## Seminars and Colloquia by Series

### The steady-state degree and mixed volume of a chemical reaction network

Series
Student Algebraic Geometry Seminar
Time
Friday, March 26, 2021 - 09:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Cvetelina HillGeorgia Tech
Chemical reaction networks (CRNs), under the assumption of mass-action kinetics, are deterministic polynomial systems commonly used in systems biology. The steady-state of a CRN is the number of complex steady-states (solutions to the polynomial system), which is a measure of the algebraic complexity of solving the steady-state system. In general, the steady-state degree may be difficult to compute. Using three case studies of infinite families of networks, each generated by joining smaller networks to create larger ones, we give an upper bound to the steady-state degree of a CRN by utilizing the underlying polyhedral geometry associated with the corresponding system. In this talk I will give an overview of the necessary background for CRNs and the associated polyhedral geometry, and I will discuss the results on one of the case studies through examples.

### Another interpretation of tropical rank.

Series
Student Algebraic Geometry Seminar
Time
Friday, February 26, 2021 - 09:00 for 1 hour (actually 50 minutes)
Location
Online
Speaker
Tianyi ZhangGeorgia Tech

Tropical rank is defined in terms of determinant in the literature. I will introduce a rank in terms of linear dependence and show it equals the tropical rank. This fact is nontrivial because we do not have row reduction which is a key tool to prove the equality for matrices over fields. This talk is based on the paper "the tropical rank of a tropical matrix" written by Z. Izhakian.

### Comparison between SOS and PSD via an algebraic quantity

Series
Student Algebraic Geometry Seminar
Time
Friday, February 12, 2021 - 09:00 for 1 hour (actually 50 minutes)
Location
Online
Speaker
Jaewoo JungGeorgia Tech

Even though it is not easy to determine global non-negativity of a polynomial, if the polynomial can be written as a sum of squares(SOS), we certainly see that it must be non-negative(PSD). Representability of polynomials in terms of sums of squares is a good certification for global non-negativity in the sense that any non-negative polynomials is just a sum of squares in some cases. However, there are some non-negative polynomials which cannot be written as sum of squares in general. So, one can ask about when the set of sums of squares is same as the set of non-negative polynomials or describing gap between set of sums of squares and non-negative polynomials if they are different.

In this talk, we will introduce an algebraic invariant (of variety) which can tell us when the two sets are same (or not). Moreover, we will discuss about cases that we can exactly describe structural gaps between the two sets.

URL: Microsoft Teams

### A funny thing happened on the way to infinity: homotopy continuation on a compact toric variety

Series
Student Algebraic Geometry Seminar
Time
Friday, February 5, 2021 - 09:00 for 1 hour (actually 50 minutes)
Location
SAGS Microsoft Teams
Speaker
Tim DuffGeorgia Tech

Homotopy continuation methods are numerical methods for solving polynomial systems of equations in many unknowns. These methods assume a set of start solutions to some start system. The start system is deformed into a system of interest (the target system), and the associated solution paths are approximated by numerical integration (predictor/corrector) schemes.

The most classical homotopy method is the so-called total-degree homotopy. The number of start solutions is given by Bézout's theorem. When the target system has more structure than start system, many paths will diverge, This behavior may be understood by working with solutions in a compact projective space.

In joint work with Telen, Walker, and Yahl, we describe a generalization of the total degree homotopy which aims to track fewer paths by working in a compact toric variety analagous to projective space. This allows for a homotopy that may more closely mirror the structure of the target system. I will explain what this is all about and, time-permitting, touch on a few twists we discovered in this more general setting. The talk will be accessible to a general mathematical audience -- I won't assume any knowledge of algebraic geometry.

### Representations of Sl(2,C) in combinatorics

Series
Student Algebraic Geometry Seminar
Time
Friday, January 29, 2021 - 09:00 for 1 hour (actually 50 minutes)
Location
Microsoft Teams
Speaker
Trevor GunnGeorgia Tech

There are two purposes of this talk: 1. to give an example of representation theory in algebraic combinatorics and 2. to explain some of the early work on unimodal/symmetric sequences in combinatorics related to recent work on Hodge theory in combinatorics. We will investigate the structure of graded vector spaces $\bigoplus V_j$ with two "shifting" operators $V_j \to V_{j+1}$ and $V_j → V_{j-1}$. We will see that this leads to a very rich theory of unimodal and symmetric sequences with several interesting connections (e.g. the Edge-Reconstruction Conjecture and Hard Lefschetz). The majority of this talk should be accessible to anyone with a solid knowledge of linear algebra.

### Chain conditions in power series and polynomial rings

Series
Student Algebraic Geometry Seminar
Time
Friday, November 13, 2020 - 09:00 for 1 hour (actually 50 minutes)
Location
Microsoft Teams Meeting
Speaker
Hamed MousaviGeorgia Tech

Following the Hilbert Basis theorem and its applications, there has been a vast variety of studies involving the chain conditions over the polynomial or the power series rings. One type of chain condition is the Archimedean condition, which says \cap_n Rt_n = 0for any nonunit element t in the ring R. In this talk, we start with the ascending chain condition on principal ideals (ACCP) over a larger class “skew generalized power series rings”. Then we explain the relation between ACCP rings and Archimedean rings and answer partially to the question “when these properties can be lifted from the ring R to the ring R[[x; α]]? ” In particular we show that if R is an Archimedean reduced ring and satisfy ACC on annihilators, then R[[x]] is also an Archimedean reduced ring.

### Hankel index of a projected of rational curves

Series
Student Algebraic Geometry Seminar
Time
Friday, November 6, 2020 - 09:00 for 1 hour (actually 50 minutes)
Location
Microsoft Teams Meeting
Speaker
Jaewoo JungGeorgia Tech

If we can write a (homogeneous) polynomial as a sum of squares(SOS), the polynomial is guaranteed to be a non-negative polynomial. However, every non-negative forms does not have to be written as sums of squares in general. This implies that set of sums of square is strictly contained in set of non-negative forms in general. We want to discuss about one way to describe the gaps between the two sets. Since the sets have cone structures, we can consider dual cones of each cones. In particular, the description of dual cone of non-negative polynomials is simple: convex hull of point evaluations. Therefore, we are interested in positive semi-definite quadratic forms that is not point evaluations. We call "Hankel index" the minimal rank of quadratic form (on extreme ray of the dual cone of SOS) which is not a point evaluation. In this talk, we introduce the Hankel index of variety and will discuss about a criterion to obtain the Hankel index of projected rational curves.

### Minimal problems and their monodromy groups

Series
Student Algebraic Geometry Seminar
Time
Friday, October 23, 2020 - 09:00 for 1 hour (actually 50 minutes)
Location
Microsoft Teams Meeting
Speaker
Tim DuffGeorgia Tech

### Extreme Rays of Locally PSD Cones

Series
Student Algebraic Geometry Seminar
Time
Friday, October 16, 2020 - 09:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Kevin ShuGeorgia Tech

Locally PSD matrices are a generalization of PSD matrices which appear in sparse semidefinite programming. We will try to explore some connections of extreme rays of this type of matrix with algebraic topology.

### Introduction to Kajiwara-Payne Tropicalization II

Series
Student Algebraic Geometry Seminar
Time
Friday, October 9, 2020 - 09:00 for 1 hour (actually 50 minutes)
Location