Studying symplectic structures up to deformation equivalences is a fundamental question in symplectic geometry. Donaldson asked: given two homeomorphic closed symplectic four-manifolds, are they diffeomorphic if and only if their stabilized symplectic six-manifolds, obtained by taking products with CP^1 with the standard symplectic form, are deformation equivalent? I will discuss joint work with Amanda Hirschi on showing how deformation inequivalent symplectic forms remain deformation inequivalent when stabilized, under certain algebraic conditions. This gives the first counterexamples to one direction of Donaldson�??s �??four-six�?� question and the related Stabilizing Conjecture by Ruan. In the other direction, I will also discuss more supporting evidence via Gromov-Witten invariants.

Ensembles of redistricting plans can be challenging to analyze and visualize because every plan is an unordered set of shapes, and therefore non-Euclidean in at least two ways. I will describe two methods designed to address this challenge: barycenters for partitioned datasets, and a novel dimension reduction technique based on Gromov-Wasserstein distance. I will cover some of the theory behind these methods and show how they can help us untangle redistricting ensembles to find underlying trends. This is joint work with Ranthony A. Clark and Tom Needham.

We formulate a new version of the local Langlands correspondence for discrete L-parameters which involves (Weyl orbits of) packets of representations of all twisted Levi subgroups of a connected reductive group G through which the parameter factors and prove that this version of the correspondence follows if one assumes the pre-existing local Langlands conjectures. Twisted Levi subgroups are crucial objects in the study of supercuspidal representations; this work is a step towards deepening the relationship between the representation theory of p-adic groups and the Langlands correspondence. This is joint work with David Schwein (Bonn).

We study the half-space KPZ equation with a Neumann boundary condition, starting from stationary Brownian initial data. We derive a variance identity that links the fluctuations of the height function to the transversal fluctuations of a half-space polymer model. We then establish optimal fluctuation exponents for the height function in both the subcritical and critical regimes, along with corresponding estimates for the polymer endpoint. Based on a joint work with Yu Gu.