Georgia Institute of Technology College of Sciences

Caleb McFarland: We prove a structure theorem for Γ-labeled graphs G which forbid a fixed Γ-labeled graph H as an immersion in the case that Γ is a finite abelian group. Joint work with Rose McCarty and Paul Wollan.
Richter Jordaan: In this expository talk I will give introduce an approach to the cycle double cover based on the more general problem of finding specific cycle covers of cubic graphs. After stating the basics of the cycle double cover conjecture and structure of a minimal counterexample, I'll try to describe the setup and basic intuition behind how the general cyle cover problem could be used to approach the cycle double cover conjecture.
Owen Huang: We will discuss some recent work with Rose McCarty concerning the product structure of Cayley graphs. We also introduce an integer-valued invariant of finitely generated groups and note its relevance in geometric group theory. 

Let $d\geq 2$, and $n\geq 2d+2$. Frankl and Pach initiated the study of the maximum size of a $(d+1)$-uniform set system in $[n]$ with VC-dimension at most $d$. The best-known upper bound is essentially $\binom{n}{d}$, and the best-known lower bound is $\binom{n-1}{d} + \binom{n-4}{d-2}$. In this talk, I will discuss some recent improvements on the upper bound and some interesting connections between this problem and the celebrated Erdős--Ko--Rado theorem. In particular, I will discuss our conjecture, which can be viewed as a generalization of the EKR as well as an "uniform version" of the disproved Erdős--Frankl--Pach conjecture, and highlight some of our partial progress. Joint work with Ting-Wei Chao, Zixiang Xu, and Shengtong Zhang.

N. Zabusky coined the word "soliton" in 1965 to describe a curious feature he and M. Kruskal observed in their numerical simulations of the initial-value problem for a simple nonlinear PDE. The first part of the talk will be a broad introduction to the theory of solitons/solitary waves and integrable PDEs (the Korteweg-de Vries equation in particular), describing classical results in the field. The second (and main) part of the talk will focus on some new developments and growing interest into a special case of solutions defined as "soliton gas".

We study random configurations of N soliton solutions q_N(x,t) of the KdV equation. The randomness appears in the scattering (linear) problem, which is used to solve the PDE: the complex eigenvalues are chosen to be (1) i.i.d. random variables sampled from a probability distribution with compact support on the complex plane, or (2) sampled from a random matrix law. 

Next, we consider the scattering problem for the expectation of the random measure associated to the spectral data, in the limit as N -> + infinity. The corresponding solution q(x,t) of the KdV equation is a soliton gas. 

We are then able to prove a Law of Large Numbers and a Central Limit Theorem for the differences q_N(x,t)-q(x,t).

This is a collection of works (and ongoing collaborations) done with K. McLaughlin (Tulane U.), T. Grava (SISSA/Bristol), R. Jenkins (UCF), A. Minakov (U. Karlova), J. Najnudel (Bristol).