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.