Seminars and Colloquia by Series

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.

Ranks of points via Macaulay 2 (2nd talk)

Series
Algebra Student Seminar
Time
Friday, April 22, 2022 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006 and Teams
Speaker
Jaewoo JungGeorgia Tech

The rank of a point $p$ with respect to a non-degenerate variety is the smallest number of the points in the variety that spans the point $p$. Studies about the ranks of points are important in various areas of applied mathematics and engineering in the sense that they are the smallest number of summands in the decompositions of vectors into combinations of simple vectors.

In the last talk, we discussed how to generate points of given ranks with respect to the rational normal curves. We continue to discuss some known facts via Macaulay 2 and how to find the list of all ranks of points in linear spaces.

Ranks of points via Macaulay 2

Series
Algebra Student Seminar
Time
Friday, April 8, 2022 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006 or ONLINE
Speaker
Jaewoo JungGeorgia Tech

The rank of a point $p$ with respect to a non-degenerate variety is the smallest number of the points in the variety that spans the point $p$. Studies about the ranks of points are interesting and important in various areas of applied mathematics and engineering in the sense that they are the shortest sizes of the decompositions of vectors into combinations of simple vectors.

In this talk, we focus on the ranks of points with respect to the rational normal curves, i.e. Waring ranks of binary forms. We introduce an algorithm that produces random points of given rank r. (Note that if we choose points randomly, we expect the rank of the points is just the generic rank.) Moreover, we check some known facts by Macaulay 2 computations. Lastly, we discuss the maximal and minimal rank of points in linear spaces.

A computer program for matroid representation

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

Baker and Lorscheid have developed a theory of foundation that characterize the representability of matroids. Justin Chen and I are developing a computer program that computes representations of matroids based on the theory of foundation. In this talk, I will introduce backgrounds on matroids and the foundation, then I will talk about the key algorithms in computing the morphisms of pastures. If possible, I will also show some examples of the program.

Hilbert's Tenth Problem and Generalizations

Series
Algebra Student Seminar
Time
Friday, March 18, 2022 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006 and Teams
Speaker
Ian LewisGeorgia Tech
Hilbert's Tenth Problem asks whether there exists an algorithm to determine whether an arbitrary polynomial with integer coefficients has a solution or not. This problem was resolved by Matiyasevich building of the work in of Robinson, Davis, and Putnam in the 70s. We will give an overview of how this problem was resolved and the current status of various generalizations.

Nonnegativity and Real-Rootedness

Series
Algebra Student Seminar
Time
Friday, March 11, 2022 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006 or ONLINE
Speaker
Kevin ShuGeorgia Tech

There are many interesting classes of polynomials in real algebraic geometry that are of modern interest. A polynomial is nonnegative if it only takes nonnegative values on R^n. A univariate polynomial is real-rooted if all of its complex roots are real, and a hyperbolic polynomial is a multivariate generalization of a real-rooted polynomial. We will discuss connections between these two classes of polynomials. In particular, we will discuss recent ideas of Saunderson giving new ways of proving that a polynomial is nonnegative beyond showing that it is sum-of-squares.

Nonnegative symmetric polynomials and symmetric sums of squares at the limit.

Series
Algebra Student Seminar
Time
Friday, March 4, 2022 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006 and Teams
Speaker
Jose AcevedoGeorgia Tech

Restricting to symmetric homogeneous polynomials of degree 2d we compare the cones of nonnegative polynomials with the cone of sums of squares when the number of variables goes to infinity. We consider two natural notions of limit and for each we completely characterize the degrees for which the limit cones are equal. To distinguish these limit cones we tropicalize their duals, which we compute via tropicalizing spectrahedra and tropical convexity. This gives us convex polyhedral cones which we can completely describe and from them obtain explicit examples of nonnegative symmetric polynomials that are not sums of squares (in some cases for any number >=4 of variables).

This is joint work with Grigoriy Blekherman, Sebastian Debus, and Cordian Riener.

Algebra Student Seminar homepage

Tropical and algebraic divisors and projective embeddings

Series
Algebra Student Seminar
Time
Friday, February 25, 2022 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006 and Teams
Speaker
Trevor GunnGeorgia Tech

We will review how divisors on abstract algebraic curves are connected with projective embeddings and then see how that language translates to tropical curves and tropicalization. This talk aims to explain some of the connections between tropical curves and algebraic curves that was not discussed during the seminar on tropical Brill-Noether theory.

Algebra Student Seminar homepage

Braided Monoidal Categories and Fusion Categories

Series
Algebra Student Seminar
Time
Friday, February 18, 2022 - 10:00 for 1 hour (actually 50 minutes)
Location
Skiles 006 or ONLINE
Speaker
Akash NarayananGeorgia Tech

We introduce the notion of braided monoidal categories and fusion categories, which are one way of reframing algebraic structures in a categorical context. After discussing various examples and analogies with the theory of finite groups, we build up to a classification of pointed fusion categories.

Hyperbolic generalization of linear algebra

Series
Algebra Student Seminar
Time
Friday, February 4, 2022 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006, or ONLINE
Speaker
Shengding SunGeorgia Tech

We will introduce the machinery of hyperbolic polynomial, and see how it can help us generalize classical linear algebra theorems and inequalities on symmetric matrices, including Hadamard-Fischer inequality, Koteljanskii's inequality and Schur-Horn theorem (last one is conjectured but not proved). Joint work with Greg Blekherman, Mario Kummer, Raman Sanyal and Kevin Shu.