Georgia Institute of Technology College of Sciences

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.

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.

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.

This talk contains two parts. First I will discuss our work related to causal modeling in hybrid systems. The idea is to model jump conditions as caused by impulsive inputs. While this is well defined for linear systems, the notion of impulsive inputs poses problems in the nonlinear case. We demonstrate a viable approach based on nonstandard analysis. The second part deals with dynamical systems with delays. First I will show an application of the maximum principle to a delayed resource allocation problem in population dynamics solving a problem in the model of a bee colony cycle. Next I discuss some problems regarding causality in systems with varying delays. These problems relate to the well-posedness (existence and uniqueness) and causality of the mathematical models for physical phenomena, and illustrate why one should consider the physics first and then the mathematics. Finally, I consider the post Newtonian problem as a problem with state dependent delay. Einstein’s field equations relate space time geometry to matter and energy distribution. These tensorial equations are so unwieldy that solutions are only known in some very specific cases. A semi-relativistic approximation is desirable: One where space-time may still be considered as flat, but where Newton’s equations (where gravity acts instantaneously) are replaced by a post-Newtonian theory, involving propagation of gravity at the speed of light. As this retardation depends on the geometry of the point masses, a dynamical system with state dependent delay results, where delay and state are implicitly related. We investigate several problems with the Lagrange-Bürman inversion technique and perturbation expansions. Interesting phenomena (entrainment, dynamic friction, fission and orbital speeds) not explainable by the Newtonian theory emerge. Further details on aspects of impulsive systems and delay systems will be elaborated on by Nak-seung (Patrick) Hyun and Aftab Ahmed respectively.