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.

The KPZ equation is a singular stochastic PDE arising as a scaling limit of various physically and probabilistically interesting models. Often, this equation describes the “crossover” between Gaussian and non-Gaussian fluctuation behavior in simple models of interacting particles, directed polymers, or interface growth. It is a difficult and elusive open problem to elucidate the nature of this crossover for general stochastic interface models. In this talk, I will discuss a series of recent works where we have made progress in understanding the KPZ crossover for models of random walks in dynamical random media. This was done through a tilting-based approach to study the extreme tails of the quenched probability distribution. This talk includes joint work with Sayan Das and Hindy Drillick.

Zoom link:

https://gatech.zoom.us/j/96535844666

We say that a Riemannian manifold has good higher expansion if every rationally null-homologous i-cycle bounds an i+1 chain of comparatively small volume. The interactions between expansion, spectral geometry, and topology have long been studied in the settings of graphs and surfaces. In this talk, I will explain how to construct rational homology 3-spheres which are good higher expanders. On the other hand, I will show that such higher expanders must be rather topologically complicated; in particular, we will demonstrate a super-polynomial-in-volume lower bound on their torsion homology.

The compressible Euler and Navier-Stokes equations describe the motion of compressible fluids. The defocusing nonlinear Schr\"odinger equation is a dispersive equation that has application in many physics areas. Through the Madelung transformation, the defocusing nonlinear Schr\"odinger equation is connected with the compressible Euler equation. In this colloquium I will start from the compressible Euler/Navier-Stokes equation and introduce the blow-up result called implosion. Then I will introduce the defocusing nonlinear Schr\"odinger equation and the longstanding open problem on the blow-up of its solutions in the energy supercritical regime. In the end I will talk about the Madelung transformation and its application to transfer the implosion from the compressible Euler to the defocusing nonlinear Schr\"odinger equation. During the talk I will mention our work with Gonzalo Cao-Labora, Javier Gómez-Serrano and Gigliola Staffilani on the first non-radial implosion result for those three equations.