October 3: Alex Eskin (University of Chicago)
Title: Rational billiards and the SL(2,R) action on modui space.
Abstract: I will discuss ergodic theory over the moduli space of compact Riemann surfaces and its applications to the study of polygonal billiard tables. There is an analogy between this subject and the theory of flows on homogeneous spaces; I will talk about some successes and limitations of this viewpoint. This is joint work with Maryam Mirzakhani.
October 24: Anton Zorich (Universite De Rennes 1)
Title: Variations of Hodge structures and zero Lyapunov exponents
Abstract: By the results of G. Forni and of R. Trevino, the Lyapunov spectrum of the Hodge bundle over the Teichmuller geodesic flow on the strata of Abelian and of quadratic differentials does not contain zeroes. Though for certain invariant submanifolds zero exponents are present in the Lyapunov spectrum, in all known examples the zero exponents correspond to those covariantly constant subbundles of the real Hodge bundle, for which the monodromy of the Gauss-Manin connection acts by isometries of the Hodge metric.
We present an example of an arithmetic Teichmuller curve T, for which the real Hodge bundle does not contain any covariantly constant subbundles, and nevertheless its spectrum of Lyapunov exponents contains zeroes. We describe the mechanism of this phenomenon; it covers the previously known situation as a particular case. We conjecture that this is the only way the zero exponents might appear in the Lyapunov spectrum of the Hodge bundle over the Teichmuller geodesic flow for any invariant measure. These results are obtained partly in collaboration with C.M. Santos and G. Forni, partly in collaboration with A. Eskin and M. Kontsevich.
October 31: Andy Hammerlindl (IMPA)
Title: The Consequences of Quasi-Isometry
Abstract: A foliation is quasi-isometric if distance measured along a leaf is roughly proportional to distance measured along the manifold. Recently, Brin, Burago, and Ivanov established quasi-isometry for the stable and unstable foliations of all partially hyperbolic systems on the 3-torus (when lifted to the universal cover). This strong result can then be used to give a form of classification for this family of systems.
In this talk, I will show several topological properties of the invariant foliations of Anosov and partially hyperbolic systems come as a consequence of quasi-isometry, and show that there is a large family of manifolds on which no partially hyperbolic system can have quasi-isometric stable or unstable foliations.
November 7: Alden Walker (Cal Tech)
Title: Ziggurats and rotation numbers
Abstract: (Joint with Danny Calegari) We establish the existence of new rigidity and rationality phenomena in the theory of nonabelian group actions on the circle, and introduce tools to translate questions about the existence of actions with prescribed dynamics into finite combinatorics. A special case of our theory gives a very short new proof of Naimi's theorem (i.e. the conjecture of Jankins--Neumann) which was the last step in the classification of taut foliations of Seifert fibered spaces.
November 14: Jon Fickenscher (Princeton)
Title: Minimal And Very Non-uniquely Ergodic Interval Exchange Transformations
Abstract: In 1977, Keane constructed an Interval Exchange Transformation (IET) that was minimal but supported two distinct ergodic probability measures. In 1978, Veech gave an upper bound on the number of distinct ergodic probability measures for any IET. We show that this bound is sharp by constructing minimal IETs that realize this bound in every Rauzy Class.
December 9 : Michael Boshernitzan (Rice)
Title: Local complexity of finite sets in [0,1]
Abstract: Attached