Georgia Institute of Technology College of Sciences

While gerrymandering has been widely suspected in Georgia for years, it has been difficult to quantify. We generate a large ensemble of randomly generated non-partisan maps that are sampled from a probability distribution which respects the geographical constraints of the redistricting process. Using a Markov chain Monte Carlo process and techniques involving spanning trees, we can quickly generate a robust set of plans.

Based on historical voting data, we compare the Georgia congressional redistricting plan enacted in 2021 with the non-partisan maps. We find that the 2021 plan will likely be highly non-responsive to changing opinions of the electorate, unlike the plans in the ensemble. Using additional spatial analysis, we highlight areas where the map has been redrawn to weaken the influence of Democratic voters.

This talk is based on joint work with Swati Gupta, Gregory Herschlag, Jonathan Mattingly, Dana Randall, and Zhanzhan Zhao.

(Based on paper by Fawzi, Saunderson and Parrilo in 2015) The space of complex-valued functions on a fixed abelian group has an orthonormal basis of group homomorphisms, via the well-known Discrete Fourier Transform. Given any nonnegative function with sparse Fourier support, it turns out that it’s possible to write it as a sum of squares, where the common Fourier support for all squares is not big. This can be used to prove results for the usual degree-based sum-of-squares hierarchy.

Thomassen famously showed that every planar graph is 5-choosable, and that every planar graph of girth at least five is 3-choosable.  These theorems are best possible for uniform list assignments: Voigt gave a construction of a planar graph that is not 4-choosable, and of a planar graph of girth four that is not 3-choosable. In this talk, I will introduce the concept of a local girth list assignment: a list assignment wherein the list size of each vertex depends not on the girth of the graph, but only on the length of the shortest cycle in which the vertex is contained. I will present a local list colouring theorem that unifies the two theorems of Thomassen mentioned above and discuss some of the highlights and difficulties of the proof. This is joint work with Luke Postle.

Given a divergence-free vector field on the torus, we consider the mixing properties of the associated flow. There is a rich body of work studying the dependence of the mixing scale on various norms of the vector field. We will discuss some interesting examples of vector fields that mix at the optimal rate, and an improved bound on the mixing scale under the extra assumption that the vector field is a shear at each time.