Albert Cohen, Sorbonne
Location: Stevenson 1310

Motivated by non-intrusive approaches for high-dimensional parametric PDEs, we consider the general problem of approximating an unknown arbitrary function in any dimension from the data of point samples. The approximants are picked from given or adaptively chosen finite-dimensional spaces. One principal objective is to obtain an approximation which performs as good as the best possible using a sampling budget that is linear in the dimension of the approximating space. We will show that this objective can is met by taking a random sample distributed according to a well-chosen probability measure, and reconstructing by appropriate least-squares measures. We discuss these optimal sampling strategies in the adaptive context and for general non-tensor-product multivariate domains.

Location: Stevenson 1425

Carolyn Abbott, Columbia University
Location: Stevenson 1308

The properties of a random walk on a group which acts on a hyperbolic metric space have been well-studied in recent years. In this talk, I will focus on random walks on acylindrically hyperbolic groups, a class of groups which includes mapping class groups, Out(Fn), and right-angled Artin and Coxeter groups, among many others. I will discuss how a random element of such a group interacts with fixed subgroups, especially so-called hyperbolically embedded subgroups. In particular, I will discuss when the subgroup generated by a random element and a fixed subgroup is a free product, and I will also describe some of the geometric properties of that free product. This is joint work with Michael Hull.

Marie Jameson, University of Tennessee
Location: Stevenson 1320

The study of arithmetic properties of coefficients of modular forms enjoys a rich history, including deep results regarding congruences in arithmetic progressions. Recently, work of S. Ahlgren, B. Kim, N. Andersen, and S. Loebrich have employed the q-expansion principle of P. Deligne and M. Rapoport in order to determine more about where these congruences can occur. Here, we extend the method to give additional results for a large class of modular forms, and investigate the consequences of that result. (Joint work with S. Garthwaite.)

Craig Guilbault, University of Wisconsin-Milwaukee
Location: Stevenson 1206

Connected sums and boundary connected sums play important roles in the study of closed manifolds and manifolds with boundary, respectively. When working with open manifolds, a third variety of connected sum—the “end sum”—is useful. Each of these operations involves a number of arbitrary choices, making well-definedness of the resulting manifold a significant question. With regards to the end sum operation, we will discuss familiar situations where all goes smoothly (the notion of semistability plays a role here) and others where significant problems arise. Our analysis leads naturally into the subtle and interesting theory of infinitely generated abelian groups. The work to be presented in this talk is joint with Jack Calcut and Patrick Haggerty.

Leonardo Abbrescia, Michigan State University
Location: Stevenson 1307

We will present a series of geometric ideas that are helpful to study the initial value problem of quasilinear wave equations satisfying the null condition on the (1+1)-dimensional Minkowski space. Using a double-null geometric formulation, we show how the conformal invariance of the equation semilinearizes it into a system that is decoupled from the equations governing the null geometry. This allows us to solve the wave equations independently, which we exploit to show that the null geometry is sufficiently regular to guarantee global existence. If time permits, I will explain how this ties into a global wellposedness result with “large” initial data. This is joint work with Willie Wong.

Rares Rasdeaconu, Vanderbilt University
Location: Stevenson 1320

Marek Kaluba, Adam Mickiewicz University, Poland
Location: Stevenson 1308

Peter Bonventre, University of Kentucky
Location: Stevenson 1320

Zack Tripp, Vanderbilt University
Location: Stevenson 1206

Michael Hull, University of North Carolina, Greensboro
Location: Stevenson 1308

Julio Caceres, Vanderbilt University
Location: Stevenson 1432

Juhi Jang, University of Southern California
Location: Stevenson 1206

Thomas Sinclair, Purdue University
Location: Stevenson 1432

Casey Rodriguez, Massachusetts Institute of Technology
Location: Stevenson 1307

Jonathan Campbell, Duke University
Location: Stevenson 1320

Dan Ginsberg, Princeton University
Location: Stevenson 1307

Ravshan Ashurov, Institute of Mathematics, National University of Uzbekistan
Location: Stevenson 1307

Historically progress with solving the Luzin conjecture has been made by considering easier problems. For multiple Fourier series one of such easier problems is to investigate convergence almost everywhere of the spherical sums on TN\ supp(f) (so called the generalized localization principle). For the spherical partial integrals of multiple Fourier integrals the generalized localization principle in classes Lp(RN) was investigated by many authors. In particular, in the remarkable paper of A. Carbery and F. Soria the validity of the generalized localization was proved in Lp(RN) when 2 <= p < 2N/(N – 1). In this paper the generalized localization principle for the spherical partial sums of the multiple Fourier series in L2 – class is proved. It was previously known that the generalized localization was not valid in classes Lp(TN) when 1 <= p = 1: if p >= 2 then we have the generalized localization and if p < 2, then the generalized localization fails.