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.

An important question is to non-invasively find the volume of each phase in body, by only probing its response at the boundary. Here we consider a body containing two phases arranged in any configuration, and address the inverse problem of bounding the volume fraction of each phase from electrical tomography measurements at the boundary, i.e. measurements of the current flux through the boundary produced by potentials applied at the boundary. It turns out that this problem is closely related to the extensively studied problem of bounding the effective conductivity of periodic composite materials. Those bounds can be used to bound the response of an arbitrarily shaped body, and if this response has been measured, they can be used to extract information about the volume fraction.

Numerical experiments show that for a wide range of inclusion shapes one of the bounds turns out to be close to the actual volume fraction. The bounds extend those obtained by Capdeboscq and Vogelius for asymptotically small inclusions. The same ideas can be extended to elasticity and used to incorporate thermal measurements as well as electrical measurements. The translation method for obtaining bounds on the effective conductivity can also be applied directly to bound the volume fraction of inclusions in a body. This is joint work with Hyeonbae Kang and Eunjoo Kim.