Georgia Institute of Technology College of Sciences

Join us as we define a whole new algebraic structure, starting from the axioms of the projective plane. This seminar will be aimed at students who have never seen this material and will focus on hands-on constructions of classic (and new!) algebraic structures that can arise from a projective plane. The goal of this seminar is to expose you to Desargues's theorem and hopefully even examine non-Desarguesian planes.

We'll begin with a primer on hyperbolic and stable polynomials, which have been popular in recent years due to their many surprising appearances in combinatorics and algebra. We will cover a sketch of the famous Branden Borcea characterization of univariate stability preservers in the first part of the talk. We will then discuss more our recent work on multivariate hyperbolic polynomials which are invariant under permutations of their variables and connections to this Branden Borcea characterization.

Zoom Link: https://gatech.zoom.us/j/99596774152

Chip-firing asks a simple question: Given a group of people and an initial integer distribution of dollars among the people including people in debt, can we redistribute the money so that no one ends up in debt? This simple question with its origins in combinatorics can be reformulated using concepts from graph theory, linear algebra, graph orientation algorithms, and even divisors in Riemann surfaces. This presentation will go over a summary of Part 1 of Divisors and Sandpiles by Scott Corry and David Perkinson. Moreover, we will cover three various approaches to solve this problem: a linear algebra approach with the Laplacian, an algorithmic approach with Dhar's algorithm, and an algebraic geometry approach with a graph-theoretic version of the Riemann-Roch theorem by Baker and Norine. If we have time, we will investigate additional topics from Part 2 and Part 3. As true to the title, there will be a non-alcoholic drinking game involved with this presentation and participation will be completely voluntary. Limited refreshments (leftover Coca-Cola I found in the grad student lounge) and plastic cups will be served.

I will describe the arithmetic of polynomials over idylls and various division algorithms and rules. For instance, that arithmetic might capture a total order/sign or an absolute value. These division algorithms will relate, for instance, the number of positive roots of a polynomial to the signs of the coefficients (Descartes's Rule of Signs).