Georgia Institute of Technology College of Sciences

Consider the following extremal problem: maximize the amplitude |X_T|, at time T, of a linear recurrent sequence X_1, X_2,... of order N < T, under natural constraints: (I) the initials are uniformly bounded; (II) the characteristic polynomial is R-stable, i.e., its roots are in the origin-centered disc of radius R. While the maximum at time T = N essentially follows from the classical Gautschi bound (1960), the general case T > N turns out to be way more challenging to handle. We find that for any triple (N,R,T), the amplitude is maximized when the roots coincide and have modulus R, and the initials are chosen to align the phases of fundamental solutions. This result is striking for two reasons. First, the same configuration of roots and initials is uniformly optimal for all T, i.e. the whole envelope is maximized at once. Second, we are not aware of any purely analytical proof: ours uses tools from algebraic combinatorics, namely Schur polynomials indexed by hook partitions. 

In the talk, I will sketch the proof of this result, making it as self-sufficient as possible under the circumstances. If time permits, we will discuss a related conjecture on the optimal error bounds in complex Lagrange interpolation.

The talk is based on the work https://arxiv.org/abs/2508.13554.

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.