We will present a new quantitative approach to the problem of proving hydrodynamic limits from microscopic stochastic particle systems, namely the zero-range, the simple exclusion and the Ginzburg-Landau process with Kawasaki dynamics, to macroscopic partial differential equations. Our method combines a modulated Wasserstein distance estimate comparing the law of the stochastic process to the local Gibbs measure, together with stability estimates a la Kruzhkov in weak distance and consistency estimates exploiting the regularity of the limit solution. It is simplified as it avoids the use of the block estimates.

This is a joint work with Clement Mouhot (University of Cambridge) and Daniel Marahrens.

In the past two decades there has been considerable progress on understanding the asymptotic fluctuations of random growing interfaces. These are often described by the Kardar-Parisi-Zhang equation, but this is a just one member of a huge universality class, and we have learned that special discretizations can have a high degree of solvability. In the long-time, large-space limit, there is a new universal fixed point with unexpected connections to classical integrable systems. The talk will be a gentle introduction to these developments.

What is the most correlated coupling between two Gaussian measures? How can one rearrange the mass of a Gaussian measure to recover another Gaussian measure? What is a natural notion of average for a collection of Gaussian measures? How do these questions depend on dimension? This talk will explore these questions and their interconnections, and briefly touch on their relevance in such diverse topics as computational linguistics, DNA mechanics, shape theory, and quantum information.