Georgia Institute of Technology College of Sciences

The classic facility location problem seeks to open a set of facilities to minimize the cost of opening the chosen facilities and the total cost of connecting all the clients to their nearby open facilities. Such an objective may induce an unequal cost over certain socioeconomic groups of clients (i.e., total distance traveled by clients in such a group). This is important when planning the location of socially relevant facilities such as emergency rooms and grocery stores. In this work, we consider a fair version of the problem by minimizing the L_p-norm of the total distance traveled by clients across different socioeconomic groups and the cost of opening facilities, to penalize high access costs to open facilities across r groups of clients. This generalizes classic facility location (p = 1) and the minimization of the maximum total distance traveled by clients in any group (p = infinity). However, it is often unclear how to select a specific "p" to model the cost of unfairness. To get around this, we show the existence of a small portfolio of at most (log2r + 1) solutions for r (disjoint) client groups, where for any L_p-norm, at least one of the solutions is a constant-factor approximation with respect to any L_p-norm. We also show that such a dependence on r is necessary by showing the existence of instances where at least ~ sqrt(log2r) solutions are required in such a portfolio. Moreover, we give efficient algorithms to find such a portfolio of solutions. Additionally, We introduce the notion of refinement across the solutions in the portfolio. This property ensures that once a facility is closed in one of the solutions, all clients assigned to it are reassigned to a single facility and not split across open facilities. We give poly(exp(sqrt(r))-approximation for refinement in general metrics and O(log r)-approximation for the line and tree metrics. This is joint work with Swati Gupta and Mohit Singh.

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.