Georgia Institute of Technology College of Sciences

Galois groups embody the structure of algebraic equations arising in both enumerative geometry and various scientific applications where such equations must be solved. I will describe a line of work that aims to elucidate the role of Galois groups in applications where data taken from multiple images are used to reconstruct a 3D scene. From this perspective, I will revisit two well-known solutions to camera pose estimation problems, which originate from classical photogrammetry and are still heavily used within modern 3D reconstruction systems. I will then discuss some less-classical problems, for which the insight we gleaned from computing Galois groups led to significant practical improvements over previous solutions. A key ingredient was the use of numerical homotopy continuation methods to (heuristically) compute monodromy permutations. Time-permitting, I will explain how such methods may also be used to automatically recover certain symmetries underlying enumerative problems. 

In 1989, Eremenko investigated the set of points that escape to infinity under iteration of a transcendental entire function, the so-called escaping set. He proved that every component of the closure of the escaping set is unbounded and conjectured that all the components of the escaping set are unbounded. Much of the recent work on the iteration of entire functions is involved in investigating properties of the escaping set, motivated by Eremenko's conjecture. We will begin by introducing many of the basic dynamical properties of iterates of an analytic function, and finally discuss constructing a transcendental entire function with a point connected component of the escaping set, providing a counterexample to Eremenko's conjecture. This is joint work with David Martí-Pete and Lasse Rempe.

How many different ways can we arrange n convex sets in R^d? One answer is provided by counting the number of d-representable complexes on vertex set [n]. We show that there are exp(Theta(n^d log n))-many such complexes, and provide bounds on the constants involved. As a consequence, we show that d-representable complexes comprise a vanishingly small fraction of the class of d-collapsible complexes. In the case d = 1 our results are more precise, and improve the previous best estimate for the number of interval graphs.

I will present numerical methods for solving the optimal transport (OT) problems in three settings. Firstly, I will discuss discrete OT problems from the perspective of linear programming and assignment problems. Additionally, I will provide a solution for a discrete problem with an obstacle in the domain.

Next, I will consider and compare several different numerical methods to solve the classic continuous OT problem with the squared Euclidean cost function. I will compare two numerical methods used for the fluid dynamics formulation with a direct discretization of the Monge-Ampère PDE. Furthermore, I will introduce a new class of problems called separable, for which very accurate methods can be devised. 

Lastly, I propose a novel implementation of Newton's method for solving semi-discrete OT problems for cost functions that are a positive combination of $p$-norms, $1