Large random combinatorial objects---whether or not they are required to satisfy some contraints---tend to look alike. But what do they look like? To answer these questions it helps to have limit objects and avariational principle. In the case of permutations, the limit objects---called "permutons"---seem to behave exceptionally nicely. We'll give some theorems, some examples, and much speculation.Joint work with Rick Kenyon, Dan Kral, and Charles Radin.

For many applications in science and engineering, the ability to efficiently and accurately approximate solutions to elliptic PDEs dictateswhat physical phenomena can be simulated numerically. In this talk, we present a high-order accurate discretization techniquefor variable coefficient PDEs with smooth coefficients. Thetechnique comes with a nested dissection inspired direct solver that scales linearly or nearly linearly with respect to the number of unknowns. Unlike the application of nested dissection methods to classic discretization techniques, the constant prefactors do not grow with the order of the discretization. The discretization is robust even for problems with highly oscillatory solutions. For example, a problem 100 wavelengths in size can be solved to 9 digits of accuracy with 3.7 million unknowns on a desktopcomputer. The precomputation of the direct solver takes 6 minutes on a desktop computer. Then applying the computed solver takes 3 seconds. The recent application of the algorithm to inverse media scattering also will be presented.

Motivated by some results from differential geometry, I will discuss a purely algebraic problem of minimizing the normalizedvolume functional on valuation spaces at Kawamata log-terminal (Klt) singularities. In the case when the minimizers are associated to K-semistable log Fano cones (conjectured to be always true), we can prove that theyare unique among all quasi-monomial valuations. Moreover, any K-semistable log Fano cone degenerates uniquely to a K-polystable log Fano cone. This allows us to attach canonical semistable/polystable objects and volume-type invariant to any Klt singularity. As one application, we answer Donaldson-Sun's conjecture about algebraicity of metric tangent cones on Gromov-Hausdorff limits. This talk is mostly based on joint works with Chenyang Xu.