Georgia Institute of Technology College of Sciences

In many applied fields of research, like Geophysics, Medicine, Engineering, Economy, and Finance, to name a few, classical problems are the extraction of hidden information and features, like quasi-periodicities and frequency patterns, as well as the separation of different components contained in a given signal, like, for instance, its trend.

Standard methods based on Fourier and Wavelet Transform, historically used in Signal Processing, proved to be limited when nonlinear and non-stationary phenomena are present. For this reason in the last two decades, several new nonlinear methods have been developed by many research groups around the world, and they have been used extensively in many applied fields of research.

In this talk, we will briefly review the Hilbert-Huang Transform (a.k.a. Empirical Mode Decomposition method) and discuss its known limitations. Then, we will review the Iterative Filtering technique and we will introduce newly developed generalizations to handle multidimensional, multivariate, or highly non-stationary signals, as well as their time-frequency representation, via the so-called IMFogram. We will discuss the theoretical and numerical properties of these methods and show their applications to real-life data.
We will conclude the talk by reviewing the main problems which are still open in this research field.

 Around 1997, Shub and Smale proved that sufficiently good upper bounds
on the number of integer roots of polynomials in one variable --- as a function
of the input complexity --- imply a variant of P not equal to NP. Since then,
later work has tried to go half-way: Trying to prove that easier root counts
(over fields instead) still imply interesting separations of complexity
classes. Koiran, Portier, and Tavenas have found such statements over the real
numbers.

        We present an analogous implication involving p-adic valuations:    
If the integer roots of SPS polynomials (i.e., sums of products of sparse polynomials) of size s never yield more than s^{O(1)} distinct p-adic
valuations, then the permanents of n by n matrices cannot be computed by constant-free, division-free arithmetic circuits of size n^{O(1)}. (The
implication would be a new step toward separating VP from VNP.) We also show that this conjecture is often true, in a tropical geometric sense (paralleling a similar result over the real numbers by Briquel and Burgisser). Finally, we prove a special case of our conjectured valuation bound, providing a p-adic analogue of an earlier real root count for polynomial systems supported on circuits. This is joint work with Joshua Goldstein, Pascal Koiran, and Natacha Portier.

We prove a 1973 conjecture due to Erdős on the existence of Steiner triple systems with arbitrarily high girth. Our proof builds on the method of iterative absorption for the existence of designs by Glock, Kü​hn, Lo, and Osthus while incorporating a "high girth triangle removal process". In particular, we develop techniques to handle triangle-decompositions of polynomially sparse graphs, construct efficient high girth absorbers for Steiner triple systems, and introduce a moments technique to understand the probability our random process includes certain configurations of triples.

(Joint with Matthew Kwan, Mehtaab Sawhney, and Michael Simkin) ​