## Seminars and Colloquia by Series

### Tropical geometry and applications

Series
Algebra Seminar
Time
Wednesday, October 14, 2020 - 15:30 for 1 hour (actually 50 minutes)
Location
online
Speaker
Leon ZhangUC Berkeley

I will describe results from two recent projects in tropical geometry with relevance in applications. In the first half, I will introduce and give several characterizations for flags of tropical linear spaces, in analogy to Speyer's results for tropical linear spaces. In the second half, I will discuss current work relating tropical fewnomials, vertex bounds of Minkowski sums, and linear regions of maxout neural networks.

### A Higher-Dimensional Sandpile Map

Series
Algebra Seminar
Time
Wednesday, September 30, 2020 - 15:30 for 1 hour (actually 50 minutes)
Location
Speaker
Alex McdonoughBrown University

Traditionally, the sandpile group is defined on a graph and the Matrix-Tree Theorem says that this group's size is equal to the number of spanning trees. An extension of the Matrix-Tree Theorem gives a relationship between the sandpile group and bases of an arithmetic matroid. I provide a family of combinatorially meaningful maps between these two sets.  This generalizes a bijection given by Backman, Baker, and Yuen and extends work by Duval, Klivans, and Martin.

### All lines on a smooth cubic surface in terms of three skew lines

Series
Algebra Seminar
Time
Monday, April 20, 2020 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Tianyi ZhangGeorgia Tech
Harris showed that the incidence variety of a smooth cubic surface containing 27 lines has solvable Galois group over the incidence variety of a smooth cubic surface containing 3 skew lines. It follows that for any smooth cubic surface, there exist formulas for all 27 lines in terms of any 3 skew lines. I will briefly talk about Harris' results and how Stephen, Daniel, and I compute these formulas explicitly.

The talk will be held online via Bluejeans, use the following link to join the meeting.

### Tropical convex hulls of infinite sets

Series
Algebra Seminar
Time
Monday, April 13, 2020 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Cvetelina HillGeorgia Tech

In this talk we will explore the interplay between tropical convexity and its classical counterpart. In particular, we will focus on the tropical convex hull of convex sets and polyhedral complexes. We give a vertex description of the tropical convex hull of a line segment and of a ray in Rn/R1 and show that tropical convex hull and classical convex hull commute in R3/R1. Finally, we prove results on the dimension of tropical convex fans and give an upper bound on the dimension of the tropical convex hull of tropical curves under certain hypothesis.

The talk will be held online via Bluejeans, use the following link to join the meeting.

### Polynomials over real valued fields and other stuff about hyperfields

Series
Algebra Seminar
Time
Monday, April 6, 2020 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Trevor GunnGeorgia Tech

The main goal of this talk is to discuss my proof of a multiplicity formula for polynomials over a real valued field. I also want to talk about some of the raisons d’être for hyperfields and polynomials over hyperfields. This talk is based on my paper “A Newton Polygon Rule for Formally-Real Valued Fields and Multiplicities over the Signed Tropical Hyperfield” which is in turn based on a paper of Matt Baker and Oliver Lorscheid “Descartes' rule of signs, Newton polygons, and polynomials over hyperfields.”

The talk will be held online via Bluejeans. Use the following link to join the meeting.

### Linear and rational factorization of tropical polynomials

Series
Algebra Seminar
Time
Monday, March 30, 2020 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Bo LinGeorgia Tech

Already for bivariate tropical polynomials, factorization is an NP-Complete problem.In this talk, we will introduce a rich class of tropical polynomials in n variables, which admit factorization and rational factorization into well-behaved factors. We present efficient algorithms of their factorizations with examples. Special families of these polynomials have appeared in economics,discrete convex analysis, and combinatorics. Our theorems rely on an intrinsic characterization of regular mixed subdivisions of integral polytopes, and lead to open problems of interest in discrete geometry.

The talk will be held online via Bluejeans. Use the following link to join the meeting.

### Cancelled - A refined Brill-Noether theory over Hurwitz spaces

Series
Algebra Seminar
Time
Monday, March 23, 2020 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Hannah LarsonStanford University

This talk was cancelled due to the current status. The following is the original abstract for the talk. The celebrated Brill-Noether theorem says that the space of degree $d$ maps of a general genus $g$ curve to $\mathbb{P}^r$ is irreducible. However, for special curves, this need not be the case. Indeed, for general $k$-gonal curves (degree $k$ covers of $\mathbb{P}^1$), this space of maps can have many components, of different dimensions (Coppens-Martens, Pflueger, Jensen-Ranganathan). In this talk, I will introduce a natural refinement of Brill-Noether loci for curves with a distinguished map $C \rightarrow \mathbb{P}^1$, using the splitting type of push forwards of line bundles to $\mathbb{P}^1$. In particular, studying this refinement determines the dimensions of all irreducible components of Brill-Noether loci of general $k$-gonal curves.

### Generating functions for induced characters of the hyperoctahedral group

Series
Algebra Seminar
Time
Monday, March 9, 2020 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Mark SkanderaLehigh University

Merris and Watkins interpreted results of Littlewood to give generating functions for symmetric group characters induced from one-dimensional characters of Young subgroups.  Beginning with an n by n matrix X of formal variables, one obtains induced sign and trivial characters by expanding sums of products of certain determinants and permanents, respectively. We will look at a new analogous result which holds for hyperoctahedral group characters induced from the four one-dimensional characters of its Young subgroups.  This requires a 2n by 2n matrix of formal variables and four combinations of determinants and permanents.  This is joint work with Jongwon Kim.

### Toric Vector Bundles and the tropical geometry of piecewise-linear functions

Series
Algebra Seminar
Time
Monday, March 2, 2020 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Chris ManonUniversity of Kentucky

Like toric varieties, toric vector bundles are a rich class of algebraic varieties that can be described with combinatorial data.  Klyachko gave a classification of toric vector bundles in terms of certain systems of filtrations in a vector space.  I'll talk about some recent work with Kiumars Kaveh showing that Klyachko's data has an interesting interpretation in terms of tropical geometry.  In particular, we show that toric vector bundles can be classified by points on tropicalized linear spaces over a semifield of piecewise-linear functions.   I'll discuss how to use this recipe and a closely related tropicalization map to produce toric vector bundles and more general flat toric families.

### Differential Invariant Algebras

Series
Algebra Seminar
Time
Monday, February 24, 2020 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Peter OlverUniversity of Minnesota

A classical theorem of Lie and Tresse states that the algebra of differential invariants of a Lie group or (suitable) Lie pseudo-group action is finitely generated.  I will present a fully constructive algorithm, based on the equivariant method of moving frames, that reveals the full structure of such non-commutative differential algebras, and, in particular, pinpoints generating sets of differential invariants as well as their differential syzygies. Some applications and outstanding issues will be discussed.