## Seminars and Colloquia Schedule

### Effective bounds for the measure of rotations

Series
CDSNS Colloquium
Time
Monday, October 28, 2019 - 11:15 for 1 hour (actually 50 minutes)
Location
Skiles 05
Speaker
Alex HaroUniv. de Barcelona

A fundamental question in Dynamical Systems is to identify regions of
phase/parameter space satisfying a given property (stability,
linearization, etc).  In this talk, given a family of analytic circle
diffeomorphisms depending on a parameter, we obtain effective (almost
optimal) lower bounds of the Lebesgue measure of the set of parameters
for which that diffeomorphism is conjugate to a rigid rotation.
We estimate this measure using an a-posteriori KAM
scheme that relies on quantitative conditions that
are checkable using computer-assistance. We carefully describe
how the hypotheses in our theorems are reduced to a finite number of
computations, and apply our methodology to the case of the
Arnold family, in the far-from-integrable regime.

This is joint work with Jordi Lluis Figueras and Alejandro Luque.

### Heegaard Floer homology and Seifert manifolds

Series
Geometry Topology Seminar Pre-talk
Time
Monday, October 28, 2019 - 12:45 for 1 hour (actually 50 minutes)
Location
Skile 006
Speaker
Sungkyung KangChinese University of Hong Kong

Heegaard Floer homology gives a powerful invariant of closed 3-manifolds. It is always computable in the purely combinatorial sense, but usually computing it needs a very hard work. However, for certain graph 3-manifolds, its minus-flavored Heegaard Floer homology can be easily computed in terms of lattice homology, due to Nemethi. I plan to give the basic definitions and constructions of Heegaard Floer theory and lattice homology, as well as the isomorphism between those two objects.

### Nonnegative symmetric polynomials and sums of squares with many variables

Series
Student Algebraic Geometry Seminar
Time
Monday, October 28, 2019 - 13:30 for 1 hour (actually 50 minutes)
Location
Skiles 254
Speaker
Jose Gabriel Acevedo HabeychGeorgia Tech

By using the representation theory of the symmetric group we try to compare, with respect to two different bases of the vector space of symmetric forms, the cones of symmetric nonnegative forms and symmetric sums of squares of a fixed even degree when the number of variables goes to infinity.

### Analysis and Applications of Nonsmooth Bifurcations

Series
Applied and Computational Mathematics Seminar
Time
Monday, October 28, 2019 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 05
Speaker
Oleg MakarenkovUniv Texas at Dallas
In this talk I will first give a brief overview of how nonsmooth bifurcations (border-splitting, grazing, and fold-fold bifurcations) help to rigorously explain the existence of nonsmooth limit cycles in the models of anti-lock braking systems, power converters, integrate-and-fire neurons, and climate dynamics. I will then focus on one particular application that deals with nonsmooth bifurcations in dispersing billiards. In [Nonlinearity 11 (1998)] Turaev and Rom-Kedar discovered that every periodic orbit that is tangent to the boundary of the billiard produces an island of stability upon smoothening the boundary of the billiard. The result to be presented in the talk (joint work with Turaev) proves that any dispersing billiard admits such an arbitrary small perturbation that ensures the occurrence of a tangent periodic orbit.

### Connected Floer homology of covering involutions

Series
Geometry Topology Seminar
Time
Monday, October 28, 2019 - 14:00 for 1 hour (actually 50 minutes)
Location
Skile 006
Speaker
Sungkyung KangChinese University of Hong Kong

Using the covering involution on the double branched cover of S3 branched along a knot, and adapting ideas of Hendricks-Manolescu and Hendricks-Hom-Lidman, we define new knot (concordance) invariants and apply them to deduce novel linear independence results in the smooth concordance group of knots. This is a joint work with A. Alfieri and A. Stipsicz.

### Knots, Legendrian Knots, and Their Invariants

Series
Time
Monday, October 28, 2019 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 171
Speaker
Dr. Caitlin LeversonGeorgia Tech
A knot can be thought of as a piece of string tied up, that then has its ends glued together. As long as we don’t cut the string, any way we move the string in space doesn’t change the knot we are considering. A surprisingly hard and interesting problem is, when handed two knots, how to determine if they are the same knot or not. We can further give structure to our knots and thus the problem, by adding geometric constraints to our knots, yielding what are called Legendrian knots. We can once again try to determine if two Legendrian knots are the same or not. In this talk I will introduce knots, Legendrian knots, and some ways we have to try to distinguish two knots or Legendrian knots, called knot invariants.

### Quantum fate of classical solitons

Series
Math Physics Seminar
Time
Monday, October 28, 2019 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Michael PustilnikSchool of Physics, Georgia Tech
This talk will focus on one-dimensional interacting quantum systems near the classical limit described by the Korteweg–de Vries (KdV) equation. Classical excitations in this regime are the well-known solitons, i.e., localized disturbances with particle-like properties, and delocalized waves of density, or phonons. It turns out, however, that the semiclassical description inevitably breaks down at long wavelengths. In this limit, quantum effects become dominant, the system is best described in terms of weakly interacting fermions, and classical solitons and phonons reach their ultimate quantum fate of being demoted to fermionic particles and holes.

We will give simple heuristic arguments in support of this claim and present the exact solution for the spectra of elementary excitations. The results are universally applicable to all quantum one-dimensional systems with a well-defined classical limit described by the KdV equation. This includes identical bosons with a weak short-range repulsion and identical particles, either bosons or fermions, with a strong long-range repulsion.

### 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 ZakharovCentral 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.

### Degenerating Einstein spaces

Series
PDE Seminar
Time
Tuesday, October 29, 2019 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Ruobing ZhangStony 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.

### 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

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.

### 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 HanGeorgia 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!

### 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 ShenfeldPrinceton 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.

### Degenerations and examples of Einstein manifolds

Series
Geometry Topology Student Seminar
Time
Wednesday, October 30, 2019 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Ruobing ZhangSUNY Stony Brook

The talk will discuss the relationship between topology and
geometry of Einstein 4-manifolds such as K3 surfaces.

### 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 ShenfeldPrinceton 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.

### 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 HendricksRutgers

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.

### Research proposal: Matchings in hypergraphs

Series
Other Talks
Time
Thursday, October 31, 2019 - 13:30 for 30 minutes
Location
Skiles 005
Speaker
Xiaofan YuanGeorgia 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.

### 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 BerkowitzYale 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.

### Renormalization for the almost Mathieu operator and related skew products.

Series
CDSNS Colloquium
Time
Friday, November 1, 2019 - 11:15 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Hans KochUniv. of Texas, Austin

Considering SL(2,R) skew-product maps over circle rotations,
we prove that a renormalization transformation
associated with the golden mean alpha
has a nontrivial periodic orbit of length 3.
We also present some numerical results,
including evidence that this period 3 describes
scaling properties of the Hofstadter butterfly
near the top of the spectrum at alpha,
and scaling properties of the generalized eigenfunction
for this energy.

### Asymptotic normality of the $r\to p$ norm for random matrices with non-negative entries

Series
ACO Student Seminar
Time
Friday, November 1, 2019 - 13:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Debankur MukherjeeISyE, Georgia Tech

For an $n\times n$ matrix $A_n$, the $r\to p$ operator norm is defined as $\|A_n\|_{r \to p}= \sup_{\|x\|_r\leq 1 } \|A_n x\|_p$ for $r,p\geq 1$. The $r\to p$ operator norm puts a huge number of important quantities of interest in diverse disciplines under a single unified framework. The application of this norm spans a broad spectrum of areas including data-dimensionality reduction in machine learning, finding oblivious routing schemes in transportation network, and matrix condition number estimation.

In this talk, we will consider the $r\to p$ norm of a class of symmetric random matrices with nonnegative entries, which includes the adjacency matrices of the Erd\H{o}s-R\'enyi random graphs and matrices with sub-Gaussian entries. For $1< p\leq r< \infty$, we establish the asymptotic normality of the appropriately centered and scaled $\|A_n\|_{r \to p}$, as $n\to\infty$. The special case $r=p=2$, which corresponds to the largest singular value of matrices, was proved in a seminal paper by F\"uredi and Koml\'os (1981). Of independent interest, we further obtain a sharp $\ell_\infty$-approximation for the maximizer vector. The results also hold for sparse matrices and further the $\ell_\infty$-approximation for the maximizer vector also holds for a broad class of deterministic sequence of matrices with certain asymptotic `expansion' properties.

This is based on a joint work with Souvik Dhara (MIT) and Kavita Ramanan (Brown U.).