Last year (more precisely November 25, 2010), I gave a talk at the Séminaire de Théorie Spectrale et Géométrie of the Institut Fourier (Grenoble) about the Teichmuller geodesic flow, the geometry of the Gauss-Manin connection on the Hodge bundle and the detection of special (totally degenerate) orbits of the natural -action on the moduli space of Abelian differentials (based on 3 joint works with G. Forni, A. Zorich and J.-C. Yoccoz, 2 of them being already published here and here, and a third one still in preparation).
A few weeks ago, I was asked by one of the organizers of this seminar (namely, Benoît Kloeckner) to write down the lecture notes of my talk, so that they could be published in the 2010/2011 fascicle of the Actes du Séminaire de Théorie Spectrale et Géométrie. Thus, after gladly accepting Kloeckner’s invitation, I started polishing my own notes and the outcome is this text here (written in English despite of its title and abstract in French).
As the reader can imagine, this text has a significant intersection with previous posts in this blog (e.g., this post here). However, since the audience of the talk was mostly interested in the “geometrical” side of the subject of my notes, I refrained myself from discussing Lyapunov exponents of Kontsevich-Zorich cocycle and its applications to interval exchange transformations, translation flows and billiards (a topic that I started discussing in this blog here, here, here and here). On the other hand, since this “geometrical” side of the history inspired some recent developments in the study of the neutral Oseledets bundle of the Kontsevich-Zorich cocycle (see the last remark of my lecture notes above), this text contains (by its end) some “scenes / excerpts” of future posts of my series on Teichmuller dynamics (currently in its 4th chapter, as you can see here).
In any case, I hope you will enjoy the reading!