Seminars and Colloquia by Series

Couplings of Markov chain Monte Carlo and their uses

Series
Stochastics Seminar
Time
Thursday, September 10, 2020 - 15:30 for 1 hour (actually 50 minutes)
Location
https://us02web.zoom.us/j/83378796301
Speaker
Pierre Jacob – Harvard University

Markov chain Monte Carlo (MCMC) methods are state-of-the-art techniques for numerical integration. MCMC methods yield estimators that converge to integrals of interest in the limit of the number of iterations, obtained from Markov chains that converge to stationarity. This iterative asymptotic justification is not ideal. Indeed the literature offers little practical guidance on how many iterations should be performed, despite decades of research on the topic. This talk will describe a computational approach to address some of these issues. The key idea, pioneered by Glynn and Rhee in 2014, is to generate couplings of Markov chains, whereby pairs of chains contract, coalesce or even "meet" after a random number of iterations; we will see that these meeting times, which can be simulated in many practical settings, contain useful information about the finite-time marginal distributions of the chains. This talk will provide an overview of this line of research, joint work with John O'Leary, Yves Atchadé and various collaborators.
The main reference is available here: https://rss.onlinelibrary.wiley.com/doi/abs/10.1111/rssb.12336

Singularity of sparse Bernoulli matrices with$ p$ is close to $\log(n)/n$.

Series
High Dimensional Seminar
Time
Wednesday, September 9, 2020 - 15:00 for 1 hour (actually 50 minutes)
Location
Join Zoom Meeting https://us02web.zoom.us/j/88203571169 Meeting ID: 882 0357 1169
Speaker
Han Huang – Georgia Tech –

It has been conjectured that for a sufficiently large $n$, and $p = p_n \in [\log(n)/n, 1/2)$, the probability that a $n\times n$ Bernoulli($p$) matrix $A$ is singular equals to the probability that $A$ contains of a zero row or zero column up to a negligible error.

This conjecture has been recently proved by Litvak-Tikhomirov in the regime $ C\log(n)/ n < p < 1/C$ for some universal constant $C>1$ with their new tool. While for $p = (1+o(1)) \log(n) /n$, it also holds due to a result of Basak-Rudelson. In this talk, we will discuss how to extend their results to fill the gap between these two regions. ( $1\le pn/\log(n) <\infty$ )

Knot Concordance

Series
Geometry Topology Student Seminar
Time
Wednesday, September 9, 2020 - 14:00 for 1 hour (actually 50 minutes)
Location
Speaker
Hugo Zhou – Georgia Tech

Two knots are concordant to each other if they cobound an annulus in the product of S^3. We will discuss a few basic facts about knot concordance and look at J. Levine’s classical result on the knot concordance group.

Unavoidable dense induced subgraphs

Series
Graph Theory Seminar
Time
Tuesday, September 8, 2020 - 15:45 for 1 hour (actually 50 minutes)
Location
https://bluejeans.com/681348075/???? (replace ???? with password). For password, please email Anton Bernshteyn (bahtoh~at~gatech.edu)
Speaker
Rose McCarty – University of Waterloo –

Thomassen conjectures that every graph of sufficiently large average degree has a subgraph of average degree at least d and girth at least k, for any d and k. What if we want the subgraph to be induced? Large cliques and bicliques are the obvious obstructions; we conjecture there are no others. We survey results in this direction, and we prove that every bipartite graph of sufficiently large average degree has either K_{d,d} or an induced subgraph of average degree at least d and girth at least 6.

On the Helfgott-Lindenstrauss conjecture for linear groups

Series
Combinatorics Seminar
Time
Friday, September 4, 2020 - 15:00 for 1 hour (actually 50 minutes)
Location
https://bluejeans.com/751242993/PASSWORD (To receive the password, please email Lutz Warnke)
Speaker
Brendan Murphy – University of Bristol

Freiman's theorem characterizes finite subsets of abelian groups that behave "approximately" like subgroups: any such set is (roughly) a sum of arithmetic progressions and a finite subgroup. Quantifying Freiman's theorem is an important area of additive combinatorics; in particular, proving a "polynomial" Freiman theorem would be extremely useful.

The "Helfgott-Lindenstrauss conjecture" describes the structure of finite subsets of non-abelian groups that behave approximately like subgroups: any such set is (roughly) a finite extension of a nilpotent group. Breuillard, Green, and Tao proved a qualitative version of this conjecture. In general, a "polynomial" version of the HL conjecture cannot hold, but we prove that a polynomial version of the HL conjecture is true for linear groups of bounded rank.

In this talk, we will see how the "sum-product phenomenon" and its generalizations play a crucial role in the proof of this result. The amount of group theory needed is minimal.

Hyperbolic polynomial and its application to locally PSD matrices

Series
Student Algebraic Geometry Seminar
Time
Friday, September 4, 2020 - 09:00 for 1 hour (actually 50 minutes)
Location
Microsoft Teams Meeting
Speaker
Shengding Sun – Georgia Tech –

Please Note: https://teams.microsoft.com/l/meetup-join/19%3a3a9d7f9d1fca4f5b991b4029b09c69a1%40thread.tacv2/1598721101106?context=%7b%22Tid%22%3a%22482198bb-ae7b-4b25-8b7a-6d7f32faa083%22%2c%22Oid%22%3a%223eebc7e2-37e7-4146-9038-a57e56c92d31%22%7d

The definition of hyperbolic polynomials stems from stable polynomials, with many interesting properties related to convex geometry and optimization, including the construction of hyperbolicity cone. We will discuss some of these results and mention the application to locally PSD matrices.

A tale of two polytopes: The bipermutahedron and the harmonic polytope

Series
School of Mathematics Colloquium
Time
Thursday, September 3, 2020 - 11:00 for 1 hour (actually 50 minutes)
Location
https://us02web.zoom.us/j/89107379948
Speaker
Federico Ardila – San Francisco State University –

This talk's recording is available here.

The harmonic polytope and the bipermutahedron are two related polytopes which arose in our work with Graham Denham and June Huh on the Lagrangian geometry of matroids. This talk will explain their geometric origin and discuss their algebraic and geometric combinatorics.

The bipermutahedron is a (2n−2)-dimensional polytope with (2n!)/2^n vertices and 3^n−3 facets. Its f-polynomial, which counts the faces of each dimension, is given by a simple evaluation of the three variable Rogers-Ramanujan function. Its h-polynomial, which gives the dimensions of the intersection cohomology of the associated topic variety, is real-rooted, so its coefficients are log-concave.

The harmonic polytope is a (2n−2)-dimensional polytope with (n!)^2(1+1/2+...+1/n) vertices and 3^n−3 facets. Its volume is a weighted sum of the degrees of the projective varieties of all the toric ideals of connected bipartite graphs with n edges; or equivalently, a weighted sum of the lattice point counts of all the corresponding trimmed generalized permutahedra.

These two polytopes are related by a surprising fact: in any dimension, the Minkowski quotient of the bipermutahedron and the harmonic polytope equals 2.

The talk will be as self-contained as possible, and will feature joint work with Graham Denham, Laura Escobar, and June Huh.

Symplectic Fillings of Contact Structures

Series
Geometry Topology Student Seminar
Time
Wednesday, September 2, 2020 - 14:00 for 1 hour (actually 50 minutes)
Location
Speaker
Agniva Roy – Georgia Tech

Finding fillings of contact structures is a question that has been studied extensively over the last few decades. In this talk I will discuss some motivations for studying this question, and then visit a few ideas involved in the earliest results, due to Eliashberg and McDuff, that paved the way for a lot of current research in this direction.

Saturation problems in Ramsey theory, ordered sets and geometry

Series
Graph Theory Seminar
Time
Tuesday, September 1, 2020 - 15:45 for 1 hour (actually 50 minutes)
Location
https://bluejeans.com/681348075/????. (replace ???? with password) For password, please email Anton Bernshteyn (bahtoh~at~gatech.edu)
Speaker
Zhiyu Wang – Georgia Tech

A graph G is F-saturated if G is F-free and G+e is not F-free for any edge not in G. The saturation number of F, is the minimum number of edges in an n-vertex F-saturated graph. We consider analogues of this problem in other settings.  In particular we prove saturation versions of some Ramsey-type theorems on graphs and Dilworth-type theorems on posets. We also consider semisaturation problems, wherein we only require that any extension of the combinatorial structure creates new copies of the forbidden configuration.  In this setting, we prove a semisaturation version of the Erdös-Szekeres theorem on convex k-gons, as well as multiple semisaturation theorems for sequences and posets. Joint work with Gábor Damásdi, Balázs Keszegh, David Malec, Casey Tompkins, and Oscar Zamora.

Integral neural networks with weight penalization

Series
Analysis Seminar
Time
Tuesday, September 1, 2020 - 14:00 for 1 hour (actually 50 minutes)
Location
https://us02web.zoom.us/j/87104893132
Speaker
Armenak Petrosyan – Georgia Tech –

Artificial neural networks have gained widespread adoption as a powerful tool for various machine learning tasks in recent years. Training a neural network to approximate a target function involves solving an inherently non-convex problem. In practice, this is done using stochastic gradient descent with random initialization. For the approximation problem with neural networks error rate guarantees are established for different classes of functions however these rates are not always achieved in practice due to many  local minima of the resulting optimization problem. 

The challenge we address in this work is the following. We want to find small size shallow neural networks that can be trained algorithmically and which achieve guaranteed approximation speed and precision. To maintain the small size we apply penalties on the weights of the network. We show that under minimal requirements, all local minima of the resulting problem are well behaved and possess a desirable small size without sacrificing precision. We adopt the integral neural network framework and use techniques from optimization theory and harmonic analysis to prove our results. In this talk, we will discuss our existing work and possible future promising areas of interest where this approach can potentially be adopted. 

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