Seminars and Colloquia by Series

Breaking the Curse of Dimensionality: Graphs, Probability Measures, and Data

Series
Applied and Computational Mathematics Seminar
Time
Friday, March 13, 2026 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005 and https://gatech.zoom.us/j/94954654170
Speaker
James MurphyTufts University

The curse of dimensionality renders statistical and machine learning in high dimensions intractable without additional assumptions on the underlying data.  We consider geometric models for data that allow for mathematical performance guarantees and efficient algorithms that break the curse.  The first part of the talk develops a family of data-driven metrics that balance between density and geometry in the underlying data.  We consider discrete graph operators based on these metrics, and prove performance guarantees for clustering with them in the spectral graph paradigm.  Fast algorithms based on Euclidean nearest-neighbor graphs are proposed and connections with continuum operators on manifolds are developed. 
 
In the second part of the talk, we move away from Euclidean spaces and focus on representation learning of probability measures in Wasserstein space.  We introduce a general barycentric coding model in which data are represented as Wasserstein barycenters of a set of fixed reference measures.  Leveraging the geometry of Wasserstein space, we develop a tractable optimization program to learn the barycentric coordinates when given access to the densities of the underlying measures.  We provide a consistent statistical procedure for learning these coordinates when the measures are accessed only by i.i.d. samples.  Our consistency results and algorithms exploit entropic regularization of optimal transport maps, thereby allowing our barycentric modeling approach to scale efficiently.  Extensions to learning suitable reference measures and linearizations of our barycentric coding model will be discussed.  Throughout the talk, applications to synthetic and real data demonstrate the efficacy of our methods.

Joint parameter estimation of spin glasses

Series
Stochastics Seminar
Time
Thursday, March 12, 2026 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Qiang WuUniversity of Minnesota

Spin glasses are disordered statistical physics system with both ferromagnetic and anti-ferromagnetic spin interactions. The Gibbs measure belongs to the exponential family with parameters, such as inverse temperature $\beta>0$ and external field $h\in R$.  A fundamental statistical problem is to estimate the system parameters from a single sample of the ground truth. In 2007, Chatterjee first proved that under reasonable conditions, for spin glass models with $h=0$, the maximum pseudo-likelihood estimator for $\beta$ is $\sqrt{N}$-consistent. This is in contrast to the existing estimation results for classical non-disordered models. However, Chatterjee's approach has been restricted to the single parameter estimation setting.  The joint parameter estimation of $(\beta,h)$ for spin glasses has remained open since then. In this talk, I will introduce a new idea to show that under some easily verifiable conditions,  the bi-variate maximum pseudo-likelihood estimator is jointly $\sqrt{N}$-consistent for a large collection of spin glasses, including the Sherrington-Kirkpatrick model and its diluted variants. Based on joint work with Wei-Kuo Chen, Arnab Sen. 

Graph homomorphism inequalities

Series
School of Mathematics Colloquium
Time
Thursday, March 12, 2026 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
David ConlonCaltech

We discuss recent progress on the study of a range of problems relating to homomorphism counts in graphs. Perhaps the most celebrated such problem is Sidorenko's conjecture, which says that the number of copies of any fixed bipartite graph in another graph of given density is asymptotically minimised by the random graph. As well as talking about some of the recent results on this conjecture, we will touch on the positive graph conjecture and the study of norming graphs. If time permits, we will also say a little about similar problems in an arithmetic context.

Smooth forms on graphs and Berkovich curves

Series
Number Theory
Time
Wednesday, March 11, 2026 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Joe RabinoffDuke University

Chambert-Loir and Ducros have introduced a theory of real-valued smooth differential forms on Berkovich spaces that play the role of smooth forms on complex varieties.  We compute the associated Dolbeault cohomology groups of curves by reducing to the case of metric graphs.  I'll introduce smooth forms on graphs, and explain how the theory in CLD has to be modified in order to get finite-dimensional cohomology groups.

Dispersive estimates for 1D matrix Schrödinger operators with threshold resonance

Series
PDE Seminar
Time
Tuesday, March 10, 2026 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 254
Speaker
Yongming LiTexas A&M University

In this talk, we will discuss dispersive and local decay estimates for a class of non-self-adjoint Schrödinger operators that naturally arise from the linearization of nonlinear Schrödinger equations around a solitary wave. We review the spectral properties of these linearized operators, and discuss how threshold resonances may appear in their spectrum. In the presence of threshold resonances, it will be shown that the slow local decay rate can be pinned down to a finite rank operator corresponding to the threshold resonances. We will also discuss examples of non-self-adjoint operators that arise from linearizing around solitons in other contexts. 

Graham's Conjecture and Rainbow Paths in Graphs

Series
Graph Theory Seminar
Time
Tuesday, March 10, 2026 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Chase WilsonUCSD

We discuss the recent progress on Graham's Conjecture which states that for any subset $S \subseteq \mathbb{Z}_p \setminus \{0\}$, there exists an ordering of the elements $s_1, \cdots, s_m$ of $S$ such that the partial sums $\sum_{i = 1}^k s_i$ are all distinct. This was very recently proven for all sufficiently large primes by Pham and Sauermann, however our work focuses on the more general setting where $\mathbb{Z}_p$ is replaced by an arbitrary finite group, where the result is also conjectured to hold.

By considering the Cayley Graph, we can translate the problem into the purely graph theoretic problem of finding a rainbow path of length $d - 1$ in any $d$-regular properly edge-colored directed graph. We give an asymptotic result which builds on work by Bucić, Frederickson, Müyesser, Pokrovskiy, and Yepremyan, and shows that we can find a path of length $(1 - o(1)) d$. This corresponds to showing that for any subset $S \subseteq G$, there exists a dense subset $S' \subseteq G$ and an ordering $s'_1, \cdots, s'_m$ of the elements of $S'$ such that the partial products $\prod_{i  = 1}^k s'_i$ are all distinct.

The Hitchin fibration and its cohomology

Series
Representation Theory, Moduli, and Physics Seminar
Time
Tuesday, March 10, 2026 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Andres Fernandez HerreroUniversity of Pennsylvania

The moduli space of Higgs bundles lies at the crossroads of different areas of mathematics. Its cohomology plays a central role in Ngo's proof of the fundamental lemma of the Langlands program, and it is the subject of recent results such as topological mirror symmetry and the P=W conjecture. Even though these developments seem unrelated, they all ultimately rely on a (partial) understanding of the Decomposition Theorem for the associated Hitchin fibration. In this talk, I will report on a complete and uniform description of the Decomposition Theorem in the logarithmic case, fully generalizing Ngo's results beyond the elliptic locus. This is joint work in progress with Mark de Cataldo, Roberto Fringuelli, and Mirko Mauri.

Geometrizing Surface Group Representations

Series
Geometry Topology Seminar
Time
Monday, March 9, 2026 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Parker EvansWashington University of St. Louis

Abstract: The Teichmuller space T(S) of a closed surface S is a moduli space where each point represents a hyperbolic metric on the surface S. Interpreted appropriately, each of these hyperbolic metrics is encoded by a representation of the fundamental group of S to PSL(2,R), the group of isometries of the hyperbolic plane. This talk concerns a similar story with the Lie group PSL(2,R) replaced by the exceptional split real Lie group G2’ of type G2. That is, we shall “geometrize” surface group representations to G2’ as holonomies of some (explicitly constructed) locally homogenous (G,X)-manifolds. Along the way, we encounter pseudoholomorphic curves in a non-compact pseudosphere that carry a (T,N,B)-framing analogous to that of space curves in Euclidean 3-space. These curves play a key role in the construction. Time permitting, we discuss how this specific G_2’ recipe relates to a broader construction that unifies other approaches to geometrize representations in rank two. This talk concerns joint work with Colin Davalo.

Nonlocal Attention Operator: Understanding Attention Mechanism for Physical Responses

Series
Applied and Computational Mathematics Seminar
Time
Monday, March 9, 2026 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005 and https://gatech.zoom.us/j/94954654170
Speaker
Yue YuLehigh University

While foundation models have gained considerable attention in core AI fields such as natural language processing (NLP) and computer vision (CV), their application to learning complex responses of physical systems from experimental measurements remains underexplored. In physical systems, learning problems are often characterized as discovering operators that map between function spaces, using only a few samples of corresponding function pairs. For instance, in the automated discovery of heterogeneous material models, the foundation model must be capable of identifying the mapping between applied loading fields and the resulting displacement fields, while also inferring the underlying microstructure that governs this mapping. While the former task can be seen as a PDE forward problem, the later task frequently constitutes a severely ill-posed PDE inverse problem.

In this talk, we will explore the attention mechanism towards a foundation model for physical systems. Specifically, we show that the attention mechanism is mathematically equivalent to a double integral operator, enabling nonlocal interactions among spatial tokens through a data-dependent kernel that characterizes the inverse mapping from data to the hidden PDE parameter field of the underlying operator. Consequently, the attention mechanism captures global prior information from training data generated by multiple systems and suggests an exploratory space in the form of a nonlinear kernel map. Based on this theoretical analysis, we introduce a novel neural operator architecture, the Nonlocal Attention Operator (NAO). By leveraging the attention mechanism, NAO can address ill-posedness and rank deficiency in inverse PDE problems by encoding regularization and enhancing generalizability. To demonstrate the applicability of NAO to material modeling problems, we apply it to the development of a foundation constitutive law across multiple materials, showcasing its generalizability to unseen data resolutions and system states. Our work not only suggests a novel neural operator architecture for learning an interpretable foundation model of physical systems, but also offers a new perspective towards understanding the attention mechanism.

Quadratic Gromov--Witten invariants of rational del Pezzo surfaces of degree >5

Series
Algebra Seminar
Time
Monday, March 9, 2026 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Kirsten WickelgrenDuke University

Please Note: There will be a pre-seminar at 10:55-11:25 in Skiles 005.

Quadratic Gromov--Witten invariants allow one to count curves on varieties over a field k satisfying geometric constraints while keeping track of arithmetic information about those curves. In particular, k does not need to be the field of complex or real numbers. These invariants were developed in joint work with Kass, Levine, and Solomon in genus 0 for del Pezzo surfaces. In this talk we will compute these invariants for rational del Pezzo surfaces of degree >5. To do this, we give these invariants the structure of an unramified Witt invariant for any fixed surface and degree. We then construct a multivariable unramified Witt invariant which conjecturally contains all of these invariants for k-rational surfaces. We prove this conjecture in degree >5. To do this, we study the behavior of these Gromov–Witten invariants during an algebraic analogue of surgery on del Pezzo surfaces. We obtain a surprisingly simple formula when uncomputable terms cancel out with an identity in (twisted) binomial coefficients in the Grothendieck–Witt group. This is joint work with Erwan Brugallé and Johannes Rau.

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