Georgia Institute of Technology College of Sciences

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.

In this talk, I will survey the recent understandings on the motion of water waves obtained via rigorous mathematical tools, this includes the evolution of smooth initial data and some typical singular behaviors. In particular, I will present our recently results on gravity water waves with angled crests.

Compressed sensing illustrates the possibility of acquiring and reconstructing sparse signals via underdetermined (linear) systems. It is believed that iid Gaussian measurement vectors give near optimal results, with the necessary number of measurements on the order of $s \log(n/s)$ - $n$ is ambient dimension and $s$ is the sparsity threshold. The recovery algorithm used above relies on a certain quasi-isometry property of the measurement matrix.

Topics: local Hausdorff dimension, local Hausdorff measure, diffusion on compact metric spaces, prospective further research.

In this work we prove that the space of two parameter, matrix-valued BMO functions can be characterized by considering iterated commutators with the Hilbert transform. Specifically, we prove that the norm in the BMO space is equivalent to the norm of the commutator of the BMO function with the Hilbert transform, as an operator on L^2. The upper bound estimate relies on a representation of the Hilbert transform as an average of dyadic shifts, and the boundedness of certain paraproduct operators, while the lower bound follows Ferguson and

In the world of Hamiltonian partial differential equations, complete integrability is often associated to rare and peaceful dynamics, while wave turbulence rather refers to more chaotic dynamics. In this talk I will first try to give an idea of these different notions. Then I will discuss the example of the cubic Szegö equation, a nonlinear wave toy model which surprisingly displays both properties. The key is a Lax pair structure involving Hankel operators from classical analysis, leading to the inversion of large ill-conditioned matrices. .

We study the construction of exponential bases and exponential frames on general $L^2$ space with the measures supported on self-affine fractals. This problem dates back to the conjecture of Fuglede. It lies at the interface between analysis, geometry and number theory and it relates to translational tilings. In this talk, we give an introduction to this topic, and report on some of the recent advances. In particular, the possibility of constructing exponential frames on fractal measures without exponential bases will be discussed.