Georgia Institute of Technology College of Sciences

A Gaussian stationary sequence is a random function f: Z --> R, for which any vector (f(x_1), ..., f(x_n)) has a centered multi-normal distribution and whose distribution is invariant to shifts. Persistence is the event of such a random function to remain positive on a long interval [0,N]. Estimating the probability of this event has important implications in engineering , physics, and probability. However, though active efforts to understand persistence were made in the last 50 years, until recently, only specific examples and very general bounds

We study regularity properties of weak solutions of the homogeneous Boltzmann equation. While under the so called Grad cutoff assumption the homogeneous Boltzmann equation is known to propagate smoothness and singularities, it has long been suspected that the non-cutoff Boltzmann operator has similar coercivity properties as a fractional Laplace operator. This has led to the hope that the homogenous Boltzmann equation enjoys similar smoothing properties as the heat equation with a fractional Laplacian.

We consider the two-dimensional BCS functional with a radial pair interaction. We show that the translational symmetry is not broken in a certain temperature interval below the critical temperature. For this purpose, we first introduce the full BCS functional and the translation invariant BCS functional. Our main result states that theminimizers of the full BCS functional coincide with the minimizers of the translation invariant BCS functional for temperatures in the aforementioned interval.

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.

We impose standard $T1$-type assumptions on a Calderón-Zygmund operator $T$, and deduce that for bounded compactly supported functions $f,g$ there is a sparse bilinear form $\Lambda$ so that $$ |\langle T f, g \rangle | \lesssim \Lambda (f,g). $$ The proof is short and elementary. The sparse bound quickly implies all the standard mapping properties of a Calderón-Zygmund on a (weighted) $L^p$ space.

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.

We show that the multiwavelets, introduced by Alpert in 1993, are related to type I Legendre-Angelesco multiple orthogonal polynomials. We give explicit formulas for these Legendre-Angelesco polynomials and for the Alpert multiwavelets. The multiresolution analysis can be done entirely using Legendre polynomials, and we give an algorithm, using Cholesky factorization, to compute the multiwavelets and a method, using the Jacobi matrix for Legendre polynomials, to compute the matrices in the scaling relation for any size of the multiplicity of the

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.

I will present a discrete family of multiple orthogonal polynomials defined by a set of orthogonality conditions over a non-uniform lattice with respect to different q-analogues of Pascal distributions. I will obtain some algebraic properties for these polynomials (q-difference equation and recurrence relation, among others) aimed to discuss a connection with an infinite Lie algebra realized in terms of the creation and annihilation operators for a collection of independent ascillators. Moreover, if time

The thin-shell or variance conjecture asks whether the variance of the Euclidean norm, with respect to the uniform measure on an isotropic convex body, can be bounded from above by an absolute constant times the mean of the Euclidean norm (if the answer to this is affirmative, then we have as a consequence that most of the mass of the isotropic convex body is concentrated in an annulus with very small width, a "thin shell''). So far all the general bounds we know depend on the dimension of the bodies, however for a few special