Past Events

We consider the Glauber dynamics (also called Gibbs sampling) for sampling from a discrete high-dimensional space, where in each step one variable is chosen uniformly at random and gets updated conditional on all other variables. We show an optimal...

The form and function of tissues and organs emerge out of cell-to-cell interactions. Cell interaction dynamics take place on a variety of temporal and spatial scales, and reflect processes—diffusion, migration, force production/sensing, growth, and...

Dyck paths and standard Young tableaux (SYT) are two of the most central sets in combinatorics. There is a well-known bijection between Dyck paths with 2n steps and SYT of shape (n,n). In recent work, we found nine other bijections between classes of...

Moduli spaces of points on n-spheres carry natural actions of braid groups. For n=0,1, and 3, we prove that these symmetries extend to actions of mapping class groups of positive genus surfaces, through exceptional isomorphisms with certain character...

In an instance of the (metric) traveling salesperson problem (TSP), we are given a list of n cities and their pairwise symmetric distances satisfying the triangle inequality, and we want to find the shortest tour that visits all cities exactly once...

Any translation surface can be presented as a collection of polygons in the plane with sides identified. By acting linearly on the polygons, we obtain an action of GL(2,R) on moduli spaces of translation surfaces. Recent work of Eskin, Mirzakhani...

A connected matching is a matching contained in a connected component. A well-known method due to Łuczak reduces problems about monochromatic paths and cycles in complete graphs to problems about monochromatic matchings in almost complete graphs. We...

The congruent number problem is the following question. Consider a right-angled triangle; what natural numbers can arise from its area, all of whose sides have lengths that are rational numbers? Though this question is easy to frame, it is related to...

We obtain the asymptotic formula with an error term O ( X 1 2 + ε ) O ( X 1 2 + ε ) for the smoothed first moment of quadratic twists of modular L L -functions. The argument is largely based on a recursive method of Young (Acta Arith. 138(1):73–99...