Georgia Institute of Technology College of Sciences

Through the pioneering numerical computations of Fermi-Pasta-Ulam (mid 50s) and Kruskal-Zabusky (mid 60s) it was observed that nonlinear equations modeling wave propagation asymptotically decompose as a superposition of “traveling waves” and “radiation”. Since then, it has been a widely believed (and supported by extensive numerics) that “coherent structures” together with radiations describe the long-time asymptotic behavior of generic solutions to nonlinear dispersive equations. This belief has come to be known as the “soliton resolution conjecture”.  Roughly speaking it tells that, asymptotically in time, the evolution of generic solutions decouples as a sum of modulated solitary waves and a radiation term that disperses. This remarkable claim establishes a drastic “simplification” to the complex, long-time dynamics of general solutions. It remains an open problem to rigorously show such a description for most dispersive equations.  After an informal introduction to dispersive equations, I will survey some of my recent results towards understanding the long-time behavior of dispersive waves and the soliton resolution using techniques from both partial differential equations and inverse scattering transforms.

Thresholds for increasing properties of random structures are a central concern in probabilistic combinatorics and related areas. In 2006, Kahn and Kalai conjectured that for any nontrivial increasing property on a finite set, its threshold is never far from its "expectation-threshold," which is a natural (and often easy to calculate) lower bound on the threshold. In this talk, I will first introduce the Kahn-Kalai Conjecture with some motivating examples and then talk about the recent resolution of a fractional version of the Kahn-Kalai Conjecture due to Frankston, Kahn, Narayanan, and myself. Some follow-up work, along with open questions, will also be discussed.

This talk is about the arithmetic of points of small canonical height relative to dynamical systems over number fields, particularly those aspects amenable to the use of equidistribution techniques. Past milestones in the subject include the proof of the Bogomolov Conjecture given by Ullmo and Zhang, and Baker-DeMarco's work on the finiteness of common preperiodic points of unicritical maps. Recently, quantitative equidistribution techniques have emerged both as a way of improving upon some of these old results, and as an avenue to studying previously inaccessible problems, such as the Uniform Boundedness Conjecture of Morton and Silverman. I will describe the key ideas behind these developments, and raise related questions for future research. 

https://bluejeans.com/788895268/8348

I will present some recent results, obtained with D. Bresch and Z. Wang, on large stochastic many-particle or multi-agent systems. Because such systems are conceptually simple but exhibit a wide range of emerging macroscopic behaviors, they are now employed in a large variety of applications from Physics (plasmas, galaxy formation...) to the Biosciences, Economy, Social Sciences...

The number of agents or particles is typically quite large, with 10^20-10^25 particles in many Physics settings for example and just as many equations. Analytical or numerical studies of such systems are potentially very complex  leading to the key question as to whether it is possible to reduce this complexity, notably thanks to the notion of propagation of chaos (agents remaining almost uncorrelated).

To derive this propagation of chaos, we have introduced a novel analytical method, which led to the resolution of two long-standing conjectures:
        _The quantitative derivation of the 2-dimensional incompressible Navier-Stokes system from the point vortices dynamics;
       _The derivation of the mean-field limit for attractive singular interactions such as in the Keller-Segel model for chemotaxis and some Coulomb gases.