CMSA

Foundation Seminar
2024-11-04 09:30 - 10:30 (America/New_York) | 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.

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.

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

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.

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.

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

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/

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.

Formality Theorem and Webs
2024-10-25 12:00 - 13:00 | 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.

Math and Machine Learning Program Discussion
2024-10-25 10:30 - 12:00, 2024-10-23 10:30 - 12:00, 2024-10-21 10:30 - 12:00, 2024-10-18 10:30 - 12:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States

The spin-statistics theorem for TFTs
2024-10-25 09:00 - 10:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Quantum Field Theory and Physical Mathematics Seminar
Speaker: Luuk Stehouwer, Dalhousie University
Title: The spin-statistics theorem for TFTs
Abstract: In quantum field theory (QFT) the spin-statistics theorem says that in a unitary QFT, a particle has half-integer spin if and only if it is a fermion. I show how to phrase this statement in the language of functorial field theories. More precisely, I explain when a functorial field theory “has fermions” and “has spinors” and when they are “related”. I will then restrict to topological field theories (TFTs) and define unitary TFTs. There are counterexamples of the spin-statistics theorem for non-unitary TFTs. I will prove that every unitary TFT satisfies the spin-statistics theorem.

Topics in Deep Learning Theory
2024-10-24 14:30 - 15:45, 2024-10-22 14:30 - 15:45, 2024-10-17 14:30 - 15:45 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Topics in Deep Learning Theory
Eli Grigsby

Heterotic Little String Theories and Inequivalent Genus-One Fibrations
2024-10-24 10:00 - 11:00 | Virtual
Mathematical Physics and Algebraic Geometry Seminar
Speaker: Hamza Ahmed, Northeastern University
Title: Heterotic Little String Theories and Inequivalent Genus-One Fibrations
Abstract: Little String Theories (LSTs) are 6D Supersymmetric quantum field theories (SQFTs) with an additional physical relation called T-duality. This enables us to arrange them into equivalence classes, where each equivalence class has 6D LSTs that lead to the same 5D effective theory when compactified on a circle. The problem of finding T-dual LSTs can be mapped to the problem of finding inequivalent genus-one fibrations of the same non-compact Calabi-Yau (CY) threefold. For T-dual theories, certain field theory data is expected to match, which then implies certain invariants of inequivalent fibrations. Focusing on theories with 8 supercharges (Heterotic LSTs), we use this geometry-field theory equivalence to study the T-duality landscape, particularly in the case where the genus-one fiber does not have a section, leading to what are called twisted T-dual theories. Based on the excellent agreement we find between the geometry and field theory arguments, we conjecture the existence of a new class of twisted T-duals for which no geometric construction is known.

How Far Can Transformers Reason? The Globality Barrier and Inductive Scratchpad
2024-10-23 14:00 - 15:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
New Technologies in Mathematics Seminar
Speaker: Aryo Lotfi (EPFL)
Title: How Far Can Transformers Reason? The Globality Barrier and Inductive Scratchpad
Abstract: Can Transformers predict new syllogisms by composing established ones? More generally, what type of targets can be learned by such models from scratch? Recent works show that Transformers can be Turing-complete in terms of expressivity, but this does not address the learnability objective. This paper puts forward the notion of ‘globality degree’ of a target distribution to capture when weak learning is efficiently achievable by regular Transformers, where the latter measures the least number of tokens required in addition to the tokens histogram to correlate nontrivially with the target. As shown experimentally and theoretically under additional assumptions, distributions with high globality cannot be learned efficiently. In particular, syllogisms cannot be composed on long chains. Furthermore, we show that (i) an agnostic scratchpad cannot help to break the globality barrier, (ii) an educated scratchpad can help if it breaks the globality at each step, however not all such scratchpads can generalize to out-of-distribution (OOD) samples, (iii) a notion of ‘inductive scratchpad’, that composes the prior information more efficiently, can both break the globality barrier and improve the OOD generalization. In particular, some inductive scratchpads can achieve length generalizations of up to 6x for some arithmetic tasks depending on the input formatting.

CMSA Q&A Seminar: Dan Freed
2024-10-23 12:00 - 13:00 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA Q&A Seminar
Speaker: Dan Freed, Harvard Mathematics & CMSA
Topic: What are topological phases of matter?

Fusion 2-Categories and their Classification
2024-10-22 16:15 - 18:15 | Science Center Hall E, 1 Oxford Street, Cambridge, MA, 02138, United States
Geometry and Quantum Theory Seminar
Speaker: Thibault Décoppet, Harvard University
Title: Fusion 2-Categories and their Classification
Abstract: Categorifying the classical notion of fusion (1-)category, fusion 2-categories were recently introduced. These objects have found many applications in Physics, most notably to the classification of topological orders, but also to the description of non-invertible symmetries in 2+1 dimensions. The first part of this talk will be devoted to reviewing the definition of a fusion 2-category and giving many examples. In the second half, I will present a remarkable result concerning the Morita theory of fusion 2-categories and explain how it can be used to give a homotopy coherent classification of fusion 2-categories.

Math and Machine Learning Program Open Discussion/Tea
2024-10-22 16:00 - 17:00 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Open Discussion/Tea

Higher Vapnik–Chervonenkis theory
2024-10-21 16:30 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Colloquium
Speaker: Artem Chernikov, University of Maryland
Title: Higher Vapnik–Chervonenkis theory
Abstract: Finite VC-dimension, a combinatorial property of families of sets, was discovered simultaneously by Vapnik and Chervonenkis in probabilistic learning theory, and by Shelah in model theory (where it is called NIP). It plays an important role in several areas including machine learning, combinatorics, mathematical logic, functional analysis and topological dynamics. We develop aspects of higher-order VC-theory, in particular establishing a generalization of the epsilon-net theorem for families of sets (and functions) on n-fold product spaces with bounded VC_n-dimension (i.e. there is a bound on the sizes of n-dimensional boxes that can be shattered). We obtain some applications in combinatorics and in model theory, including a strong version of Szemerdi’s regularity lemma for hypergraphs omitting a fixed finite n-partite n-hypergraph. Joint work with Henry Towsner.

Foundation Seminar: Singularity Theorems, Part I
2024-10-21 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
Title: Singularity Theorems, Part I
Journal Club Discussion

Positive mass and rigidity theorems in Riemannian geometry
2024-10-18 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Puskar Mondal
Title: Positive mass and rigidity theorems in Riemannian geometry
Abstract: Positive mass theorem proved by Schoen-Yau, Witten, Taubes-Parker is one of the most important results in scalar curvature geometry in asymptotically flat settings. Since then several versions have been proven and generalized to other geometries such as asymptotically hyperbolic manifolds. The analogous theorem for strictly positive curvature geometries is absent. There have been counterexamples but a precise quantification does not exist.I prove a scalar curvature rigidity theorem for spheres. In particular, I prove that $n+1~(ngeq 2)$ dimensional spherical caps with constant positive mean curvature totally umbilic boundaries are rigid under smooth perturbations, and such rigidity results fail for the hemisphere. The assertion of this result is based on the notion of a real Killing connection and solution of the boundary value problem associated with its Dirac operator. Additionally, an improved eigenvalue estimate for the Dirac operator on hypersurfaces in positively curved manifolds is obtained.

Bosonic and fermionic 1-form symmetries and anomaly matching
2024-10-18 09:00 - 10:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Quantum Field Theory and Physical Mathematics Seminar
*via Zoom only*
Speaker: Rajath Radhakrishnan (ICTP, Trieste)
Title: Bosonic and fermionic 1-form symmetries and anomaly matching
Abstract: In this talk, I will consider bosonic and fermionic (non-invertible) 1-form symmetries in 2+1d QFTs. These are 1-form symmetries implemented by topological line operators with real spins. I will present a classification of topological quantum field theories in which all line operators have real topological spins, and use this framework to classify the anomalies associated with these 1-form symmetries. Additionally, I will discuss the anomaly matching condition for these symmetries under an RG flow. I will illustrate this condition in concrete examples.

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

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

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

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

From Word Prediction to Complex Skills: Data Flywheels for Mathematical Reasoning
2024-10-16 14:00 - 15:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
New Technologies in Mathematics Seminar
Speaker: Anirudh Goyal (University of Montreal)
Title: From Word Prediction to Complex Skills: Data Flywheels for Mathematical Reasoning
Abstract: This talk examines how large language models (LLMs) evolve from simple word prediction to complex skills, with a focus on mathematical problem solving. A major driver of AI products today is the fact that new skills emerge in language models when their parameter set and training corpora are scaled up. This phenomenon is poorly understood, and a mechanistic explanation via mathematical analysis of gradient-based training seems difficult. The first part of the talk focuses on analysing emergence using the famous (and empirical) Scaling Laws of LLMs. Then I talk about howc LLMs can verbalize these skills by assigning labels to problems and clustering them into interpretable categories. This metacognitive ability allows us to leverage skill-based prompting, significantly improving performance on mathematical reasoning. I then present a framework that combines LLMs with human oversight to generate challenging, out-of-distribution math questions. This process led to the creation of the MATH^2 dataset, which enhances both model and human performance, driving further advances in mathematical reasoning capabilities.

CMSA Q&A Seminar: Nazim Bouatta
2024-10-16 12:00 - 13:00 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA Q&A Seminar
Speaker: Nazim Bouatta (HMS)
Topic: What are AlphaFold2 and OpenFold
