# 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

Abstract:
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.