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.

We show how to obtain a decomposition of an arbitrary closed, smooth, orientable 4-manifold from a loop of Morse functions on a surface or as a loop in the pants complex. A nice feature of all of these decompositions is that they can be encoded on a surface so that, in principle, 4-manifold topology can be reduced to surface topology. There is a good amount to be learned from translating between the world of Morse functions and the world of pants decompositions.  We will allude to some of the applications of this translation and point the interested researcher to where they can learn more. No prior knowledge of these fields is assumed and there will be plenty of time for questions.

This talk will discuss an evolutionary de Rham-Hodge method to provide a unified paradigm for the multiscale geometric and topological analysis of evolving manifolds constructed from filtration, which induces a family of evolutionary de Rham complexes. While the present method can be easily applied to close manifolds, the emphasis is given to more challenging compact manifolds with 2-manifold boundaries, which require appropriate analysis and treatment of boundary conditions on differential forms to maintain proper topological properties. Three sets of Hodge Laplacians are proposed to generate three sets of topology-preserving singular spectra, for which the multiplicities of zero eigenvalues correspond to exact topological invariants. To demonstrate the utility of the proposed method, the application is considered for the predictions of binding free energy (BFE) changes of protein-protein interactions (PPIs) induced by mutations with machine learning modeling. It has a great application in studying the SARS-CoV-2 virus' infectivity, antibody resistance, and vaccine breakthrough, which will be presented in this talk.

We'll talk about problems of optimizing a quadratic function subject to quadratic constraints, in addition to a sparsity constraint that requires that solutions have only a few nonzero entries. Such problems include sparse versions of linear regression and principal components analysis. We'll see that this problem can be formulated as a convex conical optimization problem over a sparse version of the positive semidefinite cone, and then see how we can approximate such problems using ideas arising from the study of hyperbolic polynomials. We'll also describe a fast algorithm for such problems, which performs well in practical situations.