Georgia Institute of Technology College of Sciences

In this talk we will discuss several ways to construct new convex bodies out of old ones. We will start by defining various methods of "averaging" convex bodies, both old and new. We will explain the relationships between the various definitions and their connections to basic conjectures in convex geometry. We will then discuss the power operation, and explain for example why every convex body has a square root, but not every convex body has a square. If time permits, we will briefly discuss more complicated constructions such as logarithms. The talk is based on joint work with Vitali Milman.

For a compact subset $A$ of $R^n$ , let $A(k)$ be the Minkowski sum of $k$ copies of $A$, scaled by $1/k$. It is well known that $A(k)$ approaches the convex hull of $A$ in Hausdorff distance as $k$ goes to infinity. A few years ago, Bobkov, Madiman and Wang conjectured that the volume of $A(k)$ is non-decreasing in $k$, or in other words, that when the volume deficit between the convex hull of $A$ and $A(k)$ goes to $0$, it actually does so monotonically. While this conjecture holds true in dimension $1$, we show that it fails in dimension $12$ or greater. Then we consider whether one can have monotonicity of convergence of $A(k)$ when its non-convexity is measured in alternate ways. Our main positive result is that Schneider’s index of non-convexity of $A(k)$ converges monotonically to $0$ as $k$ increases; even the convergence does not seem to have been known before. We also obtain some results for the Hausdorff distance to the convex hull, along the way clarifying various properties of these notions of non-convexity that may be of independent interest.Joint work with Mokshay Madiman, Matthieu Fradelizi and Arnaud Marsiglietti.

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.

The Linear Independence of Time-Frequency Translates Conjecture, also known as the HRT conjecture, states that any finite set of time-frequency translates of a given $L^2$ function must be linearly independent. This conjecture, which was first stated in print in 1996, remains open today. We will discuss this conjecture, its relation to the Zero Divisor Conjecture in abstract algebra, and the (frustratingly few) partial results that are currently available.