Georgia Institute of Technology College of Sciences

We begin this talk by discussing four different problems that arenumber theoretic or combinatorial in nature. Two of these problems remainopen and the other two have known solutions. We then explain how these seeminglyunrelated problems are connected to each other. To disclose a little more information,one of the problems with a known solution is the following: Is it possible to find anirrational number $q$ such that the infinite geometric sequence $1, q, q^{2}, \dots$has 4 terms in arithmetic progression?

Deducing the state or structure of a system from partial, noisy measurements is a fundamental task throughout the sciences and engineering. The resulting inverse problems are often ill-posed because there are fewer measurements available than the ambient dimension of the model to be estimated. In practice, however, many interesting signals or models contain few degrees of freedom relative to their ambient dimension: a small number of genes may constitute the signature of a disease, very few parameters may specify the correlation structure of a time series, or a sparse collection of geometric constraints may determine a molecular configuration. Discovering, leveraging, or recognizing such low-dimensional structure plays an important role in making inverse problems well-posed. In this talk, I will propose a unified approach to transform notions of simplicity and latent low-dimensionality into convex penalty functions. This approach builds on the success of generalizing compressed sensing to matrix completion, and greatly extends the catalog of objects and structures that can be recovered from partial information. I will focus on a suite of data analysis algorithms designed to decompose general signals into sums of atoms from a simple---but not necessarily discrete---set. These algorithms are derived in a convex optimization framework that encompasses previous methods based on l1-norm minimization and nuclear norm minimization for recovering sparse vectors and low-rank matrices. I will provide sharp estimates of the number of generic measurements required for exact and robust recovery of a variety of structured models. These estimates are based on computing certain Gaussian statistics related to the latent model geometry. I will detail several example applications and describe how to scale the corresponding inference algorithms to very large data sets. (Joint work with Venkat Chandrasekaran, Pablo Parrilo, and Alan Willsky)

I will describe recent work with Marina Chugunovaand Ben Stephens on the evolution of a thin-filmequation that models a "coating flow" on a horizontalcylinder. Formally, the equation defines a gradientflow with respect to an energy that controls theH^1-norm.We show that for each given mass there exists aunique steady state, given by a droplet hanging from thebottom of the cylinder that meets the dry region withzero contact angle. The droplet minimizes the energy andattracts all strong solutions that satisfy certain energyand entropy inequalities. (Such solutions exist for arbitraryinitial values of finite energy and entropy, but it is notknown if they are unique.) The distance of any solutionfrom the steady state decays no faster than a power law.

Aim of the talk is to make a survey on some recent results concerning analysis over spaces with Ricci curvature bounded from below. I will show that the heat flow in such setting can be equivalently built either as gradient flow of the natural Dirichlet energy in L^2 or as gradient flow if the relative entropy in the Wasserstein space. I will also show how such identification can lead to interesting analytic and geometric insights on the structures of the spaces themselves. From a collaboration with L.Ambrosio and G.Savare