Title: Convexity, nonlinear PDE and principal/agent problems

Speaker: Lawrence C Evans (UC Berkeley)

Abstract: I will explain a simple convexity argument that provides an easy derivation of Sannikov’s optimality condition for continuous time principal/agent problems in economics.

Speaker: Boaz Haberman (UC Berkeley)

Title: Calderón’s problem for rough conductivities

Abstract: Calderon’s problem asks whether the coefficients of an elliptic equation can be recovered from its Dirichlet-to-Neumann map. Sylvester and Uhlmann introduced the method of complex geometrical optics solutions to solve this problem. In this talk we will discuss how to use some methods from dispersive equations to construct these solutions under more general regularity conditions for the coefficients.

Speaker: Michael Christ (UC Berkeley)

Title: An extremal problem concerning Fourier coefficients

Abstract:
Consider a set in Euclidean space, and consider the $L^q$ norm of its Fourier transform. Among sets of specified measure, what is the largest value of this norm? Is it attained? If so, by which sets?

These natural questions seem to have received little attention. I will state several partial results, and indicate some of the ideas in the proofs. One ingredient is a compactness theorem, whose proof relies on an inverse theorem of additive combinatorics.

Speaker: Ben Harrop-Griffiths (UC Berkeley)

Title: The lifespan of small solutions to the KP-I

Abstract: We show that for small, localized initial data there exists a global solution to the KP-I equation in a Galilean-invariant space using the method of testing by wave packets. This is joint work with Mihaela Ifrim and Daniel Tataru.

Speaker: Alan Hammond (UC Berkeley)

Title: Moment bounds and mass-conservation in PDE modelling coalescence

Abstract: We examine the behaviour of solutions to a system of PDE
(the Smoluchowski PDE), that model the aggregation of
mass-bearing particles that diffuse and are prone to
coagulate in pairs at close range. Conditions under which
these solutions conserve mass for all time will be
presented, along with stronger estimates, moment bounds
that show that heavy particles are rare. Uniqueness of
solutions also follows from the moment bounds.

This is joint work with Fraydoun Rezakhanlou.