Georgia Institute of Technology School of Mathematics | Georgia Institute of Technology | Atlanta, GA

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. A surprising result is that the same order of measurements gives an analogous quasi-isometry in the extreme quantization of one-bit sensing. Bylik and Lacey deliver this result as a consequence of a certain stochastic process on the sphere. We will discuss an alternative method that relies heavily on the VC-dimension of a class of subsets on the sphere.

We will describe sufficient conditions on a set $\Lambda \subset [0,2\pi) $ so that the maximal operator below is bound on $\ell^2(Z)$. $$\sup _{\lambda \in \Lambda} \Big| \sum_{n\neq 0} e^{i \lambda n^2} f(x-n)/n\Big|$$ The integral version of this result is an influential result to E.M. Stein. Of course one should be able to take $\Lambda = [0,2\pi) $, but such a proof would have to go far beyond the already complicated one we will describe. Joint work with Ben Krause.

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.

[Special time and location] The content of this talk is joint work with Yumeng Ou. We describe a novel framework for the he analysis of multilinear singular integrals acting on Banach-valued functions.Our main result is a Coifman-Meyer type theorem for operator-valued multilinear multipliers acting on suitable tuples of UMD spaces, including, in particular, noncommutative Lp spaces. A concrete case of our result is a multilinear generalization of Weis' operator-valued Hormander-Mihlin linear multiplier theorem.Furthermore, we derive from our main result a wide range of mixed Lp-norm estimates for multi-parameter multilinear multiplier operators, as well as for the more singular tensor products of a one-parameter Coifman-Meyer multiplier with a bilinear Hilbert transform. These respectively extend the results of Muscalu et. al. and of Silva to the mixed norm case and provide new mixed norm fractional Leibnitz rules.