Cal Tech Lie Group Seminar

Contact:	Hee Oh,  Alex Gorodnik
Time:	Wed., 4-5:15pm
Place:	Sloan 153 (map)

Link to Geometry and Topology Seminar

Spring 2006
Time/Location	Speaker	Title/Abstract
Tu. Apr. 4 10AM Sloan 159	Tsachik Gelander Yale University	Uniform Tits alternative (Lecture 1)
Th. Apr. 6 11AM Sloan 159	Tsachik Gelander Yale University	Uniform Tits alternative (Lecture 2)
Tu. Apr. 11 10AM Sloan 159	Tsachik Gelander Yale University	Uniform Tits alternative (Lecture 3)

Autumn 2005
Time/Location	Speaker	Title/Abstract
Wed. Oct. 5
Wed. Oct. 12
(joint with Number Theory Seminar) Th. Oct. 20 4-5:15PM Sloan 257	Hee Oh Caltech	Manin's conjecture on the rational points on compactification of semisimple groups
Wed. Oct. 26
(joint with Number Theory Seminar) Wed. Nov. 2 4-5:15PM Sloan 257	Ramin Takloo-Bighash Princeton University	Distribution of rational points on compactifications of semi-simple groups
(joint with Number Theory Seminar) Wed. Nov. 9 4-5:15PM Sloan 257	Lior Silberman IAS	Arithmetic quantum chaos in the higher-rank case I shall discuss joint work with Akshay Venkatesh on the quantum unique ergodicity conjecture for locally symmetric spaces. In particular, in the case of the (co-compact) lattice in $PGL_3(\R)$ associated to an order in a division algebra of degree $3$ over $\mathbb{Q}$, we show that any non-degenerate sequence of Hecke-Maass eigenforms becomes equidistributed in the measure-theoretic sense. We first reduce the problem to showing the equidistribution of a limit measure on the homogeous space of the lattice which is invariant under the action of a Cartan subgroup. By recent measure rigidity results it then suffices to show that elements of the Cartan subgroup act with ``positive entropy''. I will describe this property and how we establish it using harmonic analysis on the building and a (global) diophantine argument on the group
Wed. Nov. 16
Tu. Nov. 22 4-5:15PM Sloan 153	Tamar Ziegler IAS	Configurations in sets of positive upper density in R^m We use recent developments in ergodic theory to solve a classical problem in geometric Ramsey theory. Let E be a measurable subset of R^m, with positive upper density. Let V={0,v_1,...,v_k} be a subset of R^m. We show that for r large enough, we can find an isometric copy of rV arbitrarily close to E. This is a generalization of a theorem of Furstenberg, Katznelson and Weiss showing a similar property for m=k=2.

Spring 2005
Time/Location	Speaker	Title/Abstract
Wed. March 30	Yves Benoist E'cole Normale Supe'rieure	3-dimensional projective tilings
Wed. April 6
Wed. April 13
Mon. April 18 4-5:15PM Sloan 257	Omri Sarig Penn State	Ergodic theory for the horocycle flow on periodic hyperbolic surfaces (joint with F. Ledrappier) The horocycle flow of a non-geometrically finite hyperbolic surface may have many different (infinite) ergodic invariant Radon measures. (This should be contrasted with co-compact or co-finite volume case where there is just one non-trivial measure - up to scaling). We classify these measures for the class of periodic surfaces: regular covers of surfaces of finite volume. It turns out that in this case there are as many measures as there are positive eigenfunctions for the Laplacian of the surface. In some situations, only one of these measures is "relevant" from the ergodic theoretic point of view.
Wed. April 27
Wed. May 4 4-5:15PM Sloan 257	Anders Karlsson Royal Institute of Technology, Sweden	Aspects of asymptotic geometry of metric spaces I will discuss two metric geometric notions: horoballs, which are limits of balls, and stars which are limits of halfspaces. Horoballs play for example a role in a conjectural law of large numbers for random walks on groups and the dynamics of semicontractions. Stars associate a generalized Tits geometry at infinity of metric spaces. I will discuss their relevance for the dynamics of isometries with consequences for the existence of free subgroups, a metric analog of Furstenberg's lemma, the Dirichlet problem at infinity etc. This gives in particular new results for CAT(0)-geometry, holomorphic maps, and Teichmuller theory.
Wed. May 11 4-5:15PM Sloan 257	Hee Oh CalTech	Equidistribution of rational matrices in their conjugacy class Abstract: We will discuss the equidistribution of Hecke points on homogeneous spaces of semisimple algebraic Q-group. (Joint work with Y. Benoist)
Wed. May 18
Wed. May 25 11-12:15PM Sloan 153	Alireza Salehi Golsefidy Yale	Lattices of Minimum Covolume in Classical Chevalley Groups over F_q((1/t)) Abstract

Winter 2005
Time/Location	Speaker	Title/Abstract
Wed. Jan. 5 4-5:15 PM Sloan 257	Gordan Savin University of Utah	Matrix coefficients of unitary representations of Chevalley groups. By a well known result of Howe and Moore, matrix coefficients of non-trivial unitary irreducible representations decay at infinity. Moreover, by a result of Kazhdan, the trivial representation is isolated in the unitary dual if the rank of the group is at least 2 (Kazhdan's property T). Thus, it is a natural question to find optimal bounds for the decay of matrix coefficients of unitary representations. By refining an idea of Howe, we develop a uniform approach to this problem for groups of type A-D-E. As expected, we show that matrix coefficients of the minimal representation have the slowest decay among non-trivial irreducible representations. This is a joint work with H. Y. Loke.
Wed. Jan. 12
(joint with Geometry Seminar) Wed. Jan. 19 4-5:15 PM Sloan 257	Suhyoung Choi	Real projective structures on 3-manifolds and deformations A real projective structure on a 3-manifold is given by locally modeling the manifold by real projective geometry. Hyperbolic structures canonically correspond to real projective structures. We try to find deformations of hyperbolic structures on 3-manifolds to non-hyperbolic real projective structures. We obtain numerical evidences for Dehn surgered 3-manifolds. We also obtain examples for a class of 3-dimensional hyperbolic-polyhedron reflection groups.
Wed. Jan. 26
Wed. Feb. 2
Wed. Feb. 9
Mon. Feb. 14 3-4 PM Sloan 159	Gregory Margulis Yale University	Borel-Harish-Chandra Theorem and homogenous flows
Tu. Feb. 22 3-4 PM Sloan 257	Emmanuel Breuillard IHES	The asymptotic shape of metric balls in groups of polynomial growth, and pointwise ergodic theorems. Abstract: Let $G$ be a locally compact group of polynomial growth, i.e. vol(U^n)=O(n^K) for some K>0 and some compact generating set U. We show that there is a number c(U)>0 and an integer d that can be computed explicitely such that the n^(-d)vol(U^n) converges to c(U) as n tends to infinity. We give a geometric interpretation of the asymptotic volume c(U) as the volume of the unit ball for some Carnot-Caratheodory metric on a stratified simply connected nilpotent Lie group naturally associated to G. As a consequence we get in particular that the balls {U^n}_n form a Folner sequence, thus answering Greenleaf's "localization" problem in this case. The results hold for a large class of other metrics on G, such as left invariant Riemannian metrics when G is Lie, and enable to prove pointwise ergodic theorems for such averages, thus answering a question of A. Nevo.
Wed. Feb. 23 4-5:15 PM Sloan 257	Emmanuel Breuillard IHES	Equidistribution and random walks on nilpotent Lie groups Abstract: Let G be a nilpotent Lie group and Gamma be a dense subgroup generated by a symmetric finite set S. We show that Gamma is equidistributed in G for both the ball averages associated to the word metric corresponding to S, or for the random walk averages for some symmetric probability measure on S. This generalizes the classical local limit theorem on R^d for finitely supported measures. As a consequence we obtain a probabilistic version of Ratner's equidistribution theorem for actions of unipotent groups on homogeneous spaces.
Wed. March 2

Autumn 2004
Time/Location	Speaker	Title/Abstract
(joint with Geometry Seminar) Th. Oct. 21 3-5 PM Sloan 257	Roman Muchnik University of Chicago	Stationarity of smooth measures on Furstenberg's and other boundaries. For a lattice $\Gamma$ in semisimple Lie group G, I will explain how to construct a measure $\mu \in P(\Gamma)$ so that $(G/P,\pi)$ is a $\mu$-boundary for $\Gamma, where \pi$ is fixed smooth probability measure on $G/P$. I will talk how to construct a measure on a group G that acts isometrically on CAT(-1)-space X, such that $(\partial X, \nu_p)$ is a Poisson boundary for $(G,\mu)$, where $\nu_p$ is the Patterson-Sullivan measure on $\partial X$ with respect to point $p$. This a joint work with Chris Connell.
Wed. Oct. 27 4-5:15 PM Sloan 153	Alex Gorodnik CalTech	Equidistribution of orbits of lattices on homogeneous spaces. For a lattice \Gamma in a Lie group G, we study the distribution of orbits of \Gamma in a homogeneous space G/H. We show that in many cases, every dense \Gamma-orbit is equidistributed with respect to a smooth measure. However, unlike in the case of amenable group actions, this measure is not \Gamma-invariant, and it depends non-trivially on the starting point. This is a joint work with Barak Weiss.
Wed. Nov. 3 4-5:15 PM Sloan 153	Nicolas Monod University of Chicago	Geometric splitting and superrigidity. We present a superrigidity result for actions on non-positively curved spaces. In particular, one obtains an elementary proof of Margulis' superrigidity for irreducible cocompact lattices in semi-simple (non-simple) Lie groups / algebraic groups. The proof uses a splitting theorem which is an infinite-dimensional (and singular) generalization of the Gromoll-Wolf / Lawson-Yau theorem.
Wed. Nov. 10
Wed. Nov. 17 4-5:15 PM Sloan 153	Seonhee Lim Yale University	Counting overlattices in automorphism groups of trees We give un upper bound for the number $u_\Gamma (n)$ of overlattices in the automorphism group of a tree, containing a fixed lattice $\Gamma$ with index $n$. For an example of $\Gamma$ in the automorphism group of a $2p$-regular tree whose quotient is a loop, we obtain a lower bound of the asymptotic behavior as well.
Wed. Nov. 24
Wed. Dec. 1 4-5:15 PM Sloan 153	Vitaly Bergelson Ohio State University	Many facets of weak mixing After a brief survey of the many equivalent definitions of weak mixing for unitary and measure preserving group actions, we shall discuss some recent results and applications ranging from combinatorics to "generic chaos".
Mon. Dec. 6 4-5:15 PM Sloan 153	Tsachik Gelander Yale University	A topological Tits alternative I will describe a joint work with E. Breuillard. In his celebrated 1972 paper, J. Tits proved a basic and important dichotomy for linear groups, which could be stated as follows: Tits Alternative: Let K be a field and G a finitely generated subgroup of GL(n,K), then either G contains a relatively Zariski open solvable subgroup or G contains a relatively Zariski dense non commutative free subgroup. Assume now that k is a local field. We proved the analog statement for the standard topology coming from k: Topological Tits alternative: Let k be a local field and G in GL(n,K) a linear group over k. Then either G contains a relatively open solvable subgroup or it contains a relatively dense free subgroup. This result has applications in dynamics, profinite groups and Riemannian foliations. For k=R it answers a question of Carriere and Ghys and provides a short proof for a conjecture of Connes and Sullivan on amenable actions which was first proved by Zimmer by other methods. For k non-Archimedean it implies a conjecture of Dixon, Pyber, Seress and Shalev concerning the profinite completion of linear groups. Our theorem implies also a conjecture of Carriere, that the growth of the leaves in any Riemannian foliation on a compact manifold is either polynomial or exponential.
Wed. Dec. 8 4-5:15 PM Sloan 153	Tsachik Gelander Yale University	Homotopy type and volume of locally symmetric manifolds We consider locally symmetric manifolds with a fixed universal covering, and construct for each such manifold $M$ a simplicial complex $\mathcal{R}$ whose size is proportional to the volume of $M$. When $M$ is non-compact, $\mathcal{R}$ is homotopically equivalent to $M$, while when $M$ is compact, $\mathcal{R}$ is homotopically equivalent to $M\setminus N$, where $N$ is a finite union of submanifolds of fairly smaller dimension. This reflects how the volume controls the topological structure of $M$, and yields concrete bounds for various finiteness statements which previously had no quantitative proofs. For example, it gives an explicit upper bound for the possible number of locally symmetric manifolds of volume bounded by $v>0$, and it yields an estimate for the size of a minimal presentation for the fundamental group of a manifold in terms of its volume. It also yields a number of new finiteness results.