Going to the other side .... in algebra, topology, and maybe physics
2024-09-27 09:00 - 10:00 (America/New_York) | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Quantum Field Theory and Physical Mathematics
Speaker: Sergei Gukov (Caltech)

Title: Going to the other side …. in algebra, topology, and maybe physics

Abstract: Inspired by Eugene Wigner’s reflections on the ‘unreasonable effectiveness of mathematics in the natural sciences,’ this talk is about the surprising and pervasive role of a peculiar phenomenon that, a priori, seemed to have no reason to exist. Yet, it emerges across many different areas of mathematics and theoretical physics, including:

the Kazhdan-Lusztig correspondence
quantum invariants of 3-manifolds
the study of 2d (0,2) boundary conditions in 3d N=2 theories
resurgent analysis

Although each of these fields approaches the phenomenon from a different perspective, the results align in striking and unexpected ways.

Topological Invariants of gapped states through cosheaves
2024-10-01 16:15 - 18:15 | Science Center Hall E, 1 Oxford Street, Cambridge, MA, 02138, United States
Geometry and Quantum Theory Seminar
Speaker: Bowen Yang, Harvard CMSA
Title: Topological Invariants of gapped states through cosheaves
Abstract: We provide a proper mathematical framework for the constructions of topological invariants of gapped quantum states and interpret topological invariants of gapped states as lattice analogs of ’t Hooft anomalies in Quantum Field Theory. Our secondary goal is to generalize this construction in various directions. In particular, we show how to define topological invariants of lattice spin systems living on well-behaved subsets of the lattice.

CMSA Q&A Seminar: Cliff Taubes
2024-10-02 12:00 - 13:00 | MA
CMSA Q&A Seminar
Speaker: Cliff Taubes, Harvard Mathematics
Topic: What are Z/2 harmonic 1-forms?

Hierarchical data structures through the lenses of diffusion models
2024-10-02 14:00 - 15:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
New Technologies in Mathematics Seminar
Speaker: Antonio Sclocchi, EPFL
Title: Hierarchical data structures through the lenses of diffusion models
Abstract: The success of deep learning with high-dimensional data relies on the fact that natural data are highly structured. A key aspect of this structure is hierarchical compositionality, yet quantifying it remains a challenge.
In this talk, we explore how diffusion models can serve as a tool to probe the hierarchical structure of data. We consider a context-free generative model of hierarchical data and show the distinct behaviors of high- and low-level features during a noising-denoising process. Specifically, we find that high-level features undergo a sharp transition in reconstruction probability at a specific noise level, while low-level features recombine into new data from different classes. This behavior of latent features leads to correlated changes in real-space variables, resulting in a diverging correlation length at the transition.
We validate these predictions in experiments with real data, using state-of-the-art diffusion models for both images and texts. Remarkably, both modalities exhibit a growing correlation length in changing features at the transition of the noising-denoising process.
Overall, these results highlight the potential of hierarchical models in capturing non-trivial data structures and offer new theoretical insights for understanding generative AI.

Holography and Regge Phases at Large U(1) Charge
2024-10-04 09:00 - 10:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Quantum Field Theory and Physical Mathematics Seminar
Speaker: Giulia Fardelli, Boston University
Title: Holography and Regge Phases at Large U(1) Charge
Abstract: A single Conformal Field Theory (CFT) can have a rich phase diagram with qualitatively different emergent behaviors in a range of different regimes parameterized by the conserved charges of the theory. In this talk, I will consider a CFT with a global U(1) current and explore the phase diagram as a function of the U(1) charge Q and angular momentum J, particularly at large J and Q. By taking the large J limit first, we are able to employ a dual holographic interpretation in AdS_{d+1} to predict the energy spectrum of Q-particle states. This limit has been studied in detail for Q=2, yielding very general results applicable to unitary CFTs in d>2. When Q is also taken to be large, the description is more complicated; nevertheless, we can draw interesting conclusions about the energy spectrum under certain assumptions. I will conclude with a concrete example, the O(2) model in 3d, highlighting interesting connections with recent (and less recent) results in this context.

High-dimensional learning of narrow neural networks
2024-10-04 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Hugo Cui, CMSA
Title: High-dimensional learning of narrow neural networks
Abstract: This talk explores the interplay between neural network architectures and data structure through the lens of high-dimensional asymptotics. We focus on a class of narrow neural networks, namely networks possessing a finite number of hidden units, while operating in high dimensions. In the limit of large data dimension and comparably large number of samples, we derive a tight asymptotic characterization of the learning of these architectures. As an illustration, we discuss how this characterization enables the analysis of a solvable model of dot-product attention. We show how the latter can learn to implement either a positional attention mechanism (with tokens attending to each other based on their respective positions), or a semantic attention mechanism (with tokens attending to each other based on their meaning), and evidence a phase transition with sample complexity from positional to semantic learning.

Local complexity measures in modern parameterized function classes for supervised learning
2024-10-07 16:30 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Colloquium
Speaker: Elisenda Grigsby, Boston College
Title: Local complexity measures in modern parameterized function classes for supervised learning
Abstract: The parameter space for any fixed architecture of neural networks serves as a proxy during training for the associated class of functions – but how faithful is this representation? For any fixed feedforward ReLU network architecture, it is well-known that many different parameter settings can determine the same function. It is less well-known that the degree of this redundancy is inhomogeneous across parameter space. I’ll discuss two locally-applicable complexity measures for ReLU network classes and what we know about the relationship between them: (1) the local functional dimension, and (2) a local version of VC dimension called persistent pseudodimension. The former is easy to compute on finite batches of points, the latter should give local bounds on the generalization gap. I’ll speculate about how this circle of ideas might help guide our understanding of the double descent phenomenon. All of the work described in this talk is joint with Kathryn Lindsey. Some portions are also joint with Rob Meyerhoff, David Rolnick, and Chenxi Wu.

Skein traces and curve counting
2024-10-08 16:15 - 18:15 | Science Center Hall E, 1 Oxford Street, Cambridge, MA, 02138, United States
Geometry and Quantum Theory Seminar
Speaker: Sunghyuk Park, Harvard CMSA
Title: Skein traces and curve counting
Abstract: Skein modules are vector space-valued invariants of 3-manifolds describing the space of line defects modulo skein relations (determined by a choice of a ribbon category). When the 3-manifold is S x I for some surface S, the skein module has a natural algebra structure and is called the skein algebra of S.
In 2010, Bonahon and Wong constructed an algebra embedding (named “quantum trace”) of the sl_2 skein algebra into a quantum cluster variety called the “quantum Teichmuller space” for punctured surfaces, which has applications to the representation theory of skein algebras.
In the first half of this talk, I will give an overview of these concepts and explain how the quantum trace map can be generalized to the 3-dimensional setup.
In the second half, I will discuss how everything above can be generalized to HOMFLYPT skeins and has natural interpretation in terms of counts of holomorphic curves.

CMSA Q&A Seminar: Laura DeMarco
2024-10-09 12:00 - 13:00 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA Q&A Seminar
Speaker: Laura DeMarco, Harvard University
Topic: What is Teichmuller geometry and why is it important?

Dolbeault Virasoro algebra and M5 branes
2024-10-11 09:00 - 10:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Quantum Field Theory and Physical Mathematics Seminar
Speaker: Brian Williams, Boston University
Title: Dolbeault Virasoro algebra and M5 branes
Abstract: The worldvolume theory on a stack of M5 branes in M-theory is superconformal. We propose a conjecture that in the holomorphic twist of the theory on a stack of M5 branes an infinite-dimensional enhancement of the (twisted) superconformal algebra is a symmetry. This algebra is closely related to the exceptional infinite-dimensional Lie superalgebra called E(3|6). We show that under the usual AGT correspondence this enhanced algebra degenerates to the Virasoro algebra at a particular central charge.

Scattering Amplitude from a Twistor Point of View
2024-10-11 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Keyou Zeng
Title: Scattering Amplitude from a Twistor Point of View
Abstract: Scattering amplitude is a key quantity in quantum field theory. Although challenging to compute at higher loops and for large particle numbers, physicists have developed various tools to gain a deeper understanding of amplitudes. In this seminar, I will introduce a novel approach, initiated by K. Costello and N. Paquette, which makes use of twistor correspondence. This approach enables the computation of certain amplitudes using chiral algebra. I will also present an upcoming work that constructs a top-down holography model for computing tree and loop amplitudes of certain non-SUSY theories in various self-dual backgrounds.

Topological Modular Forms, its equivariant refinements and relation with supersymmetric quantum field theories
2024-10-15 16:15 - 18:15 | Science Center Hall E, 1 Oxford Street, Cambridge, MA, 02138, United States
Geometry and Quantum Theory Seminar
Speaker: Mayuko Yamashita, Kyoto University
Title: Topological Modular Forms, its equivariant refinements and relation with supersymmetric quantum field theories
Abstract: This talk is about the Segal-Stolz-Teichner program, which is one of the most deep and interesting topics relating homotopy theory and physics. Mathematically, they propose a geometric model of TMF, the spectrum (in homotopy theory) of Topological Modular Forms, in terms of supersymmetric quantum field theories. Their proposal, although far from solid formulation or a proof, has been a guiding principle leading us to many new interesting ideas and discoveries in both mathematics and physics. In this talk, I will give an overview of this topic, as well as my current works using equivariant twisted TMF.

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

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.

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.

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.

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

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.

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.

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?

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.

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.

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.

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.

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

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

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

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

Speaker: Christoph Kehle (MIT)

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

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

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