Seminars and Colloquia by Series

Determinantal Representations and the Image of the Principal Minor Map

Series
Algebra Student Seminar
Time
Friday, December 9, 2022 - 10:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Abeer Al AhmadiehGeorgia Institute of Technology

 The principal minor map takes an n  by n square matrix to the length 2^n-vector of its principal minors. A basic question is to give necessary and sufficient conditions that characterize the image of various spaces of matrices under this map. In this talk, I will describe the image of the space of complex matrices using a characterization of determinantal representations of multiaffine polynomials, based on the factorization of their Rayleigh differences. Using these techniques I will give equations and inequalities characterizing the images of the spaces of real and complex symmetric and Hermitian matrices. For complex symmetric matrices, this recovers a result of Oeding from 2011. If time permits, I will also give examples to prove that for general matrices no such finite characterization is possible. This is based on joint work with Cynthia Vinzant.

Idylls and Baker-Lorscheid Multiplicities

Series
Algebra Student Seminar
Time
Friday, November 18, 2022 - 10:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Trevor GunnGeorgia Tech

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).

Decidability in Number Theory

Series
Algebra Student Seminar
Time
Friday, November 4, 2022 - 10:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Ian Lewis

We will introduce some basic notions needed to talk about the question of decidability for roots of polynomials with coefficients in a specified ring R in the sense of Hilbert's tenth problem with an emphasis on rings of number theoretic interest. We will also attempt to give an overview of the literature on the topic and recent lines of work.

An Introductory Proof of the Compactness Theorem and Some Applications

Series
Algebra Student Seminar
Time
Friday, October 28, 2022 - 10:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Griffin EdwardsGeorgia Institute Of Technology

We will introduce the foundations of model theory, by defining languages, models, and theories. Then we will look at a couple proofs of the compactness theorem, state Gödel's completeness theorem, and prove that any planar graph is four colorable. Expect a lot of examples, and I hope everyone comes away understanding the foundations of this wonderful theory.

What is a matroid?

Series
Algebra Student Seminar
Time
Friday, October 21, 2022 - 10:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Tong JinGeorgia Institute of Technology
This is a pre-talk for the Algebra Seminar on Oct. 24. I will discuss (various) definitions of matroids, matroid minors, Tutte polynomials and characteristic polynomials, matroid basis polytopes, and Grassmannians. If time permits, I'll also discuss permutohedral varieties and the Cremona map and/or my current work. 
 

What is a Coxeter group, and why is a Coxeter group?

Series
Algebra Student Seminar
Time
Friday, October 14, 2022 - 10:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Tong JinGeorgia Institute of Technology

A Coxeter group is a (not necessarily finite) group given by certain types of generators and relations. Examples of finite Coxeter groups include dihedral groups, symmetric groups, and reflection groups. They play an important role in various areas. In this talk, I will discuss why I am interested in Coxeter groups from a combinatorial perspective - the geometric concepts associated with the finite Coxeter groups form the language of Coxeter matroids, which are generalizations of ordinary matroids. In particular, finite Coxeter groups are related to Coxeter matroids in the same way as symmetric groups are related to ordinary matroids. The main reference for this talk is Chapter 5 of Borovik-Gelfand-White's book Coxeter Matroids. I will only assume basic group theory, but not familiarity with matroids.

Sparse Quadratic Programs via Polynomial Roots

Series
Algebra Student Seminar
Time
Friday, September 23, 2022 - 10:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Kevin ShuGeorgia Institute of Technology

We'll talk about problems of optimizing a quadratic function subject to quadratic constraints, in addition to a sparsity constraint that requires that solutions have only a few nonzero entries. Such problems include sparse versions of linear regression and principal components analysis. We'll see that this problem can be formulated as a convex conical optimization problem over a sparse version of the positive semidefinite cone, and then see how we can approximate such problems using ideas arising from the study of hyperbolic polynomials. We'll also describe a fast algorithm for such problems, which performs well in practical situations.

Polynomials over ordered blueprints and tracts

Series
Algebra Student Seminar
Time
Friday, September 16, 2022 - 10:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Trevor GunnGeorgia Tech

I will introduce the concept of an ordered blueprint and a tract and discuss some algebraic and categorical properties. I will then discuss the notion of a "tropical extension" and discuss the theory of polynomials in these contexts.

Sparse Fourier sum-of-squares decomposition for nonnegative functions on abelian groups

Series
Algebra Student Seminar
Time
Friday, September 9, 2022 - 10:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Shengding SunGeorgia Institute of Technology

(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.

Log-concavity of coefficients of characteristic polynomials of matroids.

Series
Algebra Student Seminar
Time
Friday, April 29, 2022 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006 or ONLINE
Speaker
Tong JinGeorgia Tech

This is an expanded version of a 10-minute presentation in MATH 6422. I'll explain what matroids and their characteristic polynomials as well as log-concavity mean, and then sketch a proof due to Petter Brändén and Jonathan Leake (arXiv:2110.00487). If time permits, I'll describe several consequences of this and/or other existing yet different proofs.

 

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