Scott Armstrong, Université Paris-Dauphine Title: A quantitative theory of stochastic homogenization Abstract: Stochastic homogenization involves the study of solutions of partial differential equations with random coefficients, which are assumed to satisfy a "mixing" condition, for instance, an independence assumption of some sort. One typically wants information about the behavior of the solutions on very large scales, so that the ("microscopic") length scale of the correlations of the random field is comparatively small. In the asymptotic limit, one expects to see that the solutions behave like those of a constant-coefficient, deterministic equation. In this talk, we consider uniformly elliptic equations in divergence form, which has applications to the study of diffusions in random environments and effective properties of composite materials. Our interest is in obtaining quantitative results (e.g., error estimates in homogenization) and to understand the solutions on every length scale down to the microscopic scales. In joint work with Tuomo Kuusi and Jean-Christophe Mourrat, we introduce a new method for analyzing this problem, based on a higher-order regularity theory for equations with random coefficients, which, by a bootstrap argument, accelerates the exponent representing the scaling of the error the all the way to the optimal exponent given by the scaling of the central limit theorem.
Hans Lindblad, Johns Hopkins University Title: A sharp counter example to local existence for Einstein equations in wave coordinates Abstract: We are concerned with how regular initial data have to be to ensure local existence for Einstein's equations in wave coordinates. Klainerman-Rodnianski and Smith-Tataru showed that there in general is local existence for data in H^s for s>2. We give example of data in H^2 for which there is no local solution in H2. This is joint work with Boris Ettinger.
Camil Muscalu, Cornell University Title: Multiple vector valued inequalities via the helicoidal method Abstract: The goal of the talk is to describe a new method for proving vector valued inequalities in harmonic analysis, which we like to call "the helicoidal method". Some applications of it will also be discussed. This is recent joint work with Cristina Benea.
Malabika Pramanik, University of British Columbia Title: Configurations in sets big and small Abstract: Does a set of positive Lebesgue measure contain an affine copy of your favourite pattern, say a line of specially arranged points, the vertices of a polyhedron or a geometric sequence on a spiral? Would the answer change if the set is Lebesgue-null, but is still large in some quantifiable sense? Such problems, involving identification of prescribed configurations, have been vigorously pursued both in the discrete and continuous setting, often with spectacular results. Yet many deceptively simple questions remain open. I will survey the literature in this area, emphasizing some of the landmark results that focus on different aspects of the problem.
Monica Visan, UCLA Title: Symplectic non-squeezing for the cubic nonlinear Schrodinger equation on R^2 Abstract: We prove that the flow of the cubic NLS in two dimensions cannot squeeze a ball in $L^2$ into a cylinder of lesser radius. This is a PDE analogue of Gromov's non-squeezing theorem for an infinite-dimensional Hamiltonian PDE in infinite volume. This is joint work with R. Killip and X. Zhang.
