In many imaging problems such as X-ray crystallography, detectors can only record the intensity or magnitude of a diffracted wave as opposed to measuring its phase.  Phase retrieval concerns the recovery of an image from such phaseless information.  Although this problem is in general combinatorially hard, it is of great importance because it arises in many applications ranging from astronomical imaging to speech analysis. This talk discusses novel acquisition strategies and novel convex and non-convex algorithms which are provably exact, thereby allowing perfect phase recovery from a minimal number of noiseless and intensity-only measurements. More importantly, we also demonstrate that our noise-aware algorithms are stable in the sense that the reconstruction degrades gracefully as the signal-to-noise ratio decreases. This may be of special contemporary interest because phase retrieval is at the center of spectacular current research efforts collectively known under the name of coherent diffraction imaging aimed, among other things, at determining the 3D structure of large protein complexes.  

It has been a challenging problem to studying the existence of Kahler-Einstein metrics on Fano manifolds. A Fano manifold is a compact Kahler manifold with positive first Chern class. There are obstructions to the existence of Kahler-Einstein metrics on Fano manifolds. In these lectures, I will report on recent progresses on the study of Kahler-Einstein metrics on Fano manifolds. The first lecture will be a general one. I will discuss approaches to studying the existence problem. I will discuss the difficulties and tools in these approaches and results we have for studying them. In the second lecture, I will discuss the partial C^0-estimate which plays a crucial role in recent progresses on the existence of Kahler-Einstein metrics. I will show main technical aspects of proving such an estimate.

I will explain how the so-called "infinity Laplacian'' PDE arrises in sup-norm variational problems and discuss the regularity of solutions.

One-Frequency Schrödinger operators give one of the simplest models where fast transport and localization phenomena are possible. From a dynamical perspective, they can be studied in terms of certain one-parameter families of quasi-periodic co-cycles, which are similarly distinguished as simplest classes of dynamical systems compatible with both KAM phenomena and non-uniform hyperbolicity (NUH). While much studied since the 1970's, until recently the analysis was mostly confined to ''local theories'' describing the KAM and the NUH regimes in detail. In this talk we will describe some of the main aspects of the global theory that has been developed in the last few years.