Georgia Institute of Technology College of Sciences

Hilbert's Nullstellensatz about zero sets of polynomials is one of the most fundamental correspondences between algebra and geometry. More recently, there has been an emerging interest in polynomial equations and inequalities in several matrix variables, prompted by developments in control systems, quantum information theory, operator algebras and optimization. The arising problems call for a suitable version of (real) algebraic geometry in noncommuting variables; with this in mind, the talk considers matricial sets where noncommutative polynomials attain singular values, and their algebraic counterparts.

Given a polynomial f in noncommuting variables, its free (singularity) locus is the set of all matrix tuples X such that f(X) is singular.  The talk focuses on the interplay between geometry of free loci (irreducible components, inclusions, eigenlevel sets, smooth points) and factorization in the free algebra. In particular, a Nullstellensatz for free loci is given, as well as a noncommutative variant of Bertini's irreducibility theorem and its consequences.

James Anderson: Odd coloring (resp, PCF coloring) is a stricter form of proper coloring in which every nonisolated vertex is required to have a color in its neighborhood with odd multiplicity (resp, with multiplicity 1). Using the discharging method, and a new tool which we call the Forb-Flex method, we improve the bounds on the odd and PCF chromatic number of planar graphs of girth 10 and 11, respectively.

Sean Kafer: Many classical combinatorial optimization problems (e.g. max weight matching, max weight matroid independent set, etc.) have formulations as linear programs (LPs) over 0/1 polytopes on which LP solvers could be applied.  However, there often exist bespoke algorithms for these problems which, by merit of being tailored to a specific domain, are both more efficient and conceptually nicer than running a generic LP solver on the associated LP.  We will discuss recent results which show that a number of such algorithms (e.g. the shortest augmenting path algorithm, the greedy algorithm, etc.) can be "executed" by the Simplex method for solving LPs, in the sense that the Simplex method can be made to generate the same sequence of solutions when applied to the appropriate corresponding LP.

Tantan Dai: There has been extensive research on Latin Squares. It is simple to construct a Latin Square with n rows and n columns. But how do we define a Latin Triangle? What are the rows? When does a Latin Triangle exist? How can we construct them? In this talk, I will discuss two types of Latin Triangles and the construction of a countable family of Latin Triangles.

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We show new lower bounds for sphere packings in high dimensions and for independent sets in graphs with not too large co-degrees.  For dimension d, this achieves a sphere packing of density (1 + o(1)) d log d / 2^(d+1).  In general dimension, this provides the first asymptotically growing improvement for sphere packing lower bounds since Roger's bound of c*d/2^d in 1947.  The proof amounts to a random (very dense) discretization together with a new theorem on constructing independent sets on graphs with not too large co-degree.  Both steps will be discussed, and no knowledge of sphere packings will be assumed or required.  This is based on joint work with Marcelo Campos, Matthew Jenssen and Julian Sahasrabudhe.

Many numerical methods have been developed in the past years for computing weak solutions (with shock waves) to nonlinear hyperbolic conservation laws. My research, specifically, concerns the design of well-balanced numerical algorithms that preserve certain key structure of these equations in various applications, including for problems involving moving phase boundaries and other scale-dependent interfaces. In particular, in this lecture, I will focus on the evolution of a compressible fluid in spherical symmetry on a Schwarzschild curved background, for which I have designed a class of well-balanced numerical algorithms up to third-order of accuracy. Both the relativistic Burgers-Schwarzschild model and the relativistic Euler-Schwarzschild model were considered, and the proposed numerical algorithm took advantage of the explicit or implicit forms available for the stationary solutions of these models. The schemes follow the finite volume methodology and preserve the stationary solutions and, most importantly, allow us to investigate the global asymptotic behavior of such flows and determine the asymptotic behavior of the mass density and velocity field of the fluid. Blog: philippelefloch.org