Georgia Institute of Technology College of Sciences

Many important problem classes are governed by anisotropic structures such as singularities concentrated on lower dimensional embedded manifolds, for instance, edges in images or shear layers in solutions of transport

In this talk we discuss two weight estimates for well-localized operators acting on vector-valued function spaces with matrix weights. We will show that the Sawyer-type testing conditions are necessary and sufficient for the boundedness of this class of operators, which includes Haar shifts and their various generalizations. More explicitly, we will show that it is suficient to check the estimates of the operator and its adjoint only on characteristic functions of cubes. This result generalizes the work of

Recently, Awasthi et al proved that a semidefinite relaxation of the k-means clustering problem is tight under a particular data model called the stochastic ball model. This result exhibits two shortcomings: (1) naive solvers of the semidefinite program are computationally slow, and (2) the stochastic ball model prevents outliers that occur, for example, in the Gaussian mixture model. This talk will cover recent work that tackles each of these shortcomings.

We consider the following problem. Does there exist an absolute constant C such that for every natural number n, every integer 1 \leq k \leq n, every origin-symmetric convex body L in R^n, and every measure \mu with non-negative even continuous density in R^n, \mu(L) \leq C^k \max_{H \in Gr_{n-k}} \mu(L \cap H}/|L|^{k/n}, where Gr_{n-k} is the Grassmannian of (n-k)-dimensional subspaces of R^n, and |L| stands for volume?

We consider (complex) Gaussian analytic functions on a horizontal strip, whose distribution is invariant with respect to horizontal shifts (i.e., "stationary"). Let N(T) be the number of zeroes in [0,T] x [a,b]. First, we present an extension of a result by Wiener, concerning the existence and characterization of the limit N(T)/T as T approaches infinity. Secondly, we characterize the growth of the variance of N(T).

Abstract: Certain materials form geometric structures called "grains," which means that one has distinct volumes filled with the same semi-solid material but not mixing. This can happen with semi-molten copper and something like this can also happen with liquid crystals (which are used in some calculator display screens).

Compressive sensing is a (relatively) new paradigm in data analysis that is having a large impact on areas from signal processing, statistics, to scientific computing. I am teaching a special topics on the subject in the Fall term, in support of the GT-IMPACT program. The talk will list some basic principles in the subject, stating some Theorems, and using images, and sounds to illustrate these principles.

Consistent reconstruction is a method for estimating a signal from a collection of noisy linear measurements that are corrupted by uniform noise. This problem arises, for example, in analog-to-digital conversion under the uniform noise model for memoryless scalar quantization. We shall give an overview of consistent reconstruction and prove optimal mean squared error bounds for the quality of approximation. We shall also discuss an iterative alternative (due to Rangan and Goyal) to consistent reconstruction which is also able to

This talk concerns recent joint work with E. M. Stein on the extension to higher dimension of Calder\'on's andCoifman-McIntosh-Meyer's seminal results about the Cauchy integral for a Lipschitz planar curve (interpreted as the boundary of a Lipschitz domain $D\subset\mathbb C$). From the point of view of complex analysis, a fundamental feature of the 1-dimensional Cauchy kernel:\vskip-1.0em$$H(w, z) = \frac{1}{2\pi i}\frac{dw}{w-z}$$\smallskip\vskip-0.7em\noindent is that it is holomorphic (that is, analytic) as a function of $z\in D$.

In this talk we will discuss the connection between functions with bounded mean oscillation (BMO) and commutators of Calderon-Zygmund operators. In particular, we will discuss how to characterize certain BMO spaces related to second order differential operators in terms of Riesz transforms adapted to the operator and how to characterize commutators when acting on weighted Lebesgue spaces.