Georgia Institute of Technology College of Sciences

We shall explain a simple remarkable stability phenomenon regarding the centers of the group algebras of the symmetric groups in n letters, as n goes to infinity. The same type of stability phenomenon extends to a wide class of finite groups including wreath products and finite general linear groups. Such stability has connections and applications to the cohomology rings of Hilbert schemes of n points on algebraic surfaces.

Topological insulators are materials that exhibit unique physical properties due to their non-trivial topological order. One of the most notable consequences of this order is the presence of protected edge states as well as closure of bulk spectral gaps, which is known as the bulk-edge correspondence. In this talk, I will discuss the mathematical description of topological insulators and their related spectral properties. The presentation assumes only basic knowledge of spectral theory, and will begin with an overview of Floquet theory, Bloch bundles, and the Chern number. We will then examine the bulk-edge correspondence in topological insulators before delving into our research on closure of bulk spectral gaps for topological insulators with general edges. This talk is based on a joint work with Alexis Drouot.

The Fourier restriction conjecture and the Bochner-Riesz conjecture ask for Lebesgue space mapping properties of certain oscillatory integral operators. They both are central in harmonic analysis, are open in dimensions $\geq 3$, and notably have the same conjectured exponents. In the 1970s, H\"{o}rmander asked if a more general class of operators (known as H\"{o}rmander type operators) all satisfy the same $L^p$-boundedness as in the above two conjectures. A positive answer to H\"{o}rmander's question would resolve the above two conjectures and have more applications such as in the manifold setting. Unfortunately H\"{o}rmander's question is known to fail in all dimensions $\geq 3$ by the work of Bourgain and many others. It continues to fail in all dimensions $\geq 3$ even if one adds a ``positive curvature'' assumption which one does have in restriction and Bochner-Riesz settings. Bourgain showed that in dimension $3$ one always has the failure unless a derivative condition is satisfied everywhere. Joint with Shaoming Guo and Hong Wang, we generalize this condition to arbitrary dimension and call it ``Bourgain's condition''. We unify Fourier restriction and Bochner-Riesz by conjecturing that any H\"{o}rmander type operator satisfying Bourgain's condition should have the same $L^p$-boundedness as in those two conjectures. As evidence, we prove that the failure of Bourgain's condition immediately implies the failure of such an $L^p$-boundedness in every dimension. We also prove that current techniques on the two conjectures apply equally well in our conjecture and make some progress on our conjecture that consequently improves the two conjectures in higher dimensions. I will talk about some history and some interesting components in our proof.

Many results in extremal graph theory build on supersaturation of subgraphs. In other words, when a graph is dense enough, it contains many copies of a certain subgraph and these copies are then used as building blocks to force another subgraph of interest. Recently more success is found within this approach where one utilizes not only the large number of copies of a certain subgraph but a well-distributed collection of them to force the desired subgraph. We discuss some recent progress of this nature. The talk is built on joint work with Sean Longbrake, and with Sean Longbrake and Jie Ma.