A complete Kahler metric $g$ on a Kahler manifold $M$ is a ``steady gradient Kahler-Ricci soliton'' if there exists a smooth real-valued function $f:M \rightarrow R$ with $\nabla^{g}f$ holomorphic such that $Ric(g)=Hess(f)$. I will present a theorem of existence and uniqueness for such metrics. This is joint work with Alix Deruelle (Sorbonne).

When studying the cohomology of Shimura varieties and arithmetic manifolds, one encounters the following phenomenon: the same Hecke eigensystem shows up in multiple degrees around the middle dimension, and its multiplicities in these degrees resembles that of an exterior algebra.\\

In a series of recent papers, Venkatesh and his collaborators provide an explanation: they construct graded objects having a graded action on the cohomology, and show that under good circumstances this action factors through that of an explicit exterior algebra, which in turn acts faithfully and generate the entire Hecke eigenspace.\\

In this talk, we discuss joint work with Khare where we investigate the p=p situation (as opposed to the l $\neq$ p situation, which is the main object of study of Venkatesh's Derived Hecke Algebra paper): we construct a degree-raising action on the cohomology of the ordinary Hida tower and show that (under some technical assumptions), this action generates the full Hecke eigenspace under its lowest nonzero degree. Then, we bring Galois representations into the picture, and show that the derived Hecke action constructed before is in fact related to the action of a certain dual Selmer group.

A classical conjecture of Shafarevich, solved by Parshin and Arakelov, predicts that any smooth projective family of high genus curves over the complex line minus a point or an elliptic curve is isotrivial (has zero variation in its algebraic structure). A natural question then arises as to what other families of manifolds and base spaces might behave in a similar way. Kebekus and Kov\'acs conjecture that families of manifolds with good minimal models form the most natural category where Shafarevich-type conjecturesshould hold. For example, analogous to the original setting of Shafarevich Conjecture, they expect that over a base space of Kodaira dimension zero such families are always (birationally) isotrivial. In this talk I will discuss a solution to Kebekus-Kov\'acs Conjecture.