CMSA

Dimension Reduction and Adiabatic Limits of Generalized ASD Instantons
2025-11-06 16:00 - 17:00 (America/New_York) | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Differential Geometry and Physics Seminar
Speaker: Dylan Galt, Harvard
Title: Dimension Reduction and Adiabatic Limits of Generalized ASD Instantons
Abstract: I will begin by explaining a dimension reduction result for Tian’s generalized ASD instantons over product manifolds, which generalizes work of Yuanqi Wang on codimension-one reduction and includes the cases of G2 and Spin(7) instantons as well as degree zero Hermitian-Yang-Mills connections. The proof is inspired by a simple observation about the Yang-Mills energy and suggests a natural adiabatic picture for these generalized ASD connections. I will describe ongoing work towards establishing such an adiabatic scheme, emphasizing some of the complications that arise in the general case and explaining what can be said for G2 instantons. Everything I will talk about is joint work with my collaborator Langte Ma.

Classification of 2D Stabilizer States
2025-11-07 12:00 - 13:00 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Bowen Yang
Title: Classification of 2D Stabilizer States
Abstract: I will explain how translation-invariant two-dimensional stabilizer states are completely classified by finite abelian groups with nondegenerate quadratic forms—that is, by abelian anyon theories. The proof uses the algebraic structure of stabilizer codes as modules over Laurent polynomial rings, revealing how their physical features reflect in this module-theoretic framework. arXiv:2509.10418

The Moyal bracket and the BV cohomology of the spinning particle
2025-11-10 15:00 - 16:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Quantum Field Theory and Physical Mathematics Seminar
Speaker: Ezra Getzler, Northwestern
Title: The Moyal bracket and the BV cohomology of the spinning particle
Abstract: The spinning particle is the one-dimensional reduction of the Neveu-Schwartz-Ramond superstring. It consists of a supersymmetric particle moving in a one-dimensional supergravity background, and its quantization is the Hilbert superspace of harmonic spinors. (These models are classified by N, the number of copies of fermionic fields. In this talk, N=1. The extension to N=2 is work in progress with Ivo.) It is actually an AKSZ model (so a generalization of one-dimensional Chern-Simons), and so associated to a differential graded symplectic supermanifold, by which we mean a pair (ω,Q), where ω is a(n exact) symplectic form and Q is an odd function of degree 1. The cohomology of the ring of functions of this supermanifold with differential the Poisson bracket with Q determines the classical BV cohomology of the spinning particle, so is important for understanding perturbative BV quantization of this model. I calculated this cohomology in earlier work for N=1, and showed that it is somewhat bizarre, with two series of cohomology classes in arbitrary negative degrees, each a copy of the functions on the target manifold.
In the study of quantum BFV, we should instead consider the Moyal bracket on the target, and lift Q to an element Q satisfying [Q,Q]=0. The cohomology of the differential [Q,-] is the Moyal cohomology of the differential graded symplectic supermanifold. (This lift corresponds to the choice of a Spinc structure on the target manifold.) In this talk, I prove that the Moyal cohomology, unlike the Poisson cohomology, is well-behaved: in the spectral sequence from Poisson to Moyal cohomology, the extra cohomology classes of negative degree cancel each other pairwise at the E1 page.

Khovanov homology from KLRW algebras
2025-11-11 16:15 - 18:30 | Science Center 507, 1 Oxford Street, Cambridge, 02138
Geometry and Quantum Theory Seminar
Speaker: Sunghyuk Park, CMSA
Title: Khovanov homology from KLRW algebras
Abstract: This is the first in a four-part series, organized together with Vasily Krylov, Sebastian Haney, and Lorenzo Riva, aimed at understanding Aganagic’s categorification of quantum link invariants through Coulomb branches and homological mirror symmetry. In this first talk, I will lay the algebraic background for this story by explaining how Khovanov homology can be understood in terms of higher representation theory via Khovanov-Lauda-Rouquier-Webster (KLRW) algebras.

Geometry Meets Physics: Finiteness, Tameness, and Complexity
2025-11-12 09:00 - 2025-11-14 17:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Geometry Meets Physics: Finiteness, Tameness, and Complexity
Dates: November 12–14, 2025
Location: CMSA G10, 20 Garden Street, Cambridge MA 02138
(note: this event is in-person only)
Finiteness is a fundamental property in consistent physical theories. From the earliest days of quantum field theory and string theory, the drive to eliminate unphysical infinities has been a guiding principle. More recently, finiteness has emerged as a key criterion for constraining effective theories that can be embedded in quantum gravity. Formulating and testing these constraints remains a central challenge in current research.
In parallel, mathematics has made remarkable advanced in addressing finiteness questions using tame geometry. Built on the framework of o-minimal structures, tame geometry offers a precise language for describing objects of finite geometric complexity. Recent developments, such as sharp o-minimality, go further by introducing a quantitative notion of complexity, opening new directions for analyzing finiteness in mathematics and physics alike.
This workshop brings together mathematicians and physicists to exchange ideas, explore new perspectives, and spark collaborations at the interface of geometry, logic, and fundamental physics.
Invited Speakers

Vijay Balasubramanian (UPenn)
Gregorio Baldi (CNRS, IMJ-PRG & IAS)
Gal Binyamini (Weizmann Institute & IAS)
Raf Cluckers (Lille, France)
Matilda Delgado (Max Planck Institute Munich)
Bruno Klingler (Humboldt University, Berlin & IAS)
Adele Padgett (Vienna)
David Prieto (Utrecht)
Washington Taylor (MIT)
David Urbanik (IHES, France & IAS)
Cumrun Vafa (Harvard)
Mick van Vliet (Utrecht)
Benny Zak (Weizmann Institute & IAS)

Organizers: Thomas Grimm, Harvard CMSA & Utrecht University | Gal Binyamini, Weizmann Institute & IAS | Bruno Klingler, Humboldt University, Berlin & IAS

Schedule
(download pdf)
Wednesday Nov. 12, 2025
8:30–8:55 am
Morning refreshments (Common Room)
8:55–9:00 am
Introductions
9:00–10:30 am
Lecture
Speaker: Gal Binyamini, Weizmann Institute & IAS
Title: O-minimality: finiteness and complexity
Abstract: O-minimality is a mathematical formalism of “tame geometry”: a geometry where every set has finite geometric complexity. I will give an introduction to o-minimality in general, and to quantitative variants where one measures the complexity of sets in terms of some natural parameters. I’ll try to focus on the main examples that potentially come up in the interaction with physics, and describe the state of the art and some conjectures.
10:30–11:00 am
Break
11:00 am–12:00 pm
Speaker: Benny Zak, Weizmann Institute & IAS
Title: Analytic tameness – complex cells
Abstract: Complex cells are a complex anayltic version of cells from o-minimality, invented by Binyamini and Novikov. We aim to introduce complex cells, and demonstrate their usefullness in quantifying the analytic information present in a complex set. If time permits, we will discuss applications of this theory.
12:00–1:00 pm
Catered Lunch (Common Room)
1:00–2:30 pm
Lecture
Speakers: David Prieto and Mick van Vliet, Utrecht
Title: Tameness and Complexity in Physical Theories
Abstract: We give an introductory overview of recent applications of o-minimality to physics, focusing on quantum field theories and quantum gravity. In the first part of the lecture we explain how o-minimality makes a first appearance in physical theories when considering amplitudes in quantum field theory. In the second part, we concentrate on a class of theories where finiteness principles seem to be essential, namely the quantum field theories which are consistent with quantum gravity. We review some of these finiteness principles and interpret them through the lens of the o-minimal framework. Along the way, we highlight recent progress in this direction, as well as open questions to explore in the future.
2:30–3:00 pm
Break with refreshments (Common Room)
3:00–4:00 pm
Speaker: Matilda Delgado, Max Planck Institute Munich
Title: Dualities and the Compactifiability of Moduli Space
Abstract: After introducing (self-)dualities in string theory and their action on the field content & spectrum of the theory, I will present the notion of compactifiability for the moduli space of massless fields as the condition that its volume is finite or grows no faster than Euclidean space. I will argue that compactifiability generically implies the existence of non-trivial dualities by providing evidence from string theory. Moreover, I will explain how one can connect compactifiability to the condition that the spectrum of objects charged under the duality group transform in a semisimple representation. Finally, I will provide a bottom-up argument for compactifiability, and argue that it (at least in supersymmetric cases) can be explained by the finiteness of the number of massless states upon compactification to 1D. Based on arXiv:2412.03640.
5:00 PM
Millennium Lecture and Reception: Pierre Deligne (IAS) (Science Center Hall D)
Title: What is the Hodge conjecture?

Thursday, Nov. 13, 2025
8:30–9:00 am
Morning refreshments (Common Room)
9:00–10:30 am
Lecture
Speaker: Bruno Klingler, Humboldt University, Berlin & IAS
Title: Tame geometry and Hodge theory
Abstract: I will give an introduction to applications of o-minimality in complex geometry, in particular in Hodge theory.
10:30–11:00 am
Break
11:00 am–12:00 pm
Speaker: Cumrun Vafa, Harvard
Title: The Swampland Program
12:00–1:30 pm
Catered Lunch (Common Room)
1:30–2:30 pm
Speaker: Gregorio Baldi, CNRS, IMJ-PRG & IAS
Title: The Hodge locus
Abstract: We will survey various recent results around the distribution of the Hodge locus of a (mixed) variation of Hodge structures. Various concrete applications to moduli spaces will also be presented.
2:30–3:00 pm
Break with refreshments (Common Room)
3:00–4:00 pm
Speaker: Vijay Balasubramanian, U Penn
Title: Chaos and complexity in quantum dynamics
4:30–5:30
Discussion/Q&A session
6:30 PM
Dinner: Changsho Restaurant, 1712 Massachusetts Ave., Cambridge, MA 02138

Friday Nov. 14, 2025
8:30–9:00 am
Morning refreshments (Common Room)
9:00–10:00 am
Speaker: Washington Taylor, MIT
Title: Finiteness, connectivity, and the power of fibrations in the Calabi-Yau landscape
10:00–10:30am
Break
10:30–11:30 am
Speaker: Adele Padgett, Vienna
Title: Tameness of multisummable series
Abstract: There are sophisticated theories of summability that map divergent series solutions of differential or functional equations to solutions that are holomorphic in sector-like domains. Van den Dries and Speissegger proved that functions obtained from real multisummable power series have tame geometric behavior when restricted to the real numbers. It would be desirable to know that these functions are also tame on their whole sector-like domains, but recently Speissegger and I proved that these functions are in general only tame on part of their domains. I will present this result and discuss the domains on which some examples are tame, including the Stirling series which appears in the asymptotic expansion of the Gamma function. In this talk, “tame” means definable in an o-minimal structure.
11:30 am–1:00 pm
Catered Lunch (Common Room)
1:00–2:00 pm
Speaker: Raf Cluckers, Lille, France
Title: Finiteness and tameness in (non-archimedean) geometry
Abstract: Non-archimedean geometry work with orders of magnitude rather than with precise measurements. The former works for example with orders of vanishing of functions, and the latter typically works with real or complex numbers. I will discuss recent progress on non-archimedean tame geometry. I will present analogues of o-minimality, of Pila-Wilkie’s o-minimal counting results, and of other finiteness results, in non-archimedean settings.
2:00–2:30 pm
Break with refreshments (Common Room)
2:30–3:30 pm
Speaker: David Urbanik, IHES, France & IAS
Title: Degrees of Hodge Loci

Millennium Prize Problems Lecture - Pierre Deligne: What is the Hodge conjecture?
2025-11-12 17:00 - 18:00 | Harvard Science Center Hall D, 1 Oxford Street, Cambridge, MA, 02138, United States
Date: November 12, 2025
Time: 5:00–6:00 pm
Location: Harvard Science Center Hall D, 1 Oxford St., Cambridge MA
Speaker: Pierre Deligne, Institute for Advanced Study
Title: What is the Hodge conjecture?
Abstract: The Hodge conjecture is about projective non-singular complex algebraic varieties. It characterizes the cohomology classes coming from algebraic cycles. I will explain these terms, tell why the conjecture is so hard to attack, and why we care.

Series Pamphlet (pdf)
Read more about the Hodge Conjecture at the Clay Math website.
Organizers: Martin Bridson, Clay Mathematics Institute | Dan Freed, Harvard University and CMSA | Mike Hopkins, Harvard University

Millennium Prize Problems Lecture Series

Freedman Seminar: Michael Freedman, CMSA & Bowen Yang, CMSA
2025-11-14 13:00 - 16:00 | Virtual
Freedman Seminar
Speaker: Michael Freedman, Harvard CMSA
Title: Sullivan’s work on Lipschitz structures Part II (but self-contained)

Speaker: Bowen Yang, CMSA
Title: Deligne and Sullivan’s work on complex bundles with discrete structure group

Conference on Geometry and Statistics
2025-11-17 09:00 - 2025-11-19 17:00 | CMSA 20 Garden Street Cambridge, Massachusetts 02138 United States
Conference on Geometry and Statistics
Dates: November 17–19, 2025
Location: CMSA G10, 20 Garden Street, Cambridge MA & via Zoom

Speakers

Charles Fefferman, Princeton University
Stephan Huckemann, Georg-August Universität Göttingen
Sungkyu Jung, Seoul National University
Kei Kobayashi, Keio University
Clément Levrard, Université de Rennes
Ker-Chau Li, University of California, Los Angeles
Rong Ma, Harvard University
Steve Marron, University of North Carolina
Ezra Miller, Duke University
Hans-Georg Müller, University of California, Davis
Wilderich Tuschmann, Karlsruhe Institute of Technology
Melanie Weber, Harvard University
Andrew Wood, Australian National University
Horng-Tzer Yau, Harvard University

Organizer: Zhigang Yao, National University of Singapore

Youtube Playlist

SCHEDULE
download pdf
Monday, Nov. 17, 2025
9:00–9:25 am
Morning refreshments
9:25–9:30 am
Introductions
9:30–10:30 am
Speaker: Stephan Huckemann, Georg-August Universität Göttingen
Title: The Probability of the Cut Locus of a Fréchet Mean
Abstract: We show that the cut locus of a Fréchet mean of a random variable on a connected and complete Riemanian manifold has zero probability, a result known previously in special cases (Le and Barden, 2014) and conjectured in general. The proof is based on first order and second order considerations, where the latter are based on a recent result by Générau (2020) on “Laplacians in the barrier sense”. This generalizes to Fréchet p-means for p > 2. The former allow also to rule out stickiness on Riemannian manifolds, and for generalization to 1 <= p < 2, with a conjecture. We close with discussing and conjecturing extensions to noncomplete manifolds and more general metric spaces. This is joint work with Alexander Lytchak.

Générau, F. (2020). Laplacian of the distance function on the cut locus on a Riemannian manifold. Nonlinearity 33(8), 3928.
Le, H. and D. Barden (2014). On the measure of the cut locus of a Fréchet mean. Bulletin of the London Mathematical Society 46(4), 698–708.
Lytchak, A. and S. F. Huckemann (2025). Zero mass at the cut locus of a Fréchet mean on a Riemannian manifold. arXiv preprint arXiv:2508.00747.

10:30–10:45 am
break
10:45 am–11:45 am
Speaker: Hans-Georg Müller, University of California, Davis
Title: Conformal Inference for Random Objects
Abstract: The underlying probability measure of random objects, i.e., metric-space-valued random variables, can be probed by distance profiles. These are one-dimensional distributions of probability mass falling into balls of increasing radius. In a regression setting with Euclidean covariates X and responses Y that are random objects, one can consider conditional Fréchet means that can be implemented with Fréchet regression and also conditional distance profiles, conditioning on X. Conditional distance profiles can then be leveraged to obtain conditional average transport costs, the expected cost for transporting a fixed conditional distance profile to a randomly selected conditional distance profile. The conditional average transport costs can then be utilized to obtain conditional conformity scores. In conjunction with the split conformal algorithm these scores lead to conditional prediction sets located in the object space with asymptotic conditional validity and attractive finite sample behavior. Based on joint work Hang Zhou (UNC).
11:45 am–1:15 pm
Lunch (Catered)
1:15–2:15 pm
Speaker: Horng-Tzer Yau, Harvard
Title: Ramanujan property of random regular graphs and delocalization of random band matrices
Abstract: In this lecture, we review recent works on random matrices. The first result is about the normalized adjacency matrix of a random $d$-regular graph on $N$ vertices with any fixed degree $dgeq 3$ and denote its eigenvalues as $lambda_1=d/sqrt{d-1}geq lambda_2geqlambda_3cdotsgeq lambda_N$. We establish the edge universality for random $d$-regular graphs, namely, the distributions of $lambda_2$ and $-lambda_N$ converge to the Tracy-Widom$_1$ distribution associated with the Gaussian Orthogonal Ensemble. As a consequence, for sufficiently large $N$, approximately $69%$ of $d$-regular graphs on $N$ vertices.
are Ramanujan, meaning $max{lambda_2,|lambda_N|}leq 2$. This resolves a conjecture by Sarnak and Miller-Novikoff-Sabelli
The second result concerns $ N times N$ Hermitian $d$-dimensional random band matrices with band width $W$. In the bulk of the spectrum and in the large $ N $ limit, we prove that all $ L^2 $- normalized eigenvectors are delocalized in all dimensions under suitable conditions on $W$ and $N$. In addition, we proved that the eigenvalue statistics are given by those of the Gaussian unitary ensemble.
2:15–2:45 pm
break with refreshments
2:45–3:45 pm
Speaker: Clément Levrard, Université de Rennes
Title: Optimal reach estimation
Abstract: The reach of an embedded submanifold, a notion that dates back to the famous work Curvature measures of H. Federer, may be understood as a scale under which the submanifold is flat enough so that traditional Euclidean techniques in statistics locally apply, up to some approximation. I will expose several ways to estimate the reach from sample (on the submanifold), some of them being optimal from the point of view of minimax estimation theory. Along the way, intermediate estimation problems of local and global quantities will arise (curvature estimation, weak feature size estimation, distance estimation, etc.), for which various phenomenons can occur from a statistical point of view (different convergence rates, inconsistency). This will be an opportunity to provide a selective overview of the state of the art on these issues.
4:30–5:30 pm
CMSA Colloquium
Speaker: Zhigang Yao (National University of Singapore)
Title: Interaction of Statistics and Geometry: A New Landscape for Data Science
Abstract: Classical statistics views data as real numbers or vectors in Euclidean space, but modern challenges increasingly involve data with intrinsic geometric structures. A central problem in this direction is manifold fitting, with origins in H. Whitney’s work of the 1930s. The Geometric Whitney Problems ask: given a set, when can we construct a smooth 𝑑-dimensional manifold that approximates it, and how accurately can we estimate it?
In this talk, I will discuss recent progress on manifold fitting and its role in bridging geometry and data science. While many existing methods rely on restrictive assumptions, the manifold hypothesis—that data often lie near non-Euclidean structures—remains fundamental in modern statistical learning. I will highlight both theoretical insights and algorithmic challenges, drawing on recent works with, as well as ongoing research.
Youtube video

Tuesday, Nov. 18, 2025
9:00–9:30 am
Morning refreshments
9:30–10:30 am
Speaker: Charles Fefferman, Princeton University (via Zoom)
Title: Extrinsic and intrinsic manifold learning, old and new
Abstract: The talk will include an exposition of the old paper “Testing the manifold hypothesis”, joint work with S. Mitter and H. Narayanan, on extrinsic manifold learning (the manifold to be learned is assumed to be embedded in a high-dimensional Euclidean space). The talk will also include a new result on intrinsic manifold learning (the manifold to be learned is not assumed to be embedded, and the data consist of intrinsic distances corrupted by noise), provided the result is proven by the time of the conference.
10:30–10:45 am
break
10:45 am–11:45 am
Speaker: Steve Marron, University of North Carolina
Title: Data Integration Via Analysis of Manifolds (DIVAM)
Abstract: A major challenge in the age of Big Data is the integration of disparate data types into a single data analysis. That was tackled by Data Integration Via Analysis of Subspaces (DIVAS) in the context of data blocks measured on a common set of experimental cases. Joint variation was defined in terms of modes of variation having identical scores across data blocks. DIVAS allowed mathematically rigorous formulation of individual variation within each data block in terms of individual modes. The goal of DIVAM is to intrinsically extend the DIVAS approach to data objects lying in manifolds, such as shape data.
11:45 am–1:15 pm
Lunch Break
1:15–2:15 pm
Speaker: Ker-Chau Li, University of California, Los Angeles
Title: Investigation of Data clouds: From Galton’s Ellipses to Explainable AI (XAI), modeling or molding?
Abstract: Francis Galton’s seminal 1886 visualization of regression toward the mean in trait inheritance is arguably the first and most influential example of geometric thinking applied to statistical modeling. The pioneering geometric insight driving Galton’s use of elliptical contours to discover the bivariate normal distribution laid down the foundation for classic multivariate analysis (e.g., PCA, canonical correlation) and profoundly impacts modern methods like diffusion models.
Statistical models, particularly those based on parsimony, are effective for characterizing data distribution and facilitating scientific rule induction. However, the rise of unstructured big data (like images) has challenged these parsimonious approaches, necessitating the use of deep learning models. These models, containing billions of parameters, sacrifice transparency to excel in prediction. Seeking solutions to this “black-box” dilemma is now the heart of Explainable AI (XAI).
Leveraging the simplicity of elementary geometric concepts, this talk will present a new path toward interpretable and parsimonious XAI. Unstructured big data is highly plastic. Our approach moves beyond the standard data modeling perspective—which answers what the data is—and introduces a novel data molding perspective. This shift is key to unlocking the full potential of data’s plasticity, allowing us to effectively answer the crucial question: what the data can be used for.
I will first discuss a connection between manifold learning and my earlier works, helical confounding and liquid association. I will then turn to the data molding perspective and present two novel notions: mold-compliance and artificial-trait configurative-generation (ATCG). These notions guide our recent efforts in formulating novel algorithms for image data investigation, addressing issues like prediction validity and within-class heterogeneity. Data molding entails a dramatically different feature space extraction, which consequently shifts the subsequent investigation on the data clouds from out-of-distribution (OOD) to mold-violation, and from UMAP clustering to ATCG-induced hierarchical clustering.
2:15–2:45 pm
break with refreshments
2:45–3:45 pm
Speaker: Andrew Wood, Australian National University
Title: Empirical likelihood methods for Fréchet means on open books
Abstract: The open book is a simple example of a stratified space that captures some (but not all) of the properties of stratified spaces. Central limit theory for open books plus relevant background is given by Hotz et al. (2013, Annals of Applied Probability). In this talk I will describe some basic inference procedures for Fréchet means in open books based on empirical likelihood (Owen, book, 2001). Empirical likelihood (EL) is a type of nonparametric likelihood that can be useful for many types of data, including manifold-valued data and data from stratified spaces. An EL approach to basic inference for Fréchet means will be described. In particular, it will be shown how the non-regularity in the geometry of open books can result in non-regular behaviour in Wilks’s theorem (i.e. the large sample likelihood ratio test). The talk will also discuss difficulties in extending the EL inference theory from open books to more general stratified spaces, where the difference in dimension of adjacent strata can be 2 or more. For discussion of more general stratified spaces than open books, see the orthant spaces discussed in Barden and Le (2018, Proc of London Math Society) and the general stratified space setting considered by Mattingly et al. (2023, arxiv).
3:45–4:00 pm
break
4:00–5:00 pm
Speaker: Wilderich Tuschmann, Karlsruhe Institute of Technology
Title: A Spectator’s Perspective on the Manifold Hypothesis
Abstract: At its core, the Manifold Hypothesis asserts that real-world, high-dimensional data is not uniformly or randomly distributed throughout its high-dimensional “ambient” space, but concentrated on or near a low-dimensional manifold (or a collection of manifolds) embedded within that high-dimensional ambient space.
In my talk, I will discuss reasons and facts that speak for as well as against this hypothesis and also address geometric alternatives.

Wednesday, Nov. 19, 2025
9:00–9:30 am
Morning refreshments
9:30–10:30 am
Speaker: Melanie Weber, Harvard University
Title: Ricci Curvature, Ricci Flow, and the Geometry of Learning
Abstract: Geometric structure in data plays a crucial role in machine learning. In this talk, we study this observation through the lens of Ricci curvature and its associated Ricci flow. We start by reviewing a discrete notion of Ricci curvature introduced by Ollivier and the geometric flow that it induces. We further discuss the relationship between discrete Ricci curvature and its continuous counterpart via discrete-to-continuum consistency results, which imply that discrete Ricci curvature can provably characterize the geometry of a data manifold based on a finite sample. This provides a theoretical foundation for several applications of discrete Ricci curvature in machine learning, two of which we discuss in the remainder of this talk. First, we analyze learned feature representations in deep neural networks and show that they transform during training in ways that closely resemble a discrete Ricci flow. Our analysis reveals that nonlinear activations shape class separability and suggests geometry-informed training principles such as early stopping and depth selection. Second, we turn to deep learning on graphs, where we address representational limitations of state of the art graph neural networks through curvature-based data augmentations. We show that augmenting input graphs with geometric information provably increases the representational power of such models and yields performance gains in practice.
10:30–10:45 am
break
10:45 am–11:45 am
Speaker: Ezra Miller, Duke University
Title: Extracting bar lengths from multiparameter persistent homology
Abstract: Persistent homology in one parameter can be summarized using bar codes or persistence diagrams, which are elementary gadgets with many features amenable to vectorization and hence statistical analysis. For example, early work with Bendich, Marron, Pieloch, and Skwerer showed how to extract meaningful statistics from the top 100 bar lengths in persistent homology summaries of brain arteries. The story for persistent homology with multiple parameters, on the other hand, is still developing. Although it has the potential to be much more flexible and informative, multipersistence has structural issues that present fundamental mathematical challenges. There is no consensus on what might be meant by a “bar”, let alone “the top 100 bar lengths”. This talk recalls the basics of single and multiparameter persistent homology and discusses some of the mathematical issues, including obstacles and potential routes forward.
11:45 am–1:15 pm
Lunch Break
1:15–2:15 pm
Speaker: Kei Kobayashi, Keio University
Title: Metric Transformations of Data Spaces: Curvature Control and Related Developments
Abstract: We present our proposed method of increasing the accuracy of data analysis by means of two transformations of the metric of the data space. The first transformation is based on the curve length defined by the integral of the power of the density function, which can be computed approximately using an empirical graph; the second transformation can be interpreted as the extrinsic distance when the data space is embedded in a metric cone. The advantage of both distance transformations is that the hyperparameters allow the curvature to be monotonically transformed in a specific sense. Some statistical applications of these transformations and theoretical justifications are presented. Detailed analyses of the geodesics obtained by this method for several simple probability distributions will also be presented. The main part of this work is based on joint works with Henry P. Wynn.
2:15–2:45 pm
break with refreshments
2:45–3:45 pm
Speaker: Sungkyu Jung, Seoul National University
Title: Generalized Frechet means with random minimizing domains and its strong consistency
Abstract: In this talk, I will discuss a novel extension of Frechet means, referred to as generalized Frechet means, as a comprehensive framework for describing the characteristics of random elements. The generalized Frechet mean is defined as the minimizer of a cost function, and the framework encompasses various extensions of Frechet means that have appeared in the literature. The most distinctive feature of the proposed framework is that it allows the domain of minimization for the empirical generalized Frechet means to be random and different from that of its population counterpart. This flexibility broadens the applicability of the Frechet mean framework to various statistical scenarios, including sequential dimension reduction for non-Euclidean data. We establish a strong consistency theorem for generalized Frechet means. Applications such as verifying the consistency of principal geodesic analysis on the hypersphere, compositional principal component analysis on the composition space, and k-medoids clustering for data on a metric space will be discussed.
3:45–4:00 pm
break
4:00–5:00 pm
Speaker: Rong Ma, Harvard University
Title: Modern Nonlinear Embedding Methods Unpacked
Abstract: Learning and representing low-dimensional structures from noisy, high-dimensional data is a cornerstone of modern data science. Stochastic neighbor embedding algorithms, a family of nonlinear dimensionality reduction and data visualization methods, with t-SNE and UMAP as two leading examples, have become very popular in recent years. Yet despite their wide applications, these methods remain subject to points of debate, including limited theoretical understanding, ambiguous interpretations, and sensitivity to tuning parameters. In this talk, I will present our recent efforts to decipher and improve these nonlinear embedding approaches. Our key results include a rigorous theoretical framework that uncovers the intrinsic mechanisms, large-sample limits, and fundamental principles underlying these algorithms; a set of theory-informed practical guidelines for their principled use in trustworthy biological discovery; and a collection of new algorithms that address current limitations and improve performance in areas such as bias reduction and stability. Throughout the talk, I will highlight how these advances not only deepen our theoretical understanding but also open new avenues for scientific discovery.

BV and the ThimTFT
2025-11-17 15:00 - 16:00 | CMSA Room G02, 20 Garden Street, Cambridge, MA, 02138
Quantum Field Theory and Physical Mathematics Seminar
Speaker: Justin Kulp, Stony Brook
Title: BV and the ThimTFT
Abstract: The SymTFT (or “quiche”) construction relates different global forms of d-dimensional QFTs with discrete symmetry: realizing different global forms as a (d+1)-dimensional TFT on an interval, with a common “physical” boundary condition on one side, and different topological boundary conditions on the other. In my talk, I will describe an analogue of the SymTFT which relates theories with the same “perturbative equations of motion”, but different non-perturbative completions.
I will start with a lightning overview of conformal blocks and relative QFT, then explain the BV formalism in some detail—focusing on the elegant (super)geometric story in 0d for simplicity. I will argue that there is a natural 1d cohomological TFT (called the ThimTFT) associated to the classical action S, with different topological boundary conditions described by convergent path-integral contours in a complexified field space. Time permitting, I will discuss extensions to higher dimensions. Based on WIP.

Interaction of Statistics and Geometry: A New Landscape for Data Science
2025-11-17 16:30 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Colloquium
Speaker: Zhigang Yao (National University of Singapore)
Title: Interaction of Statistics and Geometry: A New Landscape for Data Science
Abstract: Classical statistics views data as real numbers or vectors in Euclidean space, but modern challenges increasingly involve data with intrinsic geometric structures. A central problem in this direction is manifold fitting, with origins in H. Whitney’s work of the 1930s. The Geometric Whitney Problems ask: given a set, when can we construct a smooth 𝑑-dimensional manifold that approximates it, and how accurately can we estimate it?
In this talk, I will discuss recent progress on manifold fitting and its role in bridging geometry and data science. While many existing methods rely on restrictive assumptions, the manifold hypothesis—that data often lie near non-Euclidean structures—remains fundamental in modern statistical learning. I will highlight both theoretical insights and algorithmic challenges, drawing on recent works with, as well as ongoing research.

Differentials and Singularities
2025-11-20 14:00 - 15:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Algebra Seminar
Speaker: Dawei Chen, Boston College
Title: Differentials and Singularities
Abstract: Given a holomorphic differential on a smooth algebraic curve, we associate to it a Gorenstein curve singularity with Gm-action. Conversely, we show that every isolated Gorenstein curve singularity with Gm-action appears in this way. This construction reveals a fascinating relation between differentials and singularities, where the zero orders of the differentials determine the combinatorial data of the singularities. In this talk, I’ll provide many concrete examples of such singularities, and explain how the study of deformations of these singularities can help us better understand the geometry of moduli spaces of differentials. This is based on joint work with Fei Yu (Zhejiang University).

Tropical-Topological(Tropological) Sigma Models
2025-11-20 16:00 - 17:00 | Virtual
Differential Geometry and Physics Seminar
Speaker: Andrés Franco Valiente, UC Berkeley
Title: Tropical-Topological (Tropological) Sigma Models
Abstract: Tropical geometry provides a powerful bridge between complex and combinatorial worlds, allowing certain curve-counting invariants to be computed in a piecewise-linear “tropical” limit. Building on Mikhalkin’s insight that Gromov–Witten invariants can be recovered from tropical curves, this talk revisits Mikhalkin’s result from the viewpoint of topological field theory and functional integration. I will first review the topological sigma model and explain how the localization equations admit a natural notion of tropicalization which allows us to reproduce the Gromov-Witten invariants using standard cohomological BRST methods without having to reformulate the functional integral in terms of the tropical semifield. We find that the relevant geometries associated to the tropical limit of the topological sigma models no longer require a complex structure but they are instead based on nilpotent structures on singular foliated manifolds. We close with a discussion on recent progress on how the tropological sigma model has a close connection to Hořava’s topologicial quantum gravity for Ricci Flow in a joint work with Emil Albrychiewicz and Viola Zixin Zhao.

Optimal learning protocols via statistical physics and control theory
2025-11-21 12:00 - 13:00 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Francesco Mori, CMSA
Title: Optimal learning protocols via statistical physics and control theory
Abstract: Behind the impressive performance of modern machine learning lies a toolkit of training tricks, from tuning learning rates to curating training data. These heuristics are powerful but hard to interpret and possibly suboptimal, leaving open the challenge of finding general principles for protocol design. In this talk, I will present a framework that combines tools from statistical physics and control theory to identify optimal training strategies in simple yet insightful neural network models. In the high-dimensional limit, the training dynamics can be reduced to closed-form ordinary differential equations for a small set of order parameters that track learning. This reduction allows us to pose the design of training protocols as an optimal control problem directly on the order-parameter dynamics, with the objective of minimizing the generalization error. This formulation encompasses a variety of learning scenarios and yields principled training strategies that clarify, and in some cases improve upon, standard heuristic practices.

Geometric Simplicity in Quantum Field Theory and Gravity
2025-11-24 16:30 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Colloquium
Speaker: Thomas Grimm, Utrecht University
Title: Geometric Simplicity in Quantum Field Theory and Gravity
Abstract: In physics we attribute much value to the emergence of simplicity, both conceptually and for computations. Familiar examples include algebraic relations among Feynman amplitudes, the surprising descriptions arising in large-N or duality limits, and the central role played by symmetries. In this colloquium we discuss how tame geometry allows one to quantitatively describe such simplifications by introducing a measure of complexity. This framework relies on finiteness: the information content of the functions and domains required to specify a theory, or an observable is finite. A key strength of the proposal is its generality as it applies to any physical quantity and can therefore be used both to analyze complexities within an individual Quantum Field Theory and to study the entire space of such theories. We present several applications and explain how this perspective ties in with our understanding of the expected properties of effective theories that can be coupled to Quantum Gravity.

Coulomb branches and KLRW algebras
2025-11-25 16:15 - 18:30 | Science Center 507, 1 Oxford Street, Cambridge, 02138
Geometry and Quantum Theory Seminar
Speaker:Vasily Krylov, CMSA
Title: Coulomb branches and KLRW algebras
Abstract: I will introduce Coulomb branches associated to a pair of a reductive group G and its complex representation N. We will discuss their main geometric properties and examine explicit examples. I will also highlight the connection to the moduli space of monopoles. Time permitting, we will see how KLRW naturally arise in this context.

Asymptotic Theory of Attention: In-Context Learning and Sparse Token Detection
2025-12-01 16:30 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Colloquium
Speaker: Yue M. Lu, Harvard University
Title: Asymptotic Theory of Attention: In-Context Learning and Sparse Token Detection
Abstract: Attention-based architectures exhibit striking emergent abilities—from learning tasks directly from context to detecting rare, weak features in long sequences—yet a rigorous theory explaining these behaviors remains limited. In this talk, I will present two recent exactly solvable models that develop a high-dimensional asymptotic theory of attention.
(i) In-context learning. For linear attention pretrained on linear regression tasks, we derive sharp asymptotics in a regime where token dimension, context length, and task diversity all scale proportionally, while the number of pretraining examples scales quadratically. The resulting learning curve exhibits double descent and a phase transition separating a low-diversity memorization regime from a high-diversity regime of genuine in-context generalization. These predictions closely track empirical behavior in both linear-attention models and nonlinear Transformer architectures.
(ii) Sparse-token classification. For detecting weak signals embedded in a small, randomly located subset of tokens, we analyze a single-layer attention classifier and determine its representational and learnability thresholds. Attention succeeds with only logarithmic signal scaling in the sequence length L, outperforming linear baselines that require √L scaling. In a proportional high-dimensional regime, we prove that two gradient descent steps yield nontrivial alignment between the query vector and the hidden signal, leading to signal-adaptive attention. Exact formulas for the test error, training loss, and separability capacity quantify this advantage.

Homological mirror symmetry for Coulomb branches
2025-12-02 16:15 - 18:30 | Science Center 507, 1 Oxford Street, Cambridge, 02138
Geometry and Quantum Theory Seminar
Speaker: Sebastian Haney, Harvard
Title: Homological mirror symmetry for Coulomb branches
Abstract: I will describe a result of Aganagic, Danilenko, Li, Shende, and Zhou which constructs a embeddings of certain cylindrical KLRW categories into Fukaya-Seidel categories of multiplicative Coulomb branches. This can be thought of as a homological mirror symmetry statement relating the Fukaya category of a multiplicative Coulomb branch to the derived category of a resolved additive Coulomb branch. I will describe the construction of the relevant Fukaya–Seidel categories, and explain how the KRLW relations are realized by counts of holomorphic disks in symmetric products of surfaces.

Machine learning tools for mathematical discovery
2025-12-03 14:00 - 15:00 | Virtual
New Technologies in Mathematics Seminar
Speaker: Adam Zsolt Wagner, Google DeepMind
Title: Machine learning tools for mathematical discovery
Abstract: I will discuss various ML tools we can use today to try to find interesting constructions to various mathematical problems. I will briefly mention simple reinforcement learning setups and PatternBoost, but the talk will mainly focus on LLM-based tools such as FunSearch and AlphaEvolve. We will discuss the pros and cons of several of these methods, and try to figure out which one is best for the problems we care about.
Joint work with François Charton, Jordan Ellenberg, Bogdan Georgiev, Javier Gómez-Serrano, Terence Tao, and Geordie Williamson.

Millennium Prize Problems Lecture - Madhu Sudan: P vs NP Problem
2025-12-03 17:00 - 18:00 | Harvard Science Center Hall D, 1 Oxford Street, Cambridge, MA, 02138, United States
Pamphlet (pdf)
Slides (pdf)
Date: December 3, 2025
Time: 5:00–6:00 pm
Location: Harvard Science Center Hall D, 1 Oxford St., Cambridge MA
Speaker: Madhu Sudan, Harvard University
Title: The P vs. NP problem: An Existential Question for Mathematics
At the beginning of the twentieth century, in response to questions raised by Hilbert, illustrious mathematicians such as Godel, Church and Turing formalized the notion of theorems and proofs. Proofs were automatically verifiable while theorems are logical propositions for which proofs exist. The formal definition of a computer, a definition that had strong influence on the later development of the technology, was a by-product of the effort to define the phrase “automatically verifiable”!
While the resulting theory had major implications already, one notion was however missing in the early definitions. Proofs were meant to be easily verifiable, while determining the truth of a proposition/conjecture (arguably a core task of mathematics) was not necessarily so. But what is “easiness” and how is it to be defined? While this was already hinted at by Godel in the 50s, the notion was finally formalized in seminal works of Cook, Levin and Karp in the early 70s. Central notions here included the adoption of the notion that polynomial time algorithms are (the only) tractable ones, and the realization that algorithms seeking to remove the existential quantifier in the definition of a “theorem” lead naively to exponential time algorithms. But are there no sophisticated algorithms to search for proofs? This is the profound “Is P = NP?” question.
In this talk we will introduce the question and explain implications of resolutions of this question to the modern computing infrastructure, to mathematics and other sciences. We will briefly describe the state of progress on this question and recent progress on weaker forms of this question. Finally we will also aim to connect this question, and why one may believe that P != NP (proof search can not be automated) even in the face of accumulating evidence on the ability of computers to solve more and more complex mathematical problems, which seem to implement brute force search in less than polynomial time.

Read more about the P vs NP Problem at the Clay Math website.

Organizers: Martin Bridson, Clay Mathematics Institute | Dan Freed, Harvard University and CMSA | Mike Hopkins, Harvard University

Millennium Prize Problems Lecture Series

Towards a Dolbeault AGT correspondence
2025-12-04 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Differential Geometry and Physics Seminar
Speaker: Surya Raghavendran, Yale
Title: Towards a Dolbeault AGT correspondence
Abstract: The AGT correspondence and its extensions propose geometric constructions of vertex algebras and their modules from the cohomology of various moduli spaces of sheaves on surfaces. Physically, the correspondence is illuminated throgh the holomorphic–topological twist of the six-dimensional N=(2,0) superconformal field theories. In this talk, I will describe a variant of AGT arising instead from the so-called minimal twist of these theories. In this setting, the natural algebraic structures are holomorphic factorization algebras in three complex dimensions. From these, one can extract an associative algebra together with a natural module, which we conjecture to coincide with a quantization of the moduli of Higgs sheaves on surfaces. In examples, this pair is furthermore expected to admit a Hodge–de Rham deformation to the Heisenberg algebra and its action on the cohomology of Hilbert schemes of surfaces, as constructed by Grojnowski and Nakajima.

A combinatorial formula for interpolation Macdonald polynomials
2025-12-05 12:00 - 13:00 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Houcine Ben Dali, Harvard CMSA
Title: A combinatorial formula for interpolation Macdonald polynomials
Abstract: In 1996, Knop and Sahi introduced a remarkable family of inhomogeneous symmetric polynomials, defined via vanishing conditions, whose top homogeneous parts are exactly the Macdonald polynomials. Like the Macdonald polynomials, these interpolation Macdonald polynomials are closely connected to the Hecke algebra, and admit nonsymmetric versions, which generalize the nonsymmetric Macdonald polynomials. I will present a combinatorial formula for interpolation Macdonald polynomials in terms of signed multiline queues. This formula generalizes the combinatorial formula for Macdonald polynomials in terms of multiline queues given by Corteel–Mandelshtam–Williams. This is based on a joint work with Lauren Williams.

Computing WKB periods
2025-12-08 15:00 - 16:00 | CMSA Room G02, 20 Garden Street, Cambridge, MA, 02138
Quantum Field Theory and Physical Mathematics Seminar
Speaker: Max Meynig, University of Connecticut
Title: Computing WKB periods
Abstract: In one dimensional quantum mechanics, the all-orders WKB method leads to ‘quantum periods’ which are formal power series in hbar whose coefficients are certain period integrals. These periods, which limelight in supersymmetric/string theories, have rich structure and can be computed in a number of ways. I will discuss a new perspective on them and their computation.

Recent Advances in Probabilistically Checkable Proofs
2025-12-08 16:30 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Colloquium
Speaker: Dor Minzer (MIT)
Title: Recent Advances in Probabilistically Checkable Proofs
Abstract: The PCP Theorem is a cornerstone of computer science, with applications to hardness of approximation, verification, interactive protocols and more. It asserts a witness for the satisfiability of a given 3CNF formula can be encoded in a robust way that allows local checking.In this talk we discuss recent developments in PCPs, and their connection with distributed protocols, high-dimensional expanders and discrete Fourier analysis. Based on joint works with Kai Zhe Zheng, Mitali Bafna, Noam Lifshitz, Nikhil Vyas.

Geometry and Quantum Theory Seminar
2025-12-09 16:15 - 18:30 | Science Center 507, 1 Oxford Street, Cambridge, 02138
Geometry and Quantum Theory Seminar
Speaker: Lorenzo Riva, CMSA
Title: Aganagic’s invariant is Khovanov homology
Abstract: Webster computed the Khovanov homology of (the closure of) a braid in terms of the action of that braid on a certain KLRW category. Aganagic proposed that the same computation could be done in the Fukaya-Seidel category of the multiplicative Coulomb branch associated to a weighted quiver. In this talk we will recap the story so far and then sketch LePage and Shende’s proof of Aganagic’s proposal.

Speaker: Bowen Yang, CMSA
Title: Some groups from Condensed matter physics
Abstract: I would like to talk about some groups coming from the study of quantum spin systems. They inspired a construction of generalized homology theories related to Azumaya algebras.

CMSA Q&A Seminar: Dan Freed, CMSA
2025-12-10 12:00 - 13:00 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA Q&A Seminar
Speaker: Dan Freed, CMSA
Topic: Constructions of homotopy types in geometry and physics

Covers of curves, Ceresa cycles, and unlikely intersections
2025-12-11 14:00 - 15:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Algebra Seminar
Speaker: Padamavathi Srinivasan, Boston University
Title: Covers of curves, Ceresa cycles, and unlikely intersections
Abstract: The Ceresa cycle is a canonical homologically trivial algebraic cycle associated to a curve in its Jacobian. In his 1983 thesis, Ceresa showed that this cycle is algebraically nontrivial for a very general complex curve of genus at least 3. In the last few years, there have been many new results shedding light on the locus in the moduli space of genus g curves where the Ceresa cycle becomes torsion. We will survey these recent results and provide new examples of positive dimensional families of curves where only finitely many members of the family have torsion Ceresa cycle. The main idea is to study covers of curves with many automorphisms, and we will explain how we use the covering maps together with results on unlikely intersections in abelian varieties to construct such families. This is joint work with Tejasi Bhatnagar, Sheela Devadas and Toren D’Nelly Warady.

The active Young-Dupré equation
2025-12-15 16:30 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Colloquium
Speaker: Julien Tailleur, MIT
Title: The active Young-Dupré equation
Abstract: The Young-Dupré equation is a cornerstone of the equilibrium theory of capillary and wetting phenomena. In the biological world, interfacial phenomena are ubiquitous, from the spreading of bacterial colonies to tissue growth and flocking of birds, but the description of such active systems escapes the realm of equilibrium physics. I will show how a microscopic, mechanical definition of surface tension allows building an Active Young-Dupré equation able to account for the partial wetting observed in simulations of active particles interacting via pairwise forces. Remarkably, the equation shows that the corresponding steady interfaces do not result from a simple balance between the surface tensions at play but instead emerge from a complex feedback mechanism. The interfaces are indeed stabilized by a drag force due to the emergence of steady currents, which are themselves a by-product of the symmetry breaking induced by the interfaces. These currents also lead to new physics by selecting the sizes and shapes of adsorbed droplets, breaking the equilibrium scale-free nature of the problem. Finally, I will demonstrate a spectacular consequence of the negative value of the liquid-gas surface tensions in systems undergoing motility-induced phase separation: partially-immersed objects are expelled from the liquid phase, in stark contrast with what is observed in passive systems. These results lay the foundations for a theory of wetting in active systems.

Electrical networks, Grassmannians, and cluster algebras
2025-12-16 14:00 - 15:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Algebra Seminar
Speaker: Lazar Guterman, Hebrew University of Jerusalem
Title: Electrical networks, Grassmannians, and cluster algebras
Abstract: An electrical network with $n$ boundary vertices induces a matrix called the response matrix which measures the electrical properties of the network. The set of response matrices of all electrical networks has a characterization in terms of positivity of circular minors. Alman, Lian and Tran constructed a cluster algebra on the set of circular minors, which encodes the tests for positivity of these minors. Lam established the embedding of the set of electrical networks with $n$ boundary vertices into the totally nonnegative Grassmannian $Gr_{ge0}(n-1,2n)$. The coordinate ring of the Grassmannian has a cluster algebra structure as was proved by Scott. Given an electrical network, we find a relation between circular minors of its response matrix and Plücker coordinates of its image in the Grassmannian. Using this property, we prove that for an odd $n$ the two cluster algebras, on circular minors and on the Grassmanian, become isomorphic after a natural freezing and subsequent trivialization of certain variables in their initial seeds. We apply this isomorphism in order to relate the tests for positivity of circular minors to tests for positivity in the Grassmannian. The talk is based on a joint work with Boris Bychkov and Anton Kazakov.

Geometry and Quantum Theory Seminar
2025-12-16 16:15 - 18:30 | Science Center 507, 1 Oxford Street, Cambridge, 02138
Geometry and Quantum Theory Seminar
Speaker: tba

The Quasi-Adiabatic Theorem and All That
2026-01-27 16:15 - 18:30 | Science Center 507, 1 Oxford Street, Cambridge, 02138
Geometry and Quantum Theory Seminar
Speaker: Daniel Spiegel, Harvard
Title: The Quasi-Adiabatic Theorem and All That
Abstract: Yosuke Kubota has recently made progress on understanding Kitaev’s conjecture by constructing a loop spectrum consisting of spaces of quantum spin systems, indexed by spatial dimension of the lattice (arXiv: 2503.12618). After a brief reminder on the C*-algebraic formalism of quantum spin systems, I will discuss Section 2 of this paper, which covers some of the more analytical tools used to construct the loop spectrum. In particular, I will focus on the quasi-adiabatic theorem which roughly speaking states that a smooth path of gapped Hamiltonians with unique ground states gives rise to a path of nice automorphisms that map the ground state at time zero to the path of ground states.
