Georgia Institute of Technology College of Sciences

Dynamical Sampling is, in a sense, a hypernym classifying the set of inverse problems arising from considering samples of a signal and its future states under the action of a linear evolution operator. In Dynamical Sampling, both the signal, $f$, and the driving operator, $A$, may be unknown. For example, let $f \in l^2(I)$ where $I=\{1, \ldots, d\}$. Suppose for $\Omega \subset I$ we know  $\{{ A^j f(i)} : j= 0, \ldots l_i, i\in \Omega\}$ for some $A: l^2(I) \to l^2(I)$. In this setting, we can obtain conditions on $\Omega, A, l_i$ that allow the stable reconstruction of $f$. Dynamical Sampling is closely related to frame theory and has applications to wireless sensor networks among other areas. In this talk, we will discuss the Dynamical Sampling problem, its motivation, related problems inspired by it, current/future work, and open problems. 

The seminar will be held on Zoom and can be found at the link

https://us02web.zoom.us/j/71579248210?pwd=d2VPck1CbjltZStURWRWUUgwTFVLZz09

Semidefinite programming (SDP) is a very well behaved class of convex optimization problems. We will introduce this class of problems, illustrate how it allows to approximate many practical nonconvex optimization problems, and discuss the role of low rank structure.

Solving systems of polynomial equations is at the heart of algebraic geometry. In this talk I will discuss a new method that improves the efficiency of equation-by-equation algorithms for solving polynomial systems. Our approach uses fewer homotopy continuation paths than the current leading method based on regeneration.  Moreover it is applicable in both projective and multiprojective settings. To motivate the approach I will also give some examples coming from applied algebraic geometry.
This is joint work with Tim Duff and Anton Leykin.

Let X_{N,n} be an iid product of N real Gaussian matrices of size n x n. In this talk, I will explain some recent joint work with G. Paouris 
(arXiv:2005.08899) about a non-asymptotic analysis of the singular values of X_{N,n} . I will begin by giving some intuition and motivation for studying such matrix products. Then, I will explain two new results. The first gives a rate of convergence for the global distribution of singular values of X_{N,n} to the so-called Triangle Law in the limit where N,n tend to infinity. The second is a kind of quantitative version of the multiplicative ergodic theorem, giving estimates at finite but large N on the distance between the joint distribution of all Lyapunov exponents of X_{N,n} and appropriately normalized independent Gaussians in the near-ergodic regime (N >> n).