BPS Algebras in Landau-Ginzburg Models
2024-09-10 16:00 - 18:00 (America/New_York) | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Speaker: Ahsan Khan (CMSA)
Title: BPS Algebras in Landau-Ginzburg Models
Abstract: The study of BPS states in supersymmetric quantum field theory has been a fruitful source of both mathematical and physical insights. In particular their study often leads to rich algebraic structures – from the “Algebra of the Infrared” of Gaiotto-Moore-Witten to the “Cohomological Hall Algebras” of Kontsevich-Soibelman. In this talk, I will provide an overview of some of these algebraic constructions, with a particular emphasis on BPS states in two-dimensional Landau-Ginzburg models. In the second half of the talk, I will discuss how these algebraic structures can be extended to more general Landau-Ginzburg models defined by closed holomorphic one-forms.

Math and Machine Learning Program Discussion
2024-09-11 10:30 - 12:00, 2024-09-13 10:30 - 12:00, 2024-09-16 10:30 - 12:00, 2024-09-18 10:30 - 12:00, 2024-09-20 10:30 - 12:00, 2024-09-23 10:30 - 12:00, 2024-09-25 10:30 - 12:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States

Twisted Tools for (Untwisted) Quantum Field Theory
2024-09-12 10:00 - 11:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Mathematical Physics and Algebraic Geometry
Speaker: Justin Kulp (Simons Center for Geometry and Physics)
Title: Twisted Tools for (Untwisted) Quantum Field Theory
Abstract: One of the most important properties of QFTs is that they can be deformed by “turning on interactions.” Essentially every observable can be viewed as coupling the theory to some external system. Famously, adding interactions (generically) breaks scale invariance, leading to familiar ideas of EFTs and RG flows in the space of QFTs. An underappreciated fact is that one can actually consider flows generated by any transformation, not just the usual scale transformations.
In my talk, I will discuss a flow in the space of QFTs coming from (an analogue of) BRST symmetry. The beta-function for this “BRST-flow” controls deformations of the QFT and is highly mathematically constrained, endowing the space of interactions with an L∞ algebra structure. The structure constants/brackets of the L∞ algebra are highly computable (requiring only a first course in QFT to compute) and contain familiar information such as anomalies and Operator Product Expansion coefficients. I will prove a non-renormalization theorem for holomorphic-topological QFTs with more than one topological direction, which can be thought of as a generalization of a formality theorem of Kontsevich. Time permitting, I will discuss how this formalism enables the systematic computation of minimal BPS operators in supersymmetric QFTs and describe the “holomorphic confinement” of N=1 SYM. Based on arXiv:2403.13049.

Topics in Deep Learning Theory
2024-09-12 14:30 - 15:45, 2024-09-17 14:30 - 15:45, 2024-09-19 14:30 - 15:45, 2024-09-24 14:30 - 15:45 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Topics in Deep Learning Theory
Eli Grigsby

Math and Machine Learning Lecture: Can AI help with hard mathematics?
2024-09-12 16:00 - 17:00 | Hybrid
Math and Machine Learning Lecture
Date: Thursday, Sep. 12, 2024
Time: 4:00 – 5:00 pm
Location: Science Center & via Zoom Webinar

Speaker: Geordie Williamson, University of Sydney
Title: Can AI help with hard mathematics?
Abstract: The last years have seen remarkable advances in what AI can do. It is perhaps surprising that its impact on research in pure mathematics has been modest. Reasoning, which is so quintessential to the mathematical process, remains a major challenge for current AI systems. I will survey some exciting recent applications of AI in mathematics research, trying to highlight what AI can do and the challenges that remain.

Abundance for mixed characteristic threefolds
2024-09-13 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Iacopo Brivio (CMSA)
Title: Abundance for mixed characteristic threefolds
Abstract: The Minimal Model Program (MMP) predicts that every algebraic variety X is birational to either a fibration in Fano varieties, or it admits a “minimal model” X’, that is a birational model with nef canonical bundle K_X’. The Abundance conjecture predicts then that K_X’ is actually semiample, in particular it endows X’ with the structure of a Calabi-Yau fibration. These conjectures were initially phrased for complex varieties, but more recently there has been a lot of interest in working over positive characteristic fields, or even mixed characteristic rings. In this talk I will give a broad overview of the subject, starting from the case of complex surfaces. In the last part I will outline a proof of the Abundance conjecture for mixed characteristic threefolds (based on joint work with F. Bernasconi and L. Stigant).

Freedman CMSA Seminar
2024-09-13 14:30 - 17:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Freedman CMSA Seminar

2:00-3:30 pm ET
Speaker: Mike Freedman, Harvard CMSA
Title: Detecting hidden structures in linear maps
Abstract: I’ll consider the problem of detecting spectral features and tensor structures within linear maps both in a quantum and classical contexts. In the quantum context there is the question of whether a Hamiltonian is local, and if so, local in distinct coordinate systems (a “duality”). Also, in the case of a unitary described by a quantum circuit, does it possess unusual spectral features or tensor structure? In ML one optimizes many linear maps. How would we know – and would we care – if the resulting maps (approximately) tensor factored?

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

4:00-5:30 pm ET
Speaker: Ryan O’Donnell, Carnegie Mellon University
Title: Quartic quantum speedups for planted inference
Abstract: Consider the following task (“noisy 4XOR”), arising in CSPs, optimization, and cryptography. There is a ‘secret’ Boolean vector x in {-1,+1}^n. One gets m randomly chosen pairs (S, b), where S is a set of 4 coordinates from [n] and b is x^S := prod_{i in S} x_i with probability 1-eps, and -x^S with probability eps. Can you tell the difference between the cases eps = 0.1 and eps = 0.5?
It depends on m. The best known algorithms use the “Kikuchi method” and run in time ~n^L when m ~ n^2/L. We will review this method, and also show that the running time can be improved to roughly n^{L/4} with a quantum algorithm.
Joint work with Alexander Schmidhuber (MIT), Robin Kothari (Google), and Ryan Babbush (Google).

Ringdown in the SYK model
2024-09-16 09:30 - 10:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Joint BHI/CMSA Foundation Seminar
Speaker: Matthew Dodelson (Harvard)
Title: Ringdown in the SYK model
Abstract: Thermal correlators in large N systems equilibrate at late times, but the precise late-time behavior is unknown away from holographic and free field limits. In this talk I will analyze this problem in the case of the SYK model away from the low-temperature limit. The basic technique is a resummation of perturbation theory which is reminiscent of the double cone construction. We will also discuss the interpretation of the result in terms of a dual stringy black hole.

Periodic pencils of flat connections and their p-curvature
2024-09-16 16:30 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Colloquium
Speaker: Pavel Etingof (MIT)
Title: Periodic pencils of flat connections and their p-curvature
A periodic pencil of flat connections on a smooth algebraic variety is a linear family of flat connections , where are local coordinates on and are matrix-valued regular functions. A pencil is periodic if it is generically invariant under the shifts up to isomorphism. I will explain that periodic pencils have many remarkable properties, and there are many interesting examples of them, e.g. Knizhnik-Zamolodchikov, Dunkl, Casimir connections and equivariant quantum connections for conical symplectic resolutions with finitely many torus fixed points. I will also explain that in characteristic , the -curvature operators of a periodic pencil are isospectral to the commuting endomorphisms , where is the Frobenius twist of . This allows us to compute the eigenvalues of the -curvature for the above examples, and also to show that a periodic pencil of connections always has regular singularites. This is joint work with Alexander Varchenko.
(Abstract link (pdf)

Math and Machine Learning Program Open Discussion/Tea
2024-09-17 16:00 - 17:00, 2024-09-24 16:00 - 17:00 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Open Discussion/Tea

Mathematics around Twisted Holography
2024-09-17 16:00 - 18:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Geometry and Quantum Theory Seminar
Speaker: Keyou Zeng (CMSA)
Title: Mathematics around Twisted Holography
Abstract: The holography principle is an important idea in physics and has been widely studied since the 90s. Twisted holography offers a way to simplify physical holography models through the procedure called twisting. In the first part of the talk, I’ll introduce some of the mathematical structures underlying this twisted version of holography, such as Koszul duality.
In the second part, I’ll discuss the concept of vertex algebras in symmetric monoidal categories, specifically in Deligne category. This framework will serve as a tool to rigorously define the “large N” algebra that emerges from twisted holography.

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

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

CMSA Q&A Seminar: Noam Elkies
2024-09-18 12:00 - 13:00 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA Q&A Seminar
Speaker: Noam Elkies, Harvard Math
Topic: How to show E8 and Leech lattices have optimal sphere packings?

Feynman graph integrals from topological holomorphic theories
2024-09-19 10:00 - 11:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Mathematical Physics and Algebraic Geometry Seminar
Speaker: Minghao Wang (Boston University)
Title: Feynman graph integrals from topological holomorphic theories
Abstract: Feynman graph integrals from topological theories were developed by M. Kontsevich, S. Axelrod and I. M. Singer in 1990s. These integrals have many mathematical applications, such as knot invariants, operad theory and formality theorems. In this talk, I will talk about Feynman graph integrals from topological-holomorphic theories. In particular, I will prove the finiteness of Feynman graph integrals when spacetime is flat spaces and a vanishing result of graph integrals. Combining the vanishing result with Batalin-Vilkovisky(BV) formalism, we can show the absence of anomalies of topological-holomorphic theories on flat spaces with at least two topological dimensions. As a consequence, we can construct factorization algebras of quantum observables. This is a joint with Brian Williams.

Communication Complexity of Combinatorial Auctions
2024-09-20 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Tomer Ezra (CMSA)
Title: Communication Complexity of Combinatorial Auctions
Abstract: We study the communication complexity of welfare maximization in combinatorial auctions with m items and two subadditive bidders. A 2-approximation can be guaranteed by a trivial randomized protocol with zero communication, or a trivial deterministic protocol with O(1) communication. We show that outperforming these trivial protocols requires exponential communication, settling an open question of [DobzinskiNS10, Feige09].
Specifically, we show that any (randomized) protocol guaranteeing a o(logm)-approximation requires communication exponential in m. We complement it by presenting an O(logm)-approximation in poly(m) communication.

Classification and Construction of crystalline topological superconductors and insulators in interacting fermion systems
2024-09-20 14:00 - 15:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Quantum Matter Seminar
Speaker: Zhengcheng Gu, Chinese University of Hong Kong
Title: Classification and construction of crystalline topological superconductors and insulators in interacting fermion systems
Abstract: The construction and classification of crystalline symmetry protected topological (SPT) phases in interacting bosonic and fermionic systems have been intensively studied in the past few years. Crystalline SPT phases are not only of conceptual importance, but also provide us great opportunities towards experimental realization since space group symmetries naturally exist for any realistic material. In this talk, I will discuss how to construct and classify crystalline topological superconductors (TSC) and topological insulators (TI) in interacting fermion systems. I will also discuss the relationship between internal symmetry protected SPT phases and crystalline symmetry protected SPT Phases.

Symmetry groups in infinite dimensions
2024-09-23 16:30 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Colloquium
Speaker: Lisa Carbone, Rutgers University
Title: Symmetry groups in infinite dimensions
Abstract: The study of many physical theories requires an understanding of symmetries of infinite dimensional Lie algebras. The construction of groups of automorphisms for infinite dimensional Lie algebras is challenging, but there is well established theory for the class of Kac-Moody algebras. A generalization of Kac-Moody algebras known as Borcherds algebras arise in string theory models, but the methods for constructing Kac-Moody groups break down for this more general class. We discuss the challenges that arise and describe several approaches to constructing groups for Borcherds algebras. Our main example is the Monster Lie algebra which plays an important role in the solution of Monstrous Moonshine and which is a symmetry algebra of a model of the compactified Heterotic String.

New Energy Inequality in AdS
2024-09-24 11:00 - 12:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
General Relativity Seminar
Speaker: Diandian Wang, Harvard University
Title: New Energy Inequality in AdS
Abstract: I will describe evidence for a new energy inequality in asymptotically AdS spacetimes whose conformal boundary contains a spatial circle. It is in some sense analogous but crucially different to the Penrose inequality. In the AdS4 case, this generalizes the Horowitz-Myers conjecture. I will show how static solutions play an interesting role in determining the shape of the function that bounds the gravitational energy.

Symplectic duality in examples
2024-09-24 16:15 - 18:15 | Science Center Hall E, 1 Oxford Street, Cambridge, MA, 02138, United States
Geometry and Quantum Theory Seminar
Speaker: Vasily Krylov, Harvard CMSA & Math
Title: Symplectic duality in examples
Abstract: Over the past twenty years, mathematicians and physicists have shown increasing interest in studying certain Poisson varieties, known as “symplectic singularities.” Many of these objects naturally arise as Higgs or Coulomb branches of certain TQFTs and, therefore, fall within the framework of 3D mirror symmetry, also known as symplectic duality. The first part of the talk will provide a gentle introduction to the theory of symplectic singularities, with an emphasis on various examples. In the second part, we will discuss how the symplectic duality works in examples, beginning with the simplest cases. We will then discuss a particular phenomenon called the Hikita-Nakajima conjecture, which predicts a deep and nontrivial relationship between dual varieties. It is particularly intriguing that this conjecture was formulated by mathematicians and still requires further understanding from a physical perspective.

Infinite Limits and Scaling Laws for Deep Neural Networks
2024-09-25 14:00 - 15:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
New Technologies in Mathematics Seminar
Speaker: Blake Bordelon
Title: Infinite Limits and Scaling Laws for Deep Neural Networks
Abstract: Scaling up the size and training horizon of deep learning models has enabled breakthroughs in computer vision and natural language processing. Empirical evidence suggests that these neural network models are described by regular scaling laws where performance of finite parameter models improves as model size increases, eventually approaching a limit described by the performance of an infinite parameter model. In this talk, we will first examine certain infinite parameter limits of deep neural networks which preserve representation learning and then describe how quickly finite models converge to these limits. Using dynamical mean field theory methods, we provide an asymptotic description of the learning dynamics of randomly initialized infinite width and depth networks. Next, we will empirically investigate how close the training dynamics of finite networks are to these idealized limits. Lastly, we will provide a theoretical model of neural scaling laws which describes how generalization depends on three computational resources: training time, model size and data quantity. This theory allows analysis of compute optimal scaling strategies and predicts how model size and training time should be scaled together in terms of spectral properties of the limiting kernel. The theory also predicts how representation learning can improve neural scaling laws in certain regimes. For very hard tasks, the theory predicts that representation learning can approximately double the training-time exponent compared to the static kernel limit.
