Georgia Institute of Technology College of Sciences

In high dimensional contact and symplectic topology, finding interesting constructions for Legendrian submanifolds is an active area of research. Further, it is desirable that the constructions lend themselves nicely to computation of invariants. The doubling construction was defined by Ekholm, which uses Lagrangian fillings of a Legendrian knot in standard contact R^{2n-1} to produce a closed Legendrian submanifold in standard contact R^{2n+1}. Later Courte-Ekholm showed that symmetric doubles of embedded fillings are "uninteresting". In recent work the symmetric doubling construction was generalised to any contact manifold, giving two isotopic constructions related to open book decompositions of the ambient manifold. In a separate joint work with James Hughes, we explore the asymmetric doubling construction through Legendrian weaves.

There is a natural notion of `degree’ for functions from the symmetric group to the complex numbers, which translates roughly to saying the function counts certain weighted patterns. Low degree class functions have a classical interpretation in terms of the cycle structure of permutations. I will explain how to translate between pattern counts to cycle structure using a novel symmetric function identity analogous to the Murnaghan-Nakayama identity. This relationship allows one to lift many probabilistic properties of permutation statistics to certain non-uniform distributions, and I will present some results in this direction. This is joint work with Brendon Rhoades.

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.