Georgia Institute of Technology College of Sciences

After providing a mathematical background for some curious optical experiments in the 19th century, I will then describe how these ideas inform our understanding of the Deift conjecture for the Korteweg--de Vries equation. Specifically, in joint work with Chapouto and Visan, we showed that the evolution of almost-periodic initial data need not remain almost periodic.

 We will survey a number of recent developments in the theory of completely integrable nonlinear dispersive PDE.  These include a priori bounds, orbital stability of multisolitons, well-posedness at optimal regularity, and the existence of dynamics for Gibbs distributed initial data. I will describe the basic objects that tie together these disparate results, as well as the diverse ideas required for each problem.

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. 

In this talk, I discuss how one can use approximate message passing (AMP) algorithms — a class of efficient, iterative algorithms that have been successfully employed in many statistical learning tasks like high-dimensional linear regression and low-rank matrix estimation — for characterizing exact statistical properties of estimators in a high-dimensional asymptotic regime, where the sample size of the data is proportional to the number of parameters in the problem. As a running example, we will study sorted L1 penalization (SLOPE) for linear regression and show how AMP theory can be used to give insights on the variable selection properties of this estimator by characterizing the optimal trade-off between measures of type I and type II error. Collaborators on this work include Zhiqi Bu, Oliver Feng, Jason Klusowski, Richard Samworth, Weijie Su, Ramji Venkataramanan, and Ruijia Wu.

I will present joint work in progress with Gero Friesecke.  We introduce a two parameter family of variational problems; varying the first parameter interpolates between a regularized version of Monge's optimal transport (OT) problem and Kantorovich's relaxed version.  The first limit problem has the advantage over Monge's original problem of always admitting a solution.  In cases where a (sufficiently regular) Monge map exists, the solution will be of such a form; if not, the limit problem essentially minimizes the transportation cost among all best approximations of the target measure by  pushforwards of the source.  When the source measure is discrete, we show that this is equivalent to the optimal quantization of the target measure, with the additional constraint that the weights of the approximating discrete masses are prescribed.  The second parameter controls the regularity of the pseudo-Monge map. In both the high and low regularity limits, the problem converges to the classical Kantorovich problem, under mild assumptions.

Part of the motivation for this problem is to understand whether the strictly correlated electron ansatz is valid in the semi-classical limit of density functional theory (DFT).  We will briefly discuss the corresponding application of OT to DFT, and outline what is known about the existence of Monge solutions (or, equivalently, the validity of the strictly correlated electron ansatz).

It is a fundamental question in Ramsey theory to determine the smallest possible independence number of an H-free hypergraph on n vertices. In the case of graphs, the problem was famously solved for H=K3 by Kim and for H=K4 (up to a logarithmic factor) by Mattheus-Verstraete in 2023. Even C4 and K5 remain wide open. We study the problem for 3-uniform hypergraphs and conjecture a full classification: the minimum independence number is poly(n) if and only if H is contained in the iterated blowup of the single-edge hypergraph. We prove this conjecture for all H with at most two tightly connected components. Based on joint work with Conlon, Fox, Gunby, Mubayi, Suk, Verstraete, and Yu.