Geometry and Quantum Theory Seminar
2025-05-06 16:15 - 18:15 (America/New_York) | Science Center 507, 1 Oxford Street, Cambridge, 02138

Residues and homotopy Lie algebras
2025-05-08 10:00 - 11:00 | Virtual
Mathematical Physics and Algebraic Geometry Seminar
Speaker: Zhenping Gui, Shanghai Institute for Mathematics and Interdisciplinary Sciences
Title: Residues and homotopy Lie algebras
Abstract: I will introduce the notion of a chiral operad for any compact Riemann surface. This operad consists of compositions of residue operations, which give rise to the Chevalley-Cousin complex and lead to the definition of chiral homology (derived conformal blocks). I will explain how to use this machinery to rigorously define certain Feynman integrals in 2D chiral CFTs. Subsequently, I will present a polysimplicial construction of a series of chain models for the configuration space of points in an affine space and study residue operations. These residue operations can be described by a homotopy Lie algebra structure, and the latter defines a higher-dimensional analog of the Chevalley-Cousin complex. This is based on joint work in progress with Charles Young and Laura Felder.

Asset pricing with heterogeneous agents
2025-05-09 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Sergiy Verstyuk, Harvard CMSA
Title: Asset pricing with heterogeneous agents
Abstract: This talk will introduce the basics of continuous-time finance, discuss important existing theories and models, as well as present some new asset pricing results in a setting with many heterogeneous investors. (Joint work with Puskar Mondal.)

An index for 2d invertible phases of quantum many-body systems
2025-05-12 15:00 - 16:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Quantum Field Theory and Physical Mathematics Seminar
Speaker: Nikita Sopenko, IAS
Title: An index for 2d invertible phases of quantum many-body systems

Factorizations for data analysis
2025-05-12 16:30 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Colloquium
Speaker: Anna Seigal, Harvard University
Title: Factorizations for data analysis
Abstract: We can find structure in data by factoring it into building blocks, which should be interpretable for the context at hand. A classical example is principal component analysis (PCA), which uses the eigendecomposition of the covariance matrix to find axes of variation in a dataset. Starting from PCA, I will discuss matrix and tensor factorizations for data analysis, and the linear and multilinear algebra that underpins their theoretical properties. We will see examples from causal inference, independent component analysis, and dimensionality reduction.

Report on the Perimeter Institute Theory+AI Workshop
2025-05-14 12:00 - 13:00 | CMSA 20 Garden Street Cambridge, Massachusetts 02138 United States
Conference Reports
Speaker: Hugo Cui, Harvard CMSA
Title: Report on the Perimeter Institute Theory+AI Workshop
Abstract: I will give a survey and brief summary of some of the talks given at the Theory+AI Workshop: Theoretical Physics for AI event organized by Perimeter Institute in April, on approaches to machine learning theory inspired from physics.
Link : https://events.perimeterinstitute.ca/event/993/

Resurgence, number theory, and quantum mirror curves
2025-05-15 10:00 - 11:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Mathematical Physics and Algebraic Geometry Seminar
Speaker: Claudia Rella (IHES)
Title: Resurgence, number theory, and quantum mirror curves
Abstract: Resurgence provides a powerful toolbox to access the non-perturbative sectors hidden within the divergent asymptotic series of quantum theories. Under some special assumptions, the non-perturbative data extracted via resurgent methods possess intrinsic number-theoretic properties that are deeply rooted in the symmetries and arithmetic of the geometry underlying the quantum theory. The framework of modular resurgence aims to formalise this observation. In this talk, after introducing the basics of modular resurgence, I will consider the TS/ST correspondence for toric Calabi-Yau threefolds and focus on the fermionic spectral traces of quantum mirror curves. Here, a complete realisation of the modular resurgence paradigm is found in the spectral theory of local P^2—where the bridge between non-perturbative physics and the arithmetic properties of the geometry takes the form of an exact strong-weak symmetry—and is now being generalised to all local weighted projective spaces. This talk is based on arXiv:2212.10606, 2404.10695, 2404.11550, and work in progress.

Echo Chamber: RL Post-training Amplifies Behaviors Learned in Pretraining
2025-05-16 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Samy Jelassi, CMSA
Title: Echo Chamber: RL Post-training Amplifies Behaviors Learned in Pretraining
Abstract: Reinforcement Learning has become a crucial step in training state-of-the-art language models such as DeepSeek-R1 for solving mathematical problems. In this talk, I will first review the mechanisms of Reinforcement Learning fine-tuning. Then, I will present a systematic end-to-end study of RL fine-tuning for mathematical reasoning, training models entirely from scratch on different mixtures of fully open datasets and fine-tuning them with RL. Doing so allows us to investigate the effects of the pretraining data mixture on the behavior of RL, and its interaction with the model size and choices of the algorithm hyperparameters. Our study reveals that RL algorithms consistently converge towards a dominant output distribution, amplifying patterns in the pretraining data. We also find that models of different scales trained on the same data mixture will converge to distinct output distributions, suggesting that there are scale-dependent biases in model generalization.
The second part of the talk is based on a joint work with Rosie Zhao, Alex Meterez, Cengiz Pehlevan, Sham Kakade and Eran Malach: https://arxiv.org/abs/2504.07912

Higher Gauge Theory and Integrability
2025-05-22 10:00 - 11:00 | Virtual
Mathematical Physics and Algebraic Geometry Seminar
Speaker: Joaquin Liniado, Instituto de Física La Plata
Title: Higher Gauge Theory and Integrability
Abstract: Integrable field theories are remarkable for possessing an infinite number of conserved quantities, which often allow for their exact solvability. In two dimensions, this structure is elegantly captured by the existence of a Lax connection, whose path ordered exponential allows for the systematic construction of an infinite number of conserved quantities. In 2019, Costello, Witten and Yamazaki introduced a four-dimensional holomorphic extension of Chern-Simons theory that provides the first attempt at explaining the appearance of the Lax connection, whose origin had remained somewhat mysterious until then.
In this talk, we present a generalization of these ideas to three-dimensional field theories, guided by the so-called “categorical ladder = dimensional ladder” principle. The central idea is that conserved quantities arise from surface-ordered exponentials of higher-rank tensors, defining a higher categorical notion of the Lax connection. We show that such a structure naturally emerges from a five-dimensional holomorphic extension of higher Chern-Simons theory. This work, carried out in collaboration with Hank Chen, provides a framework that enables the systematic construction of integrable field theories in three dimensions.

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)

Joint BHI/CMSA Foundation Seminar: The semiclassical energy outflux emerging from a collapsing shell
2025-07-31 11:00 - 12:00 | Black Hole Initiative, 20 Garden Street, Cambridge MA, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Joint BHI/CMSA Foundation Seminar
Location: BHI seminar room
Speaker: Noa Zilberman (Princeton University)
Title: The semiclassical energy outflux emerging from a collapsing shell
Abstract: When a compact object collapses to form a black hole, quantum field theory predicts the emission of an energy outflux to future null infinity, which later relaxes to Hawking radiation. Within the semiclassical framework, we derive a simple, closed form, analytical expression for the energy outflux emitted from a spherical thin null shell collapsing to form a black hole. In particular, this energy outflux vanishes (quadratically in r-2M) as the shell approaches the horizon. This result refutes claims that the Hawking energy outflux originates from the collapsing body, showing instead that it develops in a broad strong-field region. Additionally, this vanishing implies that semiclassical backreaction cannot prevent or significantly affect the classical process of gravitational collapse and horizon formation (as sometimes claimed).

Fukaya category and gauge theory
2025-09-02 16:15 - 18:30 | Science Center 507, 1 Oxford Street, Cambridge, 02138
Geometry and Quantum Theory Seminar
Speaker: Saman Habibi Esfahani, Harvard CMSA
Title: Fukaya category and gauge theory
Abstract: After setting up some background, I will discuss the Fukaya $A_ infty$-category and several instances where it appears in gauge theory, such as in the study of flat connections on Riemann surfaces, holomorphic sections of some hyperkähler bundles, and instantons and holomorphic curves in K3 surfaces. If time permits, I will also outline potential applications of these ideas to the study of 3-manifolds and manifolds with special holonomy.

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

Higher categories of cobordisms
2025-09-09 16:15 - 18:30 | Science Center 507, 1 Oxford Street, Cambridge, 02138
Geometry and Quantum Theory Seminar
Speaker: Lorenzo Riva
Title: Higher categories of cobordisms
Abstract: I will give a brief introduction to topological field theories from a higher categorical perspective. After saying a few things about higher categories, I will define a family of n-categories of bordisms and talk about their universal properties. I will try to squeeze in the canonical example — representations of the 2-dimensional oriented bordism 2-category are separable symmetric Frobenius algebras — and, time permitting, talk about my current work.

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.

Orientifolds for F-theory on K3 Surfaces
2025-09-15 15:00 - 16:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Quantum Field Theory and Physical Mathematics Seminar
Speaker: Chuck Doran (Alberta/CMSA)
Title: Orientifolds for F-theory on K3 Surfaces
Abstract: Compactification of F-theory on an elliptically fibered K3 surface provides a framework to encode type IIB string theory on elliptic curves, with the Kaehler modulus of the elliptic curve encoded in the complex structure of the elliptic fibers. In work with Malmendier, Mendez-Diez, and Rosenberg we extend that perspective by examining F-theory orientifolds on elliptically fibered K3 surfaces and connecting them to D-brane classifications using real K-theory (KR-theory). The real structures—antiholomorphic involutions—on our K3 surfaces connect the geometry with the physics, providing a natural setting for understanding the interplay between elliptic fibration structures and D-brane classifications in F-theory. We construct Real normal forms with their associated antiholomorphic involutions and use this to make explicit the 2-torsion Brauer twist that relates our normal forms to the Jacobian (Weierstrass normal form) elliptic fibration, including the realization of a representative for the twisting class as an Azumaya algebra. This all connects back to the physics by considering three families of real K3 surfaces whose string limits give the three diﬀerent type IIB theories on P1 with four type I_0^∗ Kodaira fibers.

Topological Manifolds – The First 100 Years
2025-09-15 16:30 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Colloquium
Speaker: Michael Freedman (Harvard CMSA and Logical Intelligence)
Title: Topological Manifolds – The First 100 Years
Abstract: I’ll review manifold topology in the topological category from its start with work of Rado (1925) and Kneser (1926) to the present. Work of Moise, Mazur, Kirby, Siebenmann, Sullivan, Kruskal, and the speaker will be discussed. In my view there is one pressing open question (the A-B slice problem). I will end with some thoughts on putting an AI to work on it.

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

Moduli spaces of 4d N=2 quantum field theories
2025-09-18 16:00 - 17:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Differential Geometry and Physics Seminar
Speaker: Robert Moscrop, CMSA
Title: Moduli spaces of 4d N=2 quantum field theories
Abstract: Supersymmetry endows quantum field theories with several rich algebraic and geometric structures associated to their moduli space of vacua, providing powerful tools to study such theories non-perturbatively. For example, in four-dimensional theories with eight supercharges, the low energy dynamics of the theory is captured by an algebraic completely integrable system whose base is the Coulomb branch– a particular distinguished submanifold of the moduli space. This structure is so tightly constrained, that there is an ongoing program to classify such theories purely by understanding their Coulomb branch geometry. In this talk, I will give a gentle introduction to the geometry of the moduli spaces of 4d N=2 theories and, time permitting, discuss some recent results showcasing how the geometry of the Coulomb branch can be used to constrain certain physical quantities of the theory.

Top-Down Perspectives on Symmetry Theories
2025-09-19 12:00 - 13:00 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Max Hubner
Title: Top-Down Perspectives on Symmetry Theories
Abstract: I will review the construction and utility of symmetry theories for string constructed quantum field theories. Symmetry theories are extra-dimensional auxiliary theories separating aspects of a quantum field theory’s symmetries from many of its more messy features. For QFTs with extra-dimensional string constructions the symmetry theory derives directly from the extra-dimensional geometry. This perspective allows for the study of symmetries of famously string engineered systems, such as SCFTs in 5D and 6D, which we will discuss on an example by example basis.

Non-Supersymmetric Orbifolds, Quivers and Chen-Ruan Orbifold Cohomology
2025-09-22 15:00 - 16:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Quantum Field Theory and Physical Mathematics Seminar
Speaker: Max Hübner (Uppsala & CMSA)
Title: Non-Supersymmetric Orbifolds, Quivers and Chen-Ruan Orbifold Cohomology
Abstract: We consider D3-brane probes of non-supersymmetric orbifolds and IIA on the same class of non-supersymmetric orbifolds. Both setups are characterized, in part, by quivers (which in the latter case relate for example to D0-brane probes) from which symmetries constraining the scale-dependence and tachyonic instabilities of the two systems, respectively, can be derived. We demonstrate that these considerations can be matched via a geometric analysis of the asymptotic boundary of the relevant orbifolds, in all cases, via considerations centered on Chen-Ruan orbifold cohomology.

Turbulent Mixing and Antagonistic Microorganisms
2025-09-22 16:30 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Colloquium
Speaker: David Nelson, Harvard
Title: Turbulent Mixing and Antagonistic Microorganisms
Abstract: Unlike coffee and cream that homogenize when stirred, growing micro-organisms (e.g., bacteria and baker’s yeast) can actively kill each other and avoid mixing. How do such antagonistic interactions impact the growth and survival of competing strains, while being spatially advected by turbulent flows? By using analytic arguments and numerical simulations of a continuum model, we describe the dynamics of two antagonistic strains that are dispersed by both compressible and incompressible turbulent flows in two spatial dimensions. A key parameter is the ratio of the fluid transport time to that of biological reproduction, which determines the winning organism that ultimately takes over the whole population from an initial heterogeneous state, a process known as fixation. By quantifying the probability and mean time for fixation, we discuss how turbulence raises the threshold for biological nucleation and antagonism suppresses flow-induced mixing by depleting the population at interfaces. We highlight the unusual biological consequences of the interplay of turbulent fluid flows with antagonistic population dynamics, with potential implications for marine microbial ecology and origins of biological chirality.

Degeneration of Calabi-Yau 3-folds and 3-forms
2025-09-25 16:00 - 17:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Differential Geometry and Physics Seminar
Speaker: Teng Fei, Rutgers
Title: Degeneration of Calabi-Yau 3-folds and 3-forms
Abstract: We study the geometries associated to various 3-forms on a symplectic 6-manifold of different orbital types. As an application, we demonstrate how this can be used to find Lagrangian foliations and other geometric structures of interest arising from certain degeneration of Calabi-Yau 3-folds.

Sections of fibrations onto curves in characteristic p>0
2025-09-26 12:00 - 13:00 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Iacopo Brivio
Title: Sections of fibrations onto curves in characteristic p>0
Abstract: This talk is based on joint work in progress with Ben Church. Using symplectic geometry, Pieloch showed that every smooth fibration $fcolon Xto mathbb{P}^1$ of complex projective varieties always admits a section. I will explain how this theorem can be recovered using techniques from Hodge theory and the Minimal Model Program. An advantage of this approach is that it allows for a positive characteristic generalization, by replacing the Hodge theoretic input by a crystalline one. I will also give an example showing that Pieloch’s result can fail in characteristic p>0.

Graph integrals on Kahler manifolds
2025-09-29 15:00 - 16:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Quantum Field Theory and Physical Mathematics Seminar
Speaker: Minghao Wang, Boston University
Title: Graph integrals on Kahler manifolds
Abstract: I will talk about my recent work with Junrong Yan. We proved the convergence of Graph integrals on analytic Kahler manifolds in the sense of Cauchy principal values, which are originally from holomorphic quantum field theories. In particular, this allows us to construct geometric invariants of Calabi-Yau metrics. I will also talk about some potential applications of our results. References: arXiv:2507.09170, arXiv:2401.08113

Geometry and Quantum Theory Seminar
2025-09-30 16:15 - 18:30 | Science Center 507, 1 Oxford Street, Cambridge, 02138
Geometry and Quantum Theory Seminar
Speaker 1: Max Hubner, CMSA
Title: On Topological Structures in String Theory
Abstract: Geometric engineering constructions in string theory often realize QFTs relative to an extra-dimensional geometry. This perspective parallels the symmetry TFT construction where a QFT is presented relative to its extra-dimensional symmetry quiche. Unsurprisingly, as we will discuss, these constructions are related. Topological features of the extra-dimensional geometry map onto the symmetry TFT. We discuss examples and generalization beyond purely geometric constructions in string theory.
Speaker 2: Bowen Yang, CMSA
Title: Bounded L theory
Abstract: Bounded L-groups arise in the intersection of algebraic L-theory and large-scale geometry, providing a framework for quadratic forms and automorphisms subject to uniform control conditions. These groups play a role in topology and surgery theory, especially in contexts where one needs to measure obstructions not just algebraically but also geometrically, with bounds on propagation or support. In this talk I will give a gentle introduction to the basic definitions, explain how bounded L-groups differ from classical L-groups, and outline an application to quantum many body invariants.
