CMSA

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

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.

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.

Topics in Deep Learning Theory
2024-09-19 14:30 - 15:45, 2024-09-17 14:30 - 15:45, 2024-09-12 14:30 - 15:45, 2024-09-10 14:30 - 15:45 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Topics in Deep Learning Theory
Eli Grigsby

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.

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?

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.

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.

Math and Machine Learning Program Open Discussion/Tea
2024-09-17 16:00 - 17:00, 2024-09-10 16:00 - 17:00 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Open Discussion/Tea

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)

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.

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).

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).

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.

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.

BPS Algebras in Landau-Ginzburg Models
2024-09-10 16:00 - 18:00 | 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.

Combinatorics and geometry of the amplituhedron
2024-09-09 16:30 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Colloquium
Speaker: Lauren Williams, Harvard University
Title: Combinatorics and geometry of the amplituhedron
Abstract: The amplituhedron is a geometric object introduced by Arkani-Hamed and Trnka to compute scattering amplitudes in N=4 super Yang Mills theory. It generalizes interesting objects such as cyclic polytopes and the positive Grassmannian. It has connections to tropical geometry, cluster algebras, and combinatorics (plane partitions, Catalan numbers). I’ll give a gentle introduction to the amplituhedron, then survey some recent progress on some of the main conjectures about the amplituhedron: the Magic Number Conjecture, the BCFW tiling conjecture, and the Cluster Adjacency conjecture. Based on joint works withEvan-Zohar, Lakrec, Parisi, Sherman-Bennett, and Tessler.

Big Data Conference 2024
2024-09-06 09:00 - 2024-09-07 17:00 | 20 Garden Street, Cambridge, MA 02138, MA, MA, 02138, United States
Youtube Playlist
On September 6-7, 2024, the CMSA hosted the tenth annual Conference on Big Data. 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.
Location: Harvard University CMSA, 20 Garden Street, Cambridge & via Zoom

Speakers:

Tianxi Cai, Harvard Chan School
Raj Chetty, Harvard
Bianca Dumitrascu, Columbia
Boris Hanin, Princeton
Peter Hull, Brown
Jamie Morgenstern, U Washington
Kavita Ramanan, Brown
Neil Thompson, MIT
Melanie Weber, Harvard
Kun-Hsing Yu, Harvard Medical School

Organizers:

Rediet Abebe, Harvard Society of Fellows
Morgane Austern, Harvard University Statistics
Michael R. Douglas, Harvard CMSA
Yannai Gonczarowski, Harvard University Economics and Computer Science
Sam Kou, Harvard University Statistics

SCHEDULE (downloadable pdf)
Friday, Sep. 6, 2024
9:00 am: Breakfast
9:30 am: Introductions
9:45–10:45 am
Speaker: Peter Hull, Brown University
Title: Measuring Discrimination in Multi-Phase Systems, with an Application to Child Protection
Abstract: Large racial disparities have been documented in many high-stakes settings—such as employment, health care, housing, and criminal justice—raising concerns of discrimination by individual decision-makers. At the same time, there is growing understanding that a focus on individual decisions can yield an incomplete view of discrimination; an extensive theoretical literature shows how discrimination can arise and compound across multiple decision-makers in interconnected systems. We develop new empirical tools for studying discrimination in such multi-phase systems and apply them to the setting of foster care placement by child protective services. Leveraging the quasi-random assignment of two sets of decision-makers—initial hotline call screeners and subsequent investigators—we study how unwarranted racial disparities arise and propagate through this system. Using a sample of over 200,000 maltreatment allegations, we find that calls involving Black children are 55% more likely to result in foster care placement than calls involving white children with the same potential for future maltreatment in the home. Call screeners account for up to 19% of this unwarranted disparity, with the remainder due to investigators. Unwarranted disparity is concentrated in cases with potential for future maltreatment, suggesting that white children may be harmed by “underplacement” in high-risk situations.
10:45–11:00 am: Break
11:00 am –12:00 pm
Speaker: Jamie Morgenstern, U Washington
Title: What governs predictive disparity in modern machine learning applications?
Abstract: The deployment of statistical models in impactful environments is far from new—simple correlations have been used to guide decisions throughout the sciences, health care, political campaigns, and in pricing financial instruments and other products for decades. Many such models, and the decisions they supported, were known to have different degrees of predictive power for different demographic groups. These differences had numerous sources, including: limited expressiveness of the statistical models; limited availability of data from marginalized populations; noisier measurements of both features and targets from certain populations; and features with less mutual information about the prediction target for some populations than others.
Modern decision systems which use machine learning are more ubiquitous than ever, as are their differences in performance for different populations of people. In this talk, I will discuss some similarities and differences in the sources of differing performance in contemporary ML systems including facial recognition systems and those incorporating generative AI.
12:00–1:30 pm: Lunch Break
1:30–2:30 pm
Speaker: Kavita Ramanan, Brown University
Title: Understanding High-dimensional Stochastic Dynamics on Realistic Networks
Abstract: Large collections of randomly evolving particles that interact locally with respect to an underlying network model a variety of phenomena ranging from magnetism, the spread of diseases, neural and neuronal networks, opinion dynamics and load balancing on computer networks. Due to their high-dimensional nature, these systems are typically intractable to analyze exactly. Classical work, falling under the rubric of mean-field approximations, has mostly focused on the case when this interaction graph is dense. However, most real-world networks are sparse and often random. We describe a new approach to develop principled approximations for dynamics on realistic networks that beats the curse of dimensionality, and illustrate its efficacy on a class of epidemiological models. This is based on joint works with Michel Davydov, Ankan Ganguly and Juniper Cocomello.
2:30–2:45 pm: Break
2:45–3:45 pm
Speaker: Raj Chetty, Harvard University
Title: The Science of Economic Opportunity: New Insights from Big Data
Abstract: How can we improve economic opportunities for children growing up in low-income families? This talk will present findings from a recent set of studies that use various sources of big data — ranging from anonymized tax records to social network data — to understand the science of economic opportunity. Among other topics, the talk will discuss how and why children’s chances of climbing the income ladder vary across neighborhoods, the drivers of racial disparities in economic mobility, how highly selective colleges may amplify the persistence of privilege, and the role of social capital as a driver of upward mobility. The talk will conclude by giving examples of how academic research using big data is informing policy decisions from the local to federal level to expand opportunities for all.
3:45–4:00 pm: Break
4:00–5:00 pm
Speaker: Neil Thompson
Title: How Algorithmic Progress is driving progress in Big Data and AI
Abstract: Algorithm improvement is one of the purest forms of innovation: it allows the same computational task to be achieved with far fewer resources by proposing clever new ways to do that computation. In this talk, I will discuss the work that my lab has done tracking and quantifying progress across decades of algorithm research and practice. As I will show, this algorithmic progress has often outpaced hardware improvement as the most important driver of progress in Big Data and AI.

Saturday, Sep. 7, 2024
9:00 am: Breakfast
9:30 am: Introductions
9:45–10:45 am
Speaker: Tianxi Cai, Harvard Chan School
Title: Crowdsourcing with Multi-institutional EHR to Improve Reliability of Real World Evidence – Opportunities and Challenges
Abstract: The wide adoption of electronic health records (EHR) systems has led to the availability of large clinical datasets available for discovery research. EHR data, linked with bio- repository, is a valuable new source for deriving real-word, data-driven prediction models of disease risk and progression. Yet, they also bring analytical difficulties especially when aiming to leverage multi-institutional EHR data. Synthesizing information across healthcare systems is challenging due to heterogeneity and privacy. Statistical challenges also arise due to high dimensionality in the feature space. In this talk, I’ll discuss analytical approaches for mining EHR data to improve the reliability and generalizability of real world evidence generated from the analyses. These methods will be illustrated using EHR data from Mass General Brigham and Veteran Health Administration.
10:45–11:00 am: Break
11:00 am–12:00 pm
Speaker: Bianca Dumitrascu, Columbia Data Science Institute
Title: Statistical machine learning for learning representations of embryonic development
Abstract: During embryonic development, single cells read in local information from their environments and use this information to move, divide and specialize. As a result, the environments themselves change. However, it remains unclear how gene expression programs interact with cell morphology and mechanical forces to orchestrate organogenesis in early embryos. Recent advances in single cell techniques and in toto imaging enable unique venues in exploring this link between genomics and biophysics, which dynamically maps cells to organisms.
In this talk, I will describe statistical machine learning frameworks aimed at understanding how tissue level mechanical and morphometric information impact gene expression patterns in spatio-temporal contexts. We use these tools to understand boundary formation in the early development of mouse embryos and to align data from light sheet recordings of pre-gastrulation development.
12:00–1:30 pm: Lunch Break
1:30–2:30 pm
Speaker: Melanie Weber, Harvard Mathematics
Title: Data and Model Geometry in Deep Learning
Abstract: Data with geometric structure is ubiquitous in machine learning. Often such structure arises from fundamental symmetries in the domain, such as permutation-invariance in graphs and sets, and translation-invariance in images. In this talk we discuss implications of this structure on the design and complexity of neural networks. Equivariant architectures, which encode symmetries as inductive bias, have shown great success in applications with geometric data, but can suffer from instabilities as their depths increases. We propose a new architecture based on unitary group convolutions, which allows for deeper networks with less instability. In the second part of the talk we discuss the impact of data and model geometry on the learnability of neural networks. We discuss learnability in several geometric settings, including equivariant neural networks, as well as learnability with respect to the geometry of the input data manifold.
2:30–2:45 pm: Break
2:45–3:45 pm
Speaker: Boris Hanin, Princeton University
Title: Scaling Limits of Neural Networks
Abstract: Neural networks are often studied analytically through scaling limits: regimes in which taking some structural network parameters (e.g. depth, width, number of training datapoints, and so on) to infinity results in simplified models of learning. I will motivative and discuss recent results using several such approaches. I will emphasize both new theoretical insights into how model, training data, and optimizer impact learning and their practical implications for hyperparameter transfer.
3:45–4:00 pm: Break
4:00–5:00 pm
Speaker: Kun-Hsing Yu, Harvard Medical School
Title: Foundation Models for Real-Time Cancer Diagnosis
Abstract: Artificial intelligence (AI) is transforming the landscape of medical research and practice. Recent advances in microscopic image digitization, foundation models, and scalable computing infrastructure have opened new avenues for AI-enhanced cancer diagnosis. In this talk, I will highlight recent breakthroughs in multi-modal AI systems for cancer pathology evaluation, discuss integrative biomedical informatics methods that link cell morphology with molecular profiles, and outline critical challenges in developing robust medical AI systems.

Information about the 2023 Big Data Conference can be found here.

CMSA/Math Fall Gathering
2024-09-05 16:00 - 17:00 | CMSA 20 Garden Street Cambridge, Massachusetts 02138 United States
September 5, 2024
4:00 pm
CMSA Common Room, 20 Garden Street, Cambridge MA
All CMSA and Math affiliates are invited.

Mathematics and Machine Learning Program Opening Workshop
2024-09-03 10:00 - 2024-09-05 16:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Mathematics and Machine Learning Program Opening Workshop
Dates: September 3 – 5, 2024
Location: Room G10, CMSA, 20 Garden Street, Cambridge MA & via Zoom Webinar
AI for Mathematicians, with Leon Bottou, François Charton, David McAllester, Adam Wagner, Boris Hanin, and Geordie Williamson. A series of 6 tutorial lectures introducing concepts of AI and of theorem proving, with many case studies of AI applied to mathematics, and including lectures and discussion sessions on open questions, future prospects, and ethical questions.

Speakers

Leon Bottou (Meta AI)
François Charton (Meta AI)
Michael R. Douglas (Harvard CMSA)
Boris Hanin (Princeton)
David McAllester (TTIC)
Adam Wagner (WPI)
Geordie Williamson (University of Sydney)

Schedule (link to downloadable pdf)
Videos from the Opening Workshop (Youtube Link)

Tuesday Sep. 3, 2024

9:30–10:00 am
Morning refreshments

10:00–11:30 am
Mike Douglas: Overview of AI for mathematics
Slides (pdf)

11:30 am–12:00 pm
Discussion

12:00–1:30 pm
Break

1:30–2:30 pm
David McAllester: Logic and formal methods
Slides (pdf)

2:30–3:00 pm
Coffee break

3:00–4:00 pm
Panel Discussion: Automated mathematical discovery

Wednesday Sep. 4, 2024

9:30–10:00 am
Morning refreshments

10:00–11:30 am
Boris Hanin: Theory of Machine Learning
Slides (pdf)

11:30 am–12:00 pm
Discussion

12:00–1:30 pm
Break

1:30–2:30 pm
Adam Wagner: Case studies I: Reinforcement learning and pattern finding
Slides (pdf)

2:30–3:00 pm
Coffee break

Thursday Sep. 5, 2024

9:30–10:00 am
Morning refreshments

10:00–11:30 am
François Charton (slides-pdf)
and Geordie Williamson: Case studies II

11:30 am–12:00 pm
Discussion

12:00–1:30 pm
Break

1:30–2:30 pm
Leon Bottou: Open questions in AI
Slides (pdf)

2:30–3:00 pm
Coffee break

3:00–4:00 pm
Panel Discussion: How might AI change mathematics?

Instanton in Lattice QCD from Higher Categories and Higher Anafunctors
2024-08-28 14:00 - 15:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Speaker: Jing-Yuan Chen, Tsinghua University
Title: Instanton in Lattice QCD from Higher Categories and Higher Anafunctors

Abstract: Putting continuum QFT (not just TQFT) on the lattice is important for both fundamental understandings and practical numerics. The traditional way of doing so, based on simple intuitions, however, does not admit natural definitions for general topological operators of continuous-valued fields—one such example is the long standing problem in lattice QCD of lacking a natural definition for Yang-Mills instantons.
In this talk, I will explain a more systematic way to relate continuum and lattice QFT, based on higher categories and higher anafunctors, so that the topological operators in the continuum can be naturally defined on the lattice. The idea, though formulated formally, is physically very intuitive—we want to effectively capture the different possibilities of how a lattice field may interpolate into the continuum, so the higher categories that are employed to study higher homotopy theory should be naturally involved. Via this formalism, we resolve the long-standing problem of defining instanton (as well as Chern-Simons term) in lattice Yang-Mills theory, in terms of multiplicative bundle gerbes. Moreover, when the fields become discrete, our formalism can recover the Dijkgraaf-Witten and Turaev-Viro theory, so we hope this formalism to be a good starting point towards (in the very long term) a comprehensive categorical understanding of QFT that encompass both continuous and discrete degrees of freedom, applicable both to IR and to UV.
