Seminars and Colloquia by Series

Local limit theorems for combinatorial random variables

Series
Combinatorics Seminar
Time
Friday, November 1, 2019 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 249
Speaker
Ross Berkowitz – Yale University

Let X be the number of length 3 arithmetic progressions in a random subset of Z/101Z.  Does X take the values 630 and 640 with roughly the same probability?
Let Y denote the number of triangles in a random graph on n vertices.  Despite looking similar to X, the local distribution of Y is quite different, as Y obeys a local limit theorem.  
We will talk about a method for distinguishing when combinatorial random variables obey local limit theorems and when they do not.

Research proposal: Matchings in hypergraphs

Series
Other Talks
Time
Thursday, October 31, 2019 - 13:30 for 30 minutes
Location
Skiles 005
Speaker
Xiaofan Yuan – Georgia Tech –

I will introduce a minimum l-degree threshold for the existence of a nearly perfect (i.e., covering all but a constant number of vertices) matching in a k-graph where k ≥ 3 and k/2 < l ≤ k − 1. This is joint work with Hongliang Lu and Xingxing Yu.

This improves upon an earlier result of Hàn, Person, and Schacht for the range k/2 < l ≤ k − 1. In some cases, such a matching can in fact be near perfect (i.e., covering all but at most k vertices) and our bound on the minimum l-degree is best possible.

New invariants of homology cobordism

Series
School of Mathematics Colloquium
Time
Thursday, October 31, 2019 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Kristen Hendricks – Rutgers

This is a talk about 3-manifolds and knots. We will begin by reviewing some basic constructions and motivations in low-dimensional topology, and will then introduce the homology cobordism group, the group of 3-manifolds with the same homology as the 3-dimensional sphere up to a reasonable notion of equivalence. We will discuss what is known about the structure of this group and its connection to higher dimensional topology. We will then discuss some existing invariants of the homology cobordism group coming from gauge theory and symplectic geometry, particularly Floer theory. Finally, we will introduce a new invariant of homology cobordism coming from an equivariant version of the computationally-friendly Floer-theoretic 3-manifold invariant Heegaard Floer homology, and use it to construct a new filtration on the homology cobordism group and derive some structural applications. Parts of this talk are joint work with C. Manolescu and I. Zemke; more recent parts of this talk are joint work with J. Hom and T. Lidman.

The Ehrhard-Borell inequality and hypoelliptic diffusions

Series
High Dimensional Seminar
Time
Wednesday, October 30, 2019 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Yair Shenfeld – Princeton University –

The Ehrhard-Borell inequality stands at the top of the pyramid of Gaussian inequalities. It is a powerful and delicate statement about the convexity of the Gaussian measure. In this talk I will discuss the inequality and its beautiful proof by Borell. The delicate nature of the inequality however makes the characterization of the equality cases difficult and they were left unknown. I will explain how we solved this problem. Joint work with Ramon van Handel.

Quantum graphs, convex bodies, and a century-old problem of Minkowski

Series
Analysis Seminar
Time
Wednesday, October 30, 2019 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Yair Shenfeld – Princeton University –

That the ball minimizes surface area among all sets of fixed volume, was known since antiquity; this is equivalent to the fact that the ball is the unique set which yields equality in the isoperimetric inequality. But the isoperimetric inequality is only a very special case of quadratic inequalities about mixed volumes of convex bodies, whose equality cases were unknown since the time of Minkowski. This talk is about these quadratic inequalities and their unusual equality cases which we resolved using degenerate diffusions on the sphere. No background in geometry will be assumed. Joint work with Ramon van Handel.

Spectrum of quasi-periodic Schrodinger operators

Series
Research Horizons Seminar
Time
Wednesday, October 30, 2019 - 12:20 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Rui Han – Georgia Tech

One of the simplest and, at the same time, most prominent models for the discrete quasi-periodic Schrodinger operator is the almost Mathieu operator (also called the Harper's model). This simple-looking operator is known to present exotic spectral properties. Three (out of fifteen) of Barry Simon's problems on Schrodinger operators in the 21st century concerns the almost Mathieu operator. In 2014, Artur Avila won a Fields Medal for work including the solutions to these three problems. In this talk, I will concentrate on the one concerning the Lebesgue measure of the spectrum. I will also talk about the difficulties in generalizing this result to the extended Harper's model. Students with background in numerics are especially welcome to attend!

Likelihood challenges for big trees and networks

Series
Mathematical Biology Seminar
Time
Wednesday, October 30, 2019 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Claudia Solis-Lemus – University of Wisconsin-Madison

Usual statistical inference techniques for the tree of life like maximum likelihood and bayesian inference through Markov chain Monte Carlo (MCMC) have been widely used, but their performance declines as the datasets increase (in number of genes or number of species).

I will present two new approaches suitable for big data: one, importance sampling technique for bayesian inference of phylogenetic trees, and two, a pseudolikelihood method for inference of phylogenetic networks.

The proposed methods will allow scientists to include more species into the tree of life, and thus complete a broader picture of evolution.

Degenerating Einstein spaces

Series
PDE Seminar
Time
Tuesday, October 29, 2019 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Ruobing Zhang – Stony Brook University –
In the talk we discuss singularity formation of Einstein metrics as the underlying spaces degenerate or collapse. The usual analytic tools such as uniform Sobolev inequalities and nonlinear a priori estimates are unavailable in this context. We will describe an entirely new way to handle these difficulties, and construct degenerating Ricci-flat metrics with quantitative singularity behaviors.

Tropical covers with an abelian group action

Series
Algebra Seminar
Time
Tuesday, October 29, 2019 - 13:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Dmitry Zakharov – Central Michigan University –

Given a graph X and a group G, a G-cover of X is a morphism of graphs X’ --> X together with an invariant G-action on X’ that acts freely and transitively on the fibers. G-covers are classified by their monodromy representations, and if G is a finite abelian group, then the set of G-covers of X is in natural bijection with the first simplicial cohomology group H1(X,G).

In tropical geometry, we are naturally led to consider more general objects: morphisms of graphs X’ --> X admitting an invariant G-action on X’, such that the induced action on the fibers is transitive, but not necessarily free. A natural question is to classify all such covers of a given graph X. I will show that when G is a finite abelian group, a G-cover of a graph X is naturally determined by two data: a stratification S of X by subgroups of G, and an element of a cohomology group H1(X,S) generalizing the simplicial cohomology group H1(X,G). This classification can be viewed as a tropical version of geometric class field theory, and as an abelianization of Bass--Serre theory.

I will discuss the realizability problem for tropical abelian covers, and the relationship between cyclic covers of a tropical curve C and the corresponding torsion subgroup of Jac(C). The realizability problem for cyclic covers of prime degree turns out to be related to the classical nowhere-zero flow problem in graph theory.

Joint work with Yoav Len and Martin Ulirsch.

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