Georgia Institute of Technology College of Sciences

The advancement of technology has significantly enhanced our capacity to collect data. However, in many real-world applications, certain inherent limitations, such as the precision of measurement devices, environmental conditions, or operating costs, can result in missing data. In this talk, we focus on the setting where the available data consists of pairwise distances between a set of points, with the goal of estimating the configuration of the underlying geometry from incomplete distance measurements. This is known as the Euclidean distance geometry (EDG) problem and is central to many applications.

We first start by describing the solution when all distances are given using the classical multidimensional scaling (MDS) technique and then discuss a constructive approach to interpret the key mathematical objects in MDS. Next, we introduce a mathematical framework to address the EDG problem under two sampling models of the distance matrix: global sampling (uniform sampling of the entries of the distance matrix) and structured local sampling, where the measurements are limited to a subset of rows and columns. We discuss the conditions required for the exact recovery of the point configuration and the associated algorithms. The last part of the talk will illustrate the algorithms using synthetic and real data and discuss ongoing work.

Heegaard Floer homology is a tool for studying three- and four-dimensional manifolds, using methods that are inspired by symplectic geometry. Bordered Floer homology is tool, currently under construction, for understanding how to reconstruct the Heegaard Floer homology in terms of invariants associated to its pieces. This approach has both conceptual and computational ramifications. In this talk, I will sketch the outlines of Heegaard Floer homology, with an emphasis on recent progress in bordered Floer homology. Heegaard Floer homology was developed in collaboration with Zoltan Szabo; bordered Floer homology is joint work with Robert Lipshitz and Dylan Thurston.

Ultrafilters formalize a generalized notion of convergence based on a prescribed idea of "largeness" for subsets of the natural numbers, and underlie constructions like ultraproducts. In the study of moduli spaces, they provide a clean way to encode degenerations and to establish uniformity results that are difficult to obtain using ordinary limits. This talk will discuss applications of ultrafilters to uniformity theorems in dynamics and arithmetic geometry. After introducing local results that arise from this approach, I will sketch some of the arithmetic consequences, including uniform bounds on rational torsion points on abelian varieties. This is joint work with Jit Wu Yap