Georgia Institute of Technology College of Sciences

We present a general mechanism to establish the existence of diffusing orbits in a large class of nearly integrable Hamiltonian systems. Our approach relies on successive applications of the `outer dynamics' along homoclinic orbits to a normally hyperbolic invariant manifold. The information on the outer dynamics is encoded by a geometrically defined map, referred to as the `scattering map'. We find pseudo-orbits of the scattering map that keep moving in some privileged direction. Then we use the recurrence property of the `inner dynamics', restricted to the normally hyperbolic invariant manifold, to return to those pseudo-orbits. Finally, we apply topological methods to show the existence of true orbits that follow the successive applications of the two dynamics. This method differs, in several crucial aspects, from earlier works. Unlike the well known `two-dynamics' approach, the method relies heavily on the outer dynamics alone. There are virtually no assumptions on the inner dynamics, as its invariant objects (e.g., primary and secondary tori, lower dimensional hyperbolic tori and their stable/unstable manifolds, Aubry-Mather sets) are not used at all. The method applies to unperturbed Hamiltonians of arbitrary degrees of freedom that are not necessarily convex. In addition, this mechanism is easy to verify (analytically or numerically) in concrete examples, as well as to establish diffusion in generic systems.

The theory of two variable orthogonal polynomials is not very well developed. I will discuss some recent results on two variable orthogonal polynomials on the bicircle and time permitting on the square associate with orthogonality measures that are one over a trigonometric polynomial. Such measures have come to be called Bernstein-Szego measures. This is joint work with Plamen Iliev and Greg Knese.

Wajnryb showed that the mapping class group of a surface can be generated by two elements, each given as a product of Dehn twists. We will discuss a follow-up paper by Korkmaz, "Generating the surface mapping class group by two elements." Korkmaz shows that one of the generators may be taken to be a single Dehn twist instead. He then uses his construction to further prove the striking fact that the two generators can be taken to be periodic elements, each of order 4g+2, where g is the genus of the surface.

We propose a new convex relaxation for the problem of solving (random) quadratic equations known as phase retrieval. The main advantage of the proposed method is that it operates in the natural domain of the signal. Therefore, it has significantly lower computational cost than the existing convex methods that rely on semidefinite programming and competes with the recent non-convex methods. In the proposed formulation the quadratic equations are relaxed to inequalities describing a "complex polytope". Then, using an *anchor vector* that itself can be constructed from the observations, a simple convex program estimates the ground truth as an (approximate) extreme point of the polytope. We show, using classic results in statistical learning theory, that with random measurements this convex program produces accurate estimates. I will also discuss some preliminary results on a more general class of regression problems where we construct accurate and computationally efficient estimators using anchor vectors.

In this series of talks I will descibe a general proceedure to construct homology theories using analytic/geometric techiques. We will then consider Morse homology in some detail and a simple example of this process. Afterwords we will consider other situations like Floer theory and possibly contact homology.

A graphical model encodes conditional independence relations via the Markov properties. For an undirected graph these conditional independence relations are represented by a simple polytope known as the graph associahedron, which can be constructed as a Minkowski sum of standard simplices. There is an analogous polytope for conditional independence relations coming from any regular Gaussian model, and it can be defined using relative entropy. For directed acyclic graphical models we give a construction of this polytope as a Minkowski sum of matroid polytopes. The motivation came from the problem of learning Bayesian networks from observational data. This is a joint work with Fatemeh Mohammadi, Caroline Uhler, and Charles Wang.