Seminar
Speaker: Marcelo Disconzi, Vanderbilt University
Title: The Einstein-Euler system with a physical vacuum boundary in spherical symmetry
Abstract: We establish local well-posedness for the Einstein-Euler system with a physical vacuum boundary in spherical symmetry. Our proof relies on a new way of thinking about Einstein’s equations in spherical symmetry that is well-adapted to the fluid’s characteristics on the free boundary. We also exploit the Einstein constraint equations in spherical symmetry in a new way, as a tool to understand the evolution problem. This is joint work with Jared Speck.

Factorization algebras in TQFT
2024-11-19 18:00 - 19:00 | Science Center 507, 1 Oxford Street, Cambridge, 02138
Geometry and Quantum Theory Seminar
Speakers: Mayuko Yamashita, Kyoto University
Title: Factorization algebras in TQFT
Abstract: This is the first in the series of our working seminars on factorization algebras/homologies. This talk focuses on locally constant factorization algebras, which correspond to Topological QFTs. I first explain they are equivalent to algebras over E_n operads and their variants. Then I define the factorization homology and discuss basic properties and examples. If time allows, I also mention the connection with the cobordism hypothesis.

Thinking Like Transformers - A Practical Session
2024-11-20 10:00 - 23:00 | Virtual
New Technologies in Mathematics Seminar
Speaker: Gail Weiss, EPFL
Title: Thinking Like Transformers – A Practical Session
Abstract: With the help of the RASP programming language, we can better imagine how transformers—the powerful attention based sequence processing architecture—solve certain tasks. Some tasks, such as simply repeating or reversing an input sequence, have reasonably straightforward solutions, but many others are more difficult. To unlock a fuller intuition of what can and cannot be achieved with transformers, we must understand not just the RASP operations but also how to use them effectively.
In this session, I would like to discuss some useful tricks with you in more detail. How is the powerful selector_width operation yielded from the true RASP operations? How can a fixed-depth RASP program perform arbitrary length long-addition, despite the equally large number of potential carry operations such a computation entails? How might a transformer perform in-context reasoning? And are any of these solutions reasonable, i.e., realisable in practice? I will begin with a brief introduction of the base RASP operations to ground our discussion, and then walk us through several interesting task solutions. Following this, and armed with this deeper intuition of how transformers solve several tasks, we will conclude with a discussion of what this implies for how knowledge and computations must spread out in transformer layers and embeddings in practice.

CMSA Q&A Seminar: Anurag Anshu
2024-11-20 12:00 - 13:00 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA Q&A Seminar
Speaker: Anurag Anshu, Harvard University
Topic: What is quantum complexity theory?

A new construction of c = 1 conformal blocks
2024-11-20 15:00 - 16:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Mathematical Physics and Algebraic Geometry Seminar
Speaker: Qianyu Hao, University of Geneva
Title: A new construction of c = 1 conformal blocks

Abstract: The Virasoro conformal blocks are very interesting since they have many connections to other areas of math and physics. For example, when c = 1, they are related to tau functions of Painlevé equations. I will first explain what Virasoro conformal blocks are. Then I will describe a new way to construct Virasoro blocks at c = 1 on C by using the “abelian” Heisenberg conformal blocks on a branched double cover of C. The main new idea in our work is to use a spectral network. It is closely related to the idea of nonabelianization of the flat connections in the work of Gaiotto-Moore-Neitzke and Neitzke-Hollands. This nonabelianization construction enables us to compute the harder-to-get Virasoro blocks using the simpler abelian objects. This is based on a joint work with Andrew Neitzke.

Perturbative Factorization Algebras
2024-11-20 16:30 - 17:30 | Science Center Hall E, 1 Oxford Street, Cambridge, MA, 02138, United States
Geometry and Quantum Theory Seminar
Speaker: Ahsan Khan

Title: Perturbative Factorization Algebras

Abstract: In physics the starting point in studying a QFT is to write down an appropriate action functional. My talk will aim to sketch how this connects with the framework of factorization algebras.

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.

Skein valued curve counts for the topological vertex and knot conormals
2024-11-21 10:30 - 11:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Mathematical Physics and Algebraic Geometry Seminar
Speaker: Tobias Ekholm, Uppsala University
Title: Skein valued curve counts for the topological vertex and knot conormals
Abstract: Combining the invariance of holomorphic curve counts in the skein module with a study of holomorphic curves at infinity of the vertex we find three simple skein operator polynomials that annihilates the (skein valued) topological vertex. We show that these operator polynomials together with natural initial conditions determine the partition function uniquely and then demonstrate that the original Aganagic-Klemm-Marino-Vafa formula for the topological vertex interpreted as a skein valued curve count satisfies the operator polynomials. This is joint work with Longhi and Shende. We end with a general discussion of similar ‘skein D-modules’ for knot conormals.
