Category Archives: Fall 2014

Naoki Saito (December 1)

Note: this talk will not take place in the usual room. Location TBA.

Speaker: Naoki Saito (UC Davis)

Title: Laplacian eigenfunctions that do not feel the boundary: Theory, Computation, and Applications

Abstract: I will discuss Laplacian eigenfunctions defined on a Euclidean
domain of general shape, which “do not feel the boundary.”
These Laplacian eigenfunctions satisfy the Helmholtz equation inside the domain,
and can be extended smoothly and harmonically outside of the domain.
Although these eigenfunctions do not satisfy the usual Dirichlet or Neumann
boundary conditions, they can be computed via the eigenanalysis of the
integral operator (with the potential kernel) commuting with the Laplace
operator. Compared to directly solving the Helmholtz equations on such
domains, the eigenanalysis of this integral operator has several advantages
including the numerical stability and amenability to modern fast numerical
algorithms (e.g., the Fast Multipole Method).
In this talk, I will discuss their properties, the relationship with the
Krein-von Neumann self-adjoint extension of unbounded symmetric operators, and
certain applications including image extrapolation and characterization of
biological shapes.

Alexander Volberg (November 17)

Speaker: Alexander Volberg (MSU)

Title: Beyond the scope of doubling: weighted martingale multipliers and outer measure spaces

Abstract: A new approach to characterizing the unconditional basis property of martingale differences in weighted $L^2(w d\nu)$ spaces is given for arbitrary martingales, resulting in a new version with arbitrary and in particular non-doubling reference measure $\nu$. The approach combines embeddings into outer measure spaces with a core concavity argument of Bellman function type. Specifically, we prove that finiteness of the $A_2$ characteristic of the weight (defined through averages relative to arbitrary reference measure $\nu$) is equivalent to the boundedness of martingale multipliers. Even in the case of the usual dyadic martingales based on dyadic cubes in $\mathbb{R}^d$ our result is new because it is dimension free. In the case of general measures, this result is unexpected. For example, a small change in operator breaks the result immediately. This is a joint work with Christoph Thiele and Sergei Treil.

Sung-Jin Oh (November 10)

Speaker: Sung-Jin Oh (UC Berkeley)

Title: On the energy critical Maxwell-Klein-Gordon equations

In this talk I will present a recent joint work with D. Tataru on the global regularity and scattering for the Maxwell-Klein-Gordon equations on the (4+1)-dimensional Minkowski space, which is energy critical.

Lawrence C Evans (November 3)

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.

Boaz Haberman (October 27)

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.

Michael Christ (October 20)

Speaker: Michael Christ (UC Berkeley)

Title: An extremal problem concerning Fourier coefficients

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.

Ben Harrop-Griffiths (October 13)

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.

Alan Hammond (October 6)

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.