Georgia Institute of Technology College of Sciences

This talks presents two projects at the interface of computer vision and algebraic geometry. Work with Zuzana Kukelova, Tomas Pajdla and Bernd Sturmfels introduces the distortion varieties of a given projective variety. These are parametrized by duplicating coordinates and multiplying them with monomials. We study their degrees and defining equations. Exact formulas are obtained for the case of one-parameter distortions, the case of most interest for modeling cameras with image distortion. Single-authored work determines the algebraic degree of minimal problems for the calibrated trifocal variety. Our techniques rely on numerical algebraic geometry, and the homotopy continuation software Bertini.

In this lecture series, held jointly (via video conference) with the University of Buffalo and the University of Arkansas, we aim to understand the lecture notes by Vincent Guirardel on geometric small cancellation. The lecture notes can be found here: https://perso.univ-rennes1.fr/vincent.guirardel/papiers/lecture_notes_pcmi.pdf

Parallel algorithms study ways of speeding up sequential algorithms by splitting work onto multiple processors. Theoretical studies of parallel algorithms often focus on performing a small number of operations, but assume more generous models of communication. Recent progresses led to parallel algorithms for many graph optimization problems that have proven to be difficult to parallelize. In this talk I will survey routines at the core of these results: low diameter decompositions, random sampling, and iterative methods.

Consider Hermitian matrices A, B, C on an n-dimensional Hilbert space such that C=A+B. Let a={a_1,a_2,...,a_n}, b={b_1, b_2,...,b_n}, and c={c_1, c_2,...,c_n} be sequences of eigenvalues of A, B, and C counting multiplicity, arranged in decreasing order. Such a triple of real numbers (a,b,c) that satisfies the so-called Horn inequalities, describes the eigenvalues of the sum of n by n Hermitian matrices. The Horn inequalities is a set of inequalities conjectured by A. Horn in 1960 and later proved by the work of Klyachko and Knutson-Tao. In these two talks, I will start by discussing some of the history of Horn's conjecture and then move on to its more recent developments. We will show that these inequalities are also valid for selfadjoint elements in a finite factor, for types of torsion modules over division rings, and for singular values for products of matrices, and how additional information can be obtained whenever a Horn inequality saturates. The major difficulty in our argument is the proof that certain generalized Schubert cells have nonempty intersection. In the finite dimensional case, it follows from the classical intersection theory. However, there is no readily available intersection theory for von Neumann algebras. Our argument requires a good understanding of the combinatorial structure of honeycombs, and produces an actual element in the intersection algorithmically, and it seems to be new even in finite dimensions. If time permits, we will also discuss some of the intricate combinatorics involved here. In addition, some recent work and open questions will also be presented.