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.

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.