\indent More than 40 years ago, Erdos asked to determine the maximum possible number of edges in a $k$-uniform hypergraph on $n$ vertices with no matching of size $t$ (i.e., with no $t$ disjoint edges). Although this is one of the most basic problem on hypergraphs, progress on Erdos' question remained elusive. In addition to being important in its own right, this problem has several interesting applications. In this talk we present a solution of Erdos' question for $t < \dfrac{n}{(3k^2)}$. This improves upon the best previously known range $t = O \dfrac{n}{k^3}$, which dates back to the 1970's.

Joint work with P. Loh and B. Sudakov.

\indent I will explain how $p$-adic methods (the so-called $p$-adic local Langlands correspondence) can be used to prove the existence of an analytic continuation for complex $L$-functions.

\indent In $p$-adic Hodge Theory, comparison morphisms relate $p$-adic etale cohomology of varieties over local fields of mixed characteristic $(0,p)$ with their de Rham cohomology. I will present a construction of such a morphism that uses Chern classes from motivic cohomology into etale and de Rham cohomology.