Speaker: Yaiza Canzani (Harvard)

Title: On the geometry and topology of zero sets of Schrödinger eigenfunctions.

Abstract: In this talk I will present some new results on the structure of the zero sets of Schrödinger eigenfunctions on compact Riemannian manifolds.  I will first explain how wiggly the zero sets can be by studying the number of intersections with a fixed curve as the eigenvalue grows to infinity. Then, I will discuss some results on the topology of the zero sets when the eigenfunctions are randomized. This talk is based on joint works with John Toth and Peter Sarnak.

Speaker: Tristan Buckmaster (NYU)
Title: Onsager’s Conjecture
Abstract: In 1949, Lars Onsager in his famous note on statistical hydrodynamics conjectured that weak solutions to the Euler equation belonging to Hölder spaces with Hölder exponent greater than 1/3 conserve energy; conversely, he conjectured the existence of solutions belonging to any Hölder space with exponent less than 1/3 which dissipate energy.

The first part of this conjecture has since been confirmed (cf. Eyink 1994, Constantin, E and Titi 1994). During this talk we will discuss recent work by Camillo De Lellis, László Székelyhidi Jr., Philip Isett and myself related to resolving the second component of Onsager’s conjecture. In particular, we will discuss the construction of weak non-conservative solutions to the Euler equations whose Hölder $1/3-\epsilon$ norm is Lebesgue integrable in time.

Speaker: Jeff Calder (UC Berkeley)

Title: A PDE-proof of the continuum limit of non-dominated sorting

Abstract: Non-dominated sorting is a combinatorial problem that is fundamental in multi-objective optimization, which is ubiquitous in engineering and scientific contexts. The sorting can be viewed as arranging points in Euclidean space into layers according to a partial order. It is equivalent to several well-known problems in probability and combinatorics, including the longest chain problem, and polynuclear growth. Recently, we showed that non-dominated sorting of random points has a continuum limit that corresponds to solving a Hamilton-Jacobi equation in the viscosity sense. Our original proof was based on a continuum variational problem, for which the PDE is the associated Hamilton-Jacobi-Bellman equation. In this talk, I will sketch a new proof that avoids this variational interpretation, and uses only PDE techniques. The proof borrows ideas from the Barles-Souganidis framework for proving convergence of numerical schemes to viscosity solutions. As a result, it seems this proof is more robust, and we believe it can be applied to many other problems that do not have obvious underlying variational principles. I will finish the talk by briefly sketching some current problems of interest.

Speaker: Dmitry Jakobson (McGill)

Title: Probability measures on manifolds of Riemannian metrics

Abstract: This is joint work with Y. Canzani, B. Clarke, N. Kamran, L. Silberman and J. Taylor. We construct Gaussian measures on the manifold of Riemannian metrics with the fixed volume form. We show that diameter, Laplace eigenvalue and volume entropy functionals are all integrable with respect to our measures. We also compute the characteristic function for the L^2(Ebin) distance from a random metric to the reference metric.

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.

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.

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.

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.