# Meeting Details

Title: An Introduction to Teichm\"uller dynamics; this talk will continue at the special session on Wednesday at 4:40pm Working Seminar: Dynamics and its Working Tools Giovanni Forni, University of Maryland We plan an introduction to Teichmueller dynamics from the point of view of complex analytic geometry and Hodge theory. We will describe the moduli space of holomorphic Abelian and quadratic differentials or, equivalently, of translation surfaces with conical singularities. In particular, we will introduce several structures on the moduli space (period map coordinates, affine structure, canonical measures, $SL(2,R)$-action, Teichmueller flow). We will then introduce the Hodge inner product and explain how to use it to derive conclusion about the Lyapunov spectrum of the Teichmueller flow. We will state without proof the main theorems in Teichmueller dynamics (Masur and Veech theorems about the finiteness of canonical measures, the ergodicy and mixing of Teichmueller flow, etc.) and an application to the dynamics of IET's (Masur and Veech proof of the Keane conjecture). Time permitting we would like to at least sketch somewhat different proofs of ergodicity of the Teichmueller flow with respect to a general class of invariant measures and a proof of the Keane conjecture based on the local uniform hyperbolicity of the Hodge norm under the Teichmueller flow. This talk will continue at the special session on Wednesday at 3:35pm