Holomorphic symplectic manifolds (aka hyperkahler manifolds) are complex analogues of real symplectic manifolds. They have a rich geometric structure, though few compact examples are known. In this talk I will describe attempts to construct and classify holomorphic symplectic manifolds that also admit a holomorphic fibration. In particular, we will consider examples in four dimensions that are fibred by abelian surfaces known as Prym varieties.

I will describe a newly introduced toolbox that connects two areas of Probability Theory: Schramm-Loewner Evolutions (SLE) and Random Matrix Theory. This machinery opens new avenues of research that allow the use of techniques from one field to another. One aspect of this research direction is centered in an interacting particle systems model, namely the Dyson Brownian motion. In the first part of the talk, I will introduce basic ideas of SLE theory, then I will describe the connection with Random Matrix Theory via a first application of our method. I will finish the talk with some open problems that emerge using this newly introduced toolbox. This is a joint work with A. Campbell and K. Luh.

I will discuss the problem of understanding the long-time behavior of Ricci flow on a compact Kähler manifold, assuming that a solution exists for all positive time. Inspired by an analogy with the minimal model program in algebraic geometry, Song and Tian posed several conjectures which describe this behavior. I will report on recent work (joint with Hein and Lee) which confirms these conjectures.