Georgia Institute of Technology College of Sciences

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.

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.

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.

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.

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. 

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.

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.

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.

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. 

What are the odds your team will win, and what will the spread be? Our statisticians and mathematicians will help you understand the odds in sports events and how we calculate them. Try out the hands-on demos our team has developed! Sports stats are only one of the many ways math connects with sports. Find out how the wave moves through the audience in a stadium, how math helps perfect sports equipment, how to determine how much a team is worth, and more! We will finish up the night with a round of mathy sports trivia.

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