Georgia Institute of Technology College of Sciences

The celebrated Erdos-Hajnal conjecture states that for every graph H, there is a constant c > 0 such that every graph G that does not contain H as an induced subgraph has a clique or stable set of size at least n^c, where n = |V(G)|. One approach for proving this conjecture is to prove that in every H-free graph G, there are two linear-size sets A and B such that either there are no edges between A and B, or every vertex in A is adjacent to every vertex in B. As is turns out, this is not true unless both H and its complement are trees. In the case when G contains neither H nor its complement as an induced subgraph, the conclusion above was conjectured to be true for all trees (Liebenau & Pilipczuk), and I will discuss a proof of this for a class of tree called "caterpillars". I will also talk about results and open questions for some variants, including allowing one or both of A and B to have size n^c instead of linear size, and requiring the bipartite graph between A and B to have high or low density instead of being complete or empty. In particular, our results improve the bound on the size of the largest clique or stable that must be present in a graph with no induced five-cycle. Joint work with Maria Chudnovsky, Jacob Fox, Anita Liebenau, Marcin Pilipczuk, Alex Scott, and Paul Seymour.

Among functions $f$ majorized by indicator functions $1_E$, which functions have maximal ratio $\|\widehat{f}\|_q/|E|^{1/p}$? I will briefly describe how to establish the existence of such functions via a precompactness argument for maximizing sequences. Then for exponents $q\in(3,\infty)$ sufficiently close to even integers, we identify the maximizers and prove a quantitative stability theorem.

The discrete prolate spheroidal sequences (DPSS's) provide an efficient representation for discrete signals that are perfectly timelimited and nearly bandlimited. Due to the high computational complexity of projecting onto the DPSS basis - also known as the Slepian basis - this representation is often overlooked in favor of the fast Fourier transform (FFT). In this talk I will describe novel fast algorithms for computing approximate projections onto the leading Slepian basis elements with a complexity comparable to the FFT. I will also highlight applications of this Fast Slepian Transform in the context of compressive sensing and processing of sampled multiband signals.

In this talk I will present some results concerning the existence and the stability of quasi-periodic solutions for quasi-linear and fully nonlinear PDEs. In particular, I will focus on the Water waves equation. The proof is based on a Nash-moser iterative scheme and on the reduction to constant coefficients of the linearized PDE at any approximate solution. Due to the non-local nature of the water waves equation, such a reduction procedure is achieved by using techniques from Harmonic Analysis and microlocal analysis, like Fourier integral operators and Pseudo differential operators.