Mathematics

Random Walks on Out(F_r)

Catherine Pfaff (University of California at Santa-Barbara)

Abstract: While many mathematicians hypothesized for years as to which elements of mapping class groups and the Out(F_r) are generic, there has only in the past decade been an explosion of results on the topic. This explosion began with Maher and Rivin proving that pseudo-Anosovs are indeed generic within the mapping class group. Rivin further gave that fully irrreducibles (specifically fully irreducibles not induced by surface homeomorphisms) are generic. Many results followed by Sisto, Calegari-Maher, Maher-Tiozzo, Karlsson, Horbez, and Dahmani-Horbez. Kapovich-Pfaff gave a refinement of this work in a particular Out(F_r) setting by proving specific invariant values along a ``train track directed'' random walk. Answering a question of that paper, Gadre-Maher proved that pseudo-Anosovs are generically ``principal.'' Inspired by the work of Gadre-Maher, we are expanding the ` `train track directed'' random walk work to a full random walk on Out(F_r). This is joint work in progress.

Two endpoint bounds via inverse problems

Betsy Stovall (University of Wisconsin-Madison)

Abstract: In Mathematics in general, and Analysis in particular, one strategy for solving a problem is to begin by understanding what a counter-example would look like. In this talk, we will discuss two recent (and a few older) results for Fourier restriction operators and generalized Radon transforms that use a quantitative version of this approach. One of these results is joint work with Christ, Dendrinos, and Street.

Random walks, Laplacians, and volumes in sub-Riemannian geometry

Robert Neel (Lehigh University)

Abstract: We study a variety of random walks on sub-Riemannian manifolds and their diffusion limits, which give, via their infinitesimal generators, second-order operators on the manifolds. A primary motivation is the lack of a canonical Laplacian in sub-Riemannian geometry, and thus we are particularly interested in the relationship between the limiting operators, the geodesic structure, and operators which can be obtained as divergences with respect to various choices of volume. This work is joint with Ugo Boscain (CNRS), Luca Rizzi (CNRS), and Andrei Agrachev (SISSA).

A containers-type theorem for algebraic hypergraphs

Anton Bernshteyn (Illinois Math)

Abstract: An active avenue of research in modern combinatorics is extending classical extremal results to the so-called sparse random setting. The basic hope is that certain properties that a given "dense" structure is known to enjoy should be inherited by a randomly chosen "sparse" substructure. One of the powerful general approaches for proving such results is the hypergraph containers method, developed independently by Balogh, Morris, and Samotij and Saxton and Thomason. Another major line of study is establishing combinatorial results for algebraic or, more generally, definable structures. In this talk, we will combine the two directions and consider the following problem: Given a "dense" algebraically defined hypergraph, can we show that the subhypergraph induced by a generic low-dimensional algebraic set of vertices is also fairly "dense"? This is joint work with Michelle Delcourt (University of Birmingham) and Anush Tserunyan (UIUC).

A gentle approach to the de Rham-Witt complex

Akhil Mathew (The University of Chicago)

Abstract: The de Rham-Witt complex of a smooth algebra over a perfect field provides a chain complex representative of its crystalline cohomology, a canonical characteristic zero lift of its algebraic de Rham cohomology. We describe a simple approach to the construction of the de Rham-Witt complex. This relates to a homological operation L\eta_p on the derived category, introduced by Berthelot and Ogus, and can be viewed as a toy analog of a cyclotomic structure. This is joint work with Bhargav Bhatt and Jacob Lurie.

CANCELLED

Andrea Bertozzi (UCLA)