RTG WORKSHOPS & LECTURE SERIES Random Walks on Groups March 29 - 31, 2013 Room B 844, East Hall University of Michigan Ann Arbor, MI Schedule Friday 1:00 - 2:00 pm Vadim Kaimanovich Random walks and Poisson boundaries; entropy and the Liouville property Abstract: We shall give a general introduction to the boundary theory of random walks on groups starting from general notions and illustrating them on concrete examples of groups with hyperbolic properties and self-similar groups   2:15 - 3:15 pm Anders Karlsson An ergodic theorem for noncommuting products Abstract: In this introductory lecture I will recall some classical ergodic theorems (Birkhoff, Kingman, Oseledets) and formulate a rather general ergodic theorem for noncommuting products that appeared in my joint work with F. Ledrappier. This involves the notion of horofunctions which I will spend some time explaining.   4:00 - 5:00 pm Alex Furman Products of random matrices: Lyapunov exponents and stationary measures Abstract: In these three talks we shall focus on asymptotic characteristics of products of random i.i.d. matrices, that can be viewed as random walks on matrix groups. Writing the random product in the polar form (KAK decomposition) the behavior of the A-component is described by the Lyapunov exponents, and the distribution of the K-part by the stationary measure. These notions will be introduced in the first lecture. 6:00 pm Banquet at Isalita   Saturday 9:00 - 10:00 am Vadim Kaimanovich Boundary convergence and identification; applications to the mapping class group   10:15 - 11:00 am Giulio Tiozzo Geodesic ray tracking for random walks on groups Abstract: Given a finitely generated group G acting on a geodesic space X and a probability measure on G, one can construct a random walk by choosing at each step a random group element and letting it act on X. The natural question arises whether the sample paths can be approximated by some geodesic in X. We will prove that, in a quite general setting, the sample path and the limiting geodesic lie within sublinear distance. Our argument applies to the case of the mapping class group acting on Teichmueller space, answering a question of Kaimanovich. Another application includes the statistics of excursions of random Teichmueller geodesics in the thin part of moduli space.   11:15 -12:15 am Alex Furman Lyapunov exponents: positivity of the top exponent, simplicity of the spectrum, regularity Abstract: The second lecture will examine qualitative behavior of the Lyapunov exponents, and some regularity results about Lyapunov exponents and stationary measures. 1:30 - 2:30 pm Moon Duchin Random Teichmüller geodesics Abstract: I'll review some of the features and pathologies of geodesics in the Teichmüller metric, including the phenomena that are obstructions to hyperbolicity. Then I'll discuss recent work with Dowdall and Masur in which we work out properties enjoyed by generic geodesics, concluding that these obstructions are quantifiably rare, and Teichmüller space is in this sense "statistically hyperbolic." One key tool, devised by Eskin and Mirzakhani, shows that geodesics are well-modeled by random walks on a net of points.   2:45 - 3:30 pm Andrew Zimmer The Poisson and Martin boundary of a harmonic manifold Abstract: A complete Riemannian manifold is called harmonic if each geodesic sphere of sufficiently small radii has constant mean curvature. Examples of harmonic manifolds include flat spaces and rank one locally symmetric spaces. The Lichnerowicz conjecture asks if these are the only compact harmonic manifolds. In this talk we will present some evidence that this is the case. In particular, we will discuss various compactifications of non-compact non-flat simply connected harmonic manifolds. We will show that the Martin, Poisson, Busemann, and "geometric" boundaries coincide. Moreover, in this case, the harmonic measure can be identified with "visual" measure. This leads to several corollaries concerning the fundamental group of a compact harmonic manifold and the dynamics of the geodesic flow.   4:00 - 5:00 pm Anders Karlsson An ergodic theorem for noncommuting products II Abstract: In the second talk I will give a proof of the noncommutative ergodic theorem and explain a few of its consequences, notably to the drift of random walks on finitely generated groups and the existence of non-constant bounded harmonic functions improving on a result of Varopoulos.   Sunday 9:00 - 10:00 am Vadim Kaimanovich Random walks and amenability; applications to self-similar groups   10:30 - 11:30 am Anders Karlsson An ergodic theorem for noncommuting products III Abstract: In the final lecture, I hope to describe some further concrete cases of the main theorem, such as Oseledets' multiplicative ergodic theorem and a random version of Thurston's spectral theorem for surface homeomorphisms.   12:00 -1:00 pm Alex Furman An application to product actions on manifolds In the third lecture we shall see an application of (a cocycle version) of some of the above ideas to a problem that does not involve random walks, namely constraints on actions of certain groups on compact manifolds.     Sponsors National Science Foundation, Michigan Mathematical Journal, Department of Mathematics at the University of Michigan For website problems, contact Webmaster