## Seminars and Colloquia by Series

### 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.

### Coalescence estimates for the corner growth model with exponential weights

Series
Stochastics Seminar
Time
Thursday, October 15, 2020 - 15:30 for 1 hour (actually 50 minutes)
Location
Speaker
Xiao ShenUniversity of Wisconsin

(Joint work with Timo Seppäläinen) We establish estimates for the coalescence time of semi-infinite directed geodesics in the planar corner growth model with i.i.d. exponential weights. There are four estimates: upper and lower bounds on the probabilities of both fast and slow coalescence on the correct spatial scale with exponent 3/2. Our proofs utilize a geodesic duality introduced by Pimentel and properties of the increment-stationary last-passage percolation process. For fast coalescence our bounds are new and they have matching optimal exponential order of magnitude. For slow coalescence, we reproduce bounds proved earlier with integrable probability inputs, except that our upper bound misses the optimal order by a logarithmic factor.

### 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.

### Perfect matchings in random hypergraphs

Series
Graph Theory Seminar
Time
Tuesday, October 13, 2020 - 15:45 for 1 hour (actually 50 minutes)
Location
Speaker
Matthew KwanStanford University

For positive integers $d < k$ and $n$ divisible by $k$, let $m_d(k,n)$ be the minimum $d$-degree ensuring the existence of a perfect matching in a $k$-uniform hypergraph. In the graph case (where $k=2$), a classical theorem of Dirac says that $m_1(2,n) = \lceil n/2\rceil$. However, in general, our understanding of the values of $m_d(k,n)$ is still very limited, and it is an active topic of research to determine or approximate these values. In the first part of this talk, we discuss a new "transference" theorem for Dirac-type results relative to random hypergraphs. Specifically, we prove that a random $k$-uniform hypergraph $G$ with $n$ vertices and "not too small" edge probability $p$ typically has the property that every spanning subgraph with minimum $d$-degree at least $(1+\varepsilon)m_d(k,n)p$ has a perfect matching. One interesting aspect of our proof is a "non-constructive" application of the absorbing method, which allows us to prove a bound in terms of $m_d(k,n)$ without actually knowing its value.

The ideas in our work are quite powerful and can be applied to other problems: in the second part of this talk we highlight a recent application of these ideas to random designs, proving that a random Steiner triple system typically admits a decomposition of almost all its triples into perfect matchings (that is to say, it is almost resolvable).

Joint work with Asaf Ferber.

### Mathematics of Soap Films

Series
Time
Monday, October 12, 2020 - 15:30 for 1 hour (actually 50 minutes)
Location
Bluejeans meeting https://bluejeans.com/759112674
Speaker
Dr. Ben JayeGeorgia Tech

In this talk we shall give a brief introduction to the mathematics of soap films (aka minimal surfaces). These are the surfaces that, amongst all possible surfaces with prescribed boundary, have the least area. If one dips a wire mesh into soap solution, then the surface formed is a minimal surface. We shall see how minimal surfaces arise in science and engineering, look at the physical laws that a minimal surface should obey, and see how much mathematicians understand about them.

### Numerical methods for solving nonlinear PDEs from homotopy methods to machine learning

Series
Applied and Computational Mathematics Seminar
Time
Monday, October 12, 2020 - 14:00 for 1 hour (actually 50 minutes)
Location
https://bluejeans.com/884917410
Speaker
Wenrui HaoPenn State University

Many systems of nonlinear PDEs are arising from engineering and biology and have attracted research scientists to study the multiple solution structure such as pattern formation. In this talk, I will present several methods to compute the multiple solutions of nonlinear PDEs. In specific, I will introduce the homotopy continuation technique to compute the multiple steady states of nonlinear differential equations and also to explore the relationship between the number of steady-states and parameters. Then I will also introduce a randomized Newton's method to solve the nonlinear system arising from neural network discretization of the nonlinear PDEs. Several benchmark problems will be used to illustrate these ideas.

### A contact invariant from bordered Heegaard Floer homology

Series
Geometry Topology Seminar
Time
Monday, October 12, 2020 - 14:00 for 1 hour (actually 50 minutes)
Location
https://dartmouth.zoom.us/j/98031035804?pwd=NnBpTlhVS2lzVzFWTkYyTlloeWVuQT09
Speaker

Given a contact structure on a bordered 3-manifold, we describe an invariant which takes values in the bordered sutured Floer homology of the manifold. This invariant satisfies a nice gluing formula, and recovers the Oszvath-Szabo contact class in Heegaard Floer homology. This is joint work with Alishahi, Foldvari, Hendricks, Licata, and Vertesi.

Zoom info:

Meeting ID: 980 3103 5804

Passcode: 196398

### Discrepancy Minimization via a Self-Balancing Walk

Series
Combinatorics Seminar
Time
Friday, October 9, 2020 - 15:00 for 1 hour (actually 50 minutes)
Location
Speaker
Yang P. LiuStanford University

We study discrepancy minimization for vectors in R^n under various settings. The main result is the analysis of a new simple random process in multiple dimensions through a comparison argument. As corollaries, we obtain bounds which are tight up to logarithmic factors for several problems in online vector balancing posed by Bansal, Jiang, Singla, and Sinha (STOC 2020), as well as linear time algorithms for logarithmic bounds for the Komlós conjecture.

Based on joint work with Alweiss and Sawhney, see https://arxiv.org/abs/2006.14009

### Hyperbolic Relaxations of Locally Positive Semidefinite Matrices

Series
ACO Student Seminar
Time
Friday, October 9, 2020 - 13:00 for 1 hour (actually 50 minutes)
Location
https://bluejeans.com/264244877/0166
Speaker
Kevin ShuMath, Georgia Tech

Semidefinite programming is a powerful optimization tool, which involves optimizing linear functions on a slice of the positive semidefinite matrices. Locally PSD matrices are a natural relaxation of the PSD matrices which can be useful in reducing the space required for semidefinite optimization. We use the theory of hyperbolic polynomials to give precise quantitative bounds on the quality of the approximation resulting from optimizing over the locally-psd cone instead of the PSD cone.

### 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