Georgia Institute of Technology College of Sciences

In recent years, the theories of Lorentzian polynomials and combinatorial Hodge theory have been developed and utilized to resolve long-standing conjectures in matroid theory, related to log-concavity inequalities and sampling algorithms. The overarching idea in these theories is to extract the conjectured results from basic eigenvalue bounds on certain natural matrices associated to matroids. Since then, Lorentzian polynomials have been generalized beyond matroids to simplicial complexes of various types, implying old and new results on various combinatorial structures such as linear extensions of posets. That said, many questions remain open. In this talk, we will describe this generalized theory and discuss how it can be used to prove various combinatorial results. No knowledge of matroid theory will be assumed. Joint work with Kasper Lindberg and Shayan Oveis Gharan, and also with Petter Brändén.

The existence and nonexistence of tight contact structures on the 3-manifold are interesting and important topics studied over the past thirty years. Etnyre-Honda found the first example of a 3-manifold that does not admit tight contact structure, and later Lisca-Stipsicz extended their result and showed that a Seifert fiber space admits a tight contact structure if and only if it is not the smooth (2n − 1)-surgery along the T(2,2n+1) torus knot for any positive integer n.

Surprisingly, since then no other example of a 3-manifold without tight contact structure has been found. Hence, it is interesting to study if all such manifolds, except those mentioned above, admit a tight contact structure. Towards this goal, I will discuss the joint work with Zhenkun Li and Hugo Zhou about showing any negative surgeries on any knot in S^3 admit a tight contact structure.  

The use of neural networks for solving partial differential equations (PDEs) has attracted considerable attention in recent years. In this talk, I will first highlight their advantages over traditional numerical methods, including improved approximation rates and the potential to overcome the curse of dimensionality. I will then discuss the challenges that arise when applying neural networks to PDEs, particularly in training. Because training is inherently a highly nonconvex optimization problem, it can lead to poor local minima with large training errors, especially in complex PDE settings. To address these issues, I will demonstrate how incorporating mathematical insight into the design of training algorithms and network architectures can lead to significant improvements in both accuracy and robustness.

We study causal optimal transport problems with Markovian cost and prescribed Markovian marginal laws. We show that the associated value function solves a fully nonlinear parabolic PDE, for which we establish a comparison principle and, consequently, uniqueness of its viscosity solution. This PDE characterization allows us to identify the value with that of a constrained version of the control problem for the Kushner–Stratonovich equation. We also obtain a third equivalent optimal control formulation with a state constraint, which leads to implementable numerical schemes for causal optimal transport. This is joint ongoing work with Julio Backhoff, Erhan Bayraktar, and Antonios Zitridis.

The representations of quantum groups are important in topology, namely, they can be used to construct quantum invariants of links. This relationship goes both ways: for example, the equivariant tensor category of representations of $U_q(\mathfrak{sl}_2)$ can be understood as a category of tangles. We will discuss a landmark result by Kuperberg who constructed graphical calculuses which describe the representation theory of the rank-2 simple Lie algebras.

Ultrafilters formalize a generalized notion of convergence based on a prescribed idea of "largeness" for subsets of the natural numbers, and underlie constructions like ultraproducts. In the study of moduli spaces, they provide a clean way to encode degenerations and to establish uniformity results that are difficult to obtain using ordinary limits. This talk will discuss applications of ultrafilters to uniformity theorems in dynamics and arithmetic geometry. After introducing local results that arise from this approach, I will sketch some of the arithmetic consequences, including uniform bounds on rational torsion points on abelian varieties. This is joint work with Jit Wu Yap

Heegaard Floer homology is a tool for studying three- and four-dimensional manifolds, using methods that are inspired by symplectic geometry. Bordered Floer homology is tool, currently under construction, for understanding how to reconstruct the Heegaard Floer homology in terms of invariants associated to its pieces. This approach has both conceptual and computational ramifications. In this talk, I will sketch the outlines of Heegaard Floer homology, with an emphasis on recent progress in bordered Floer homology. Heegaard Floer homology was developed in collaboration with Zoltan Szabo; bordered Floer homology is joint work with Robert Lipshitz and Dylan Thurston.