Group Actions Seminar
(aka G mod Gamma seminar)

Past Schedule for 2010 fall- 2011 spring

• Feb 10: Rick Kenyon (Brown University)
  
  Title: Integrable iterations on Teichmuller space and the Ising model
  
  Abstract: TBA
  
  
• Feb 17: Rich Schwartz (Brown University)
  
  Title: Outer billiards on the penrose kite: compactification and renormalization
  
  Abstract: I'll give a fairly completely characterization of outer billiards on the penrose kite. It turns out that this planar dynamical system has a 3 dimensional compactification and that this compactification has a renormalization scheme. These two features make it possible to give sharp statements concerning the existence, distribution, large-scale geometry, and hidden algebraic symmetry of the orbits. The features in this one example suggest more general ways that renormalization might appear in polyhedron exchange maps.
  
  
• March 3: Alireza Salehi Golsefidy (Princeton University)
  
  Title: Affine sieve and expanders
  
  Abstract: I will talk about the fundamental theorem of affine sieve (joint with Sarnak). The main black box in the proof of this result will be also explained. It is a theorem on a necessary and sufficient condition for a finitely generated subgroup of SL(n,Q) under which the Cayley graphs of such a group modulo square free integers form a family of expanders (joint with Varju).
  
  
• March 17: Amir Mohammadi (University of Chicago)
  
  Title: Inhomogeneous quadratic forms
  
  Abstract: In a joint work with G. Margulis we prove a quantitative version of the Oppenheim conjecture for inhomogeneous quadratic forms. This generalizes the previous works of Eskin, Margulis and Mozes in the homogeneous setting also the work of J. Marklof. In this talk I intend to present main ideas involved in the proof.
  
  
• March 24: Yves Cornulier (University of Orsay, France)
  
  Title: On conjugacy growth of linear groups
  
  Abstract: We prove that in any linear group of exponential growth, there are exponentially many non-conjugate elements in the ball of radius n. (Joint work with Breuillard, Lubotzky and Meiri)
  
  
• April 14: Lior Fishman (Brandeis University)
  
  Title: Schmidt's game, friendly measures and exceptional sets on fractals
  
  Abstract: In this talk I shall describe new results regarding properties of certain sets on fractals. Questions regarding these sets, often exceptional both measure and category wise, arising from number theory, dynamics and Diophantine approximation theory, have been extensively studied in recent years utilizing Schmidt's game and properties of the class of friendly measures. In order to highlight the main ideas in many of these proofs, I shall (time permitting...), reprove a slight modification of Schmidt's original result regarding the set of badly approximable numbers, pointing out where generalizations have been made using modern ideas and techniques. I wish to emphasize that the talk will be quite self contained, thus accessible to graduate students as well as anyone interested.
  
  
• April 21: Anders Sodergren (IAS)
  
  Title: Equidistribution of pieces of closed horospheres in hyperbolic manifolds.
  
  Abstract: It is a classical result that on any hyperbolic surface of finite area, closed horocycles of length L become equidistributed as L tends to infinity. In this talk I will present equidistribution results for pieces of closed horospheres in hyperbolic manifolds of arbitrary dimension. The main ingredients in the discussion will come from spectral theory.
  
  
• April 28: Javier Aramayona (National University of Ireland at Galway)
  
  Title: Homomorphisms between mapping class groups
  
  Abstract: Let X and Y be surfaces of finite topological type, where genus(X) > 5 and genus(Y) < 2*genus(X). In this talk, we will describe all homomorphisms from the mapping class group of X to the mapping class group of Y. This is joint work with Juan Souto.
  
  
• May 5: Vaibhav Suresh Gadre (Harvard University)
  
  Title: Random walks on the mapping class group.
  
  Abstract: Kaimanovich and Masur proved that a typical random walk on the mapping class group when projected into Teichmuller space, converges with probability 1 to the Thurston boundary. This defines a measure on the Thurston boundary giving the distribution of the limit points of sample paths. This measure is harmonic with respect to the initial distribution on the group. We show that if the initial distribution is finitely supported then the harmonic measure is singular with respect to the natural Lebesgue measure on Thurston boundary. This result is analogous to a result of Guivarc'h and LeJan for random walks on certain discrete subgroups acting on the hyperbolic plane.
  
  
  Past Schedule for 2010
  
  • Sep 16, 2010: Nimish Shah (Ohio State University)
    
    Title: Ratner's theorem and dynamics of linear actions
    
    Abstract: In view of Ratner's theorem several questions involving unipotent flows in homogenous dynamics reduce to questions about dynamics of actions of semisimple groups on finite dimensional vector spaces. We would like to address this method via examples, which lead to results of number theoretic interest.
    
    
  • Sep 30, 2010: Tim Austin (Brown University)
    
    Title: The quantitative ergodic theorem and embeddings of groups into Banach spaces
    
    Abstract:The classical Mean Ergodic Theorem for probability-preserving transformations asserts convergence of a sequence of ergodic averages to some invariant function, but gives no effective control over how fast is this convergence (say, in norm). Truly quantitative versions of this theorem turn out to be quite subtle: one must search not for convergence to a limit function, but for long `epochs' of time over which the ergodic averages are only approximately invariant and remain approximately constant. In this talk I will show a somewhat surprising application of these quantitative results to a problem in geometric group theory, asking for the least possible distortion to the word metric on the discrete Heisenberg group when that group is Lipschitzly embedded as a metric space into a Lebesgue space L^p for some p \in (0,\infty). We obtain essentially sharp bounds on the `compression exponent' which quantifies this distortion of the word metric after reducing to a problem about probability-preserving actions, but only by making essential use of one of the quantitative variants of the ergodic theorem. Based on joint work with Assaf Naor and Romain Tessera.
    
    
  • Oct 14,2010: Elena Fuchs (IAS)
    
    Title: Positive density in integer Apollonian circle packings
    
    Abstract: Apollonian circle packings are constructed by continuously inscribing circles into the curvilinear triangles formed in a Descartes configuration of mutually tangent circles. An observation of F. Soddy in 1937 is that if any four mutually tangent circles in the packing have integer curvature, then in fact all of the curvatures in the packing will be integers. In the past few years, this observation has led to several developments regarding the number theory of such integer Apollonian packings. In this talk, I will discuss a very generalizable proof of the positive density theorem for integer Apollonian packings -- that the integers occurring as curvatures in any given integer Apollonian circle packing make up a positive fraction of $\mathbb Z$. This is joint work with J. Bourgain.
    
    
  • Oct 28,2010: Anish Ghosh (University of East Anglia, UK)
    
    Title: Diophantine Approximation on Varieties
    
    Abstract: I will discuss the effective density of rational points on homogeneous varieties of semisimple groups. This is joint with A. Gorodnik and A. Nevo.
    
    
  • Nov 18, 2010: Bryna Kra (Northwestern University)
    
    Title: Nilsequences: ergodic, combinatorial, and topological
    
    Abstract: Nilsequences have played a role in recent developments in ergodic theory (in structure theorems for measure preserving systems) and in additive combinatorics (in finding asymptotics for certain patterns in the primes), and more recently in topological dynamics. I will give an overview of how these algebraic objects play a role in various settings and discuss a characterization of that detects if a given sequence is a nilsequence, by testing local properties
    
    
  • Dec 2, 2010: Michael Hochman (Princeton University)
    
    Title: Geometric rigidity of times-m invariant measures
    
    Abstract: Let mu be a measure on [0,1] which is invariant and ergodic under multiplication by m (mod 1), and with positive Hausdorff dimension. I'll explain why, if f is a diffeomorphism of the real line, and f(mu) and mu are equivalent on some set A, then f' is a rational power of m a.e. on f^{-1}(A). In particular this generalizes Rudolph's theorem that there are no positive dimension measures on [0,1] which are simultaneously invariant under times-2 and times-3 mod(1). I'll also discuss another generalization which shows that Rudolph's theorem is a consequence of the C^2 isomorphism types of the multiplication maps, and one does not require the abelianness of the multiplication action.
    
    
    Past Schedule for 2009
    
    
  • Feb 24, 2009: Uri Shapira (Hebrew University of Jerusalem)
    
    Title: A solution to an open problem of Cassels and Diophantine properties of cubic numbers
    
    Abstract: We prove existence of real numbers x, y possessing the following property: For any real a,b. liminf |n| ||nx-a|| ||ny-b||=0, where ||c|| denotes the distance of c to the nearest integer. This answers a 50 year old question of Cassels. The most interesting part of the result is that there are algebraic numbers with the above property.
    
    
  • March 10, 2009: Hee Oh (Brown University) (CANCELLED)
    
    Title: Integral points on symmetric varieties and Satake compactification
    
    Abstract: For a symmetric variety V over Q, we compute the asymptotic distribution of the angular components of the integral points on V. This distribution is described by a family of invariant measures concentrated on the Satake boundary of V. Describing the Satake compactification and the corresponding family of measures will be the main focus of the talk, with several concrete examples. This talk is based on joint work with Alex Gorodnik and Nimish Shah.
    
    
  • March 31, 2009: Nimish Shah (Tata Institute, Inida)
    
    Title: Khinchin theorem for integral points on quadratic varieties
    
    Abstract: TBA
    
    
  • April 7, 2009: Alex Eskin (University of Chicago) is giving the Distinguished lecture series
    
    Title: The Gromov program
    
    TBA
    
    
  • April 22, 2009: David Fisher (Indiana University) (Joint with Geometry/Topology seminar)
    
    Title: Rigidity of Anosov Z^d actions on nilmanifolds
    
    Abstract: It is conjectured by Katok and Spatizer that all irreducible Anosov Z^d actions on manifolds are conjugate to affine algebraic actions on nilmanifolds. I will discuss motivavtion and prior work on the subject before concentrating on recent joint work with Kalinin and Spatizer. There is a strong analogy between global rigidity results of this kind and work on rigidity of invariant measures. I will try to explain some of this analogy and point out how our work is analogous to ideas of Lindentrauss that led to breakthroughs work on measure rigidity.
    
    
    
    Past Schedule for 2008
    
    
  • 2/28 /2008 Nir Avni, Hebrew University/UCLA
    
    Title: Representation Growth of Groups.
    
    Abstract: For a group G, let a_n(G) be the number of n-dimensional representations of G. For certain kinds of groups (SL(n,Z) for example), the behavior of a_n(G) can be encoded in a zeta-like function. I will describe several results and open questions regarding the poles (which give the rate of growth of a_n(G)), meromorphic continuation (which gives the error terms), and functional equation (which give surprising symmetries between the a_n(G)'s) of this zeta function.
    
    
  • 3/13 /2008 Danijela Damjanovic, Harvard
    Abelian actions and rigidity
    
    Abstract: I will give an overview of recent results and of some problems concerning various rigidity properties of algebraic abelian actions.
    
    
  • 4/3/2008 Gopal Prasad, U of Michigan
    Lengths of closed geodesics and isospectral locally symmetric spaces
    
    Abstract: I will give an exposition of my recent work with Andrei Rapinchuk in which we have introduced a new notion of "weak commensurability" of Zariski-dense subgroups. We have studied weak commensurability of arithmetic subgroups of semi-simple Lie groups. The notion of weak commensurability turns out to be intimately related to the commensurability of the set of lengths of closed geodesics on, and isospectrality of, locally symmetric spaces of finite volume (and with nonpositive sectional curvatures). We are in this way able to answer Marc Kac's famous question "Can one hear the shape of a drum?" for compact arithmetic locally symmetric spaces.
    
    
  • 4/24/2008 Francois Maucourant, Rennes 1 University
    Two examples of nonhomogeneous orbit closures.
    
    Abstract: We will explain how to construct orbits of non-homogeneous closure for some subgroup of the diagonal group acting on the space of lattices of dimension at least 6. Also, a similar example for the action of *2,*3 on the four dimensionnal torus is discussed.
    
    
  • 5/1 /2008 Lisa Carbone, Rutgers University
    Tits buildings for hyperbolic Kac-Moody groups
    
    Abstract: We outline how to associate a simplicial complex X to the BN-pair data for a Kac-Moody group G. When G is of hyperbolic type, X is a hyperbolic building whose local structure is determined by the proper connected subdiagrams of the Dynkin diagram for G. For Kac-Moody groups over finite fields, we indicate how the action of G on X may be used to give structure theorems, generators and relations for G and its subgroups. When G is hyperbolic of rank 2 or 3 we show that G acts on a homogeneous or bihomogeneous tree.
    
    
  • Sep 23, 2008 (Joint with Geometry/Topology Seminar): Hee Oh, Brown university
    
    Title: Counting Apollonian circles of bounded curvature
    
    Abstract: Apollonian circle packings are arrangements of tangent circles that arise by repeatedly filling the interstices between four mutually tangent circles with further tangent circles. In a recent joint work with Alex Kontorovich, we obtain the asymptotic number of circles of curvature at most T, as T tends to infinity, in any gievn Apollonian packing. We also obtain the optimal upper bounds for the number of primes/twin primes in any integral Apollonian packing. In my talk, I will try to explain these results as well as to explain how the equidistribution of long horospheres in infinite volume hyperbolic 3-manifold are related.
    
    
  • Oct 14,2008: Dominique Hulin, Orsay University, France
    
    Iteration of quadrilateral foldings
    
    Abstract: We start with a quadrilateral q in the Euclidean plane. Replacing successively each of the four vertices by its symmetric with respect to the opposite diagonal, we obtain a new quadrilateral f(q), where f is our so-called ``cyclic folding transformation''. In a joint work with Y. Benoist, we study the dynamics of this transformation. This cyclic folding transformation was first studied by G. Darboux in 1879, who related it to translations on elliptic curves.
    
    
  • Oct 21,2008: Ludovic Marquis, Orsay University, France
    
    Properly convex projective surface of finite volume
    
    Abstract:
    
    
  • Oct 28, 2008: Tsachik Gelander, Hebrew university, Israel
    
    On deformations of free groups in compact Lie groups.
    
    Abstract:I will discuss some properties of the varieties of deformations of free groups in compact Lie groups. In particular a proof of a conjecture of Margulis and Soifer about the density of nonvirtually free points, and a proof of a conjecture of Goldman on the ergodicity of the action of Aut(F_n) on such variety when n>2. I will also try to locate these result in a more general setup involving: finite groups, topological groups and actions of Aut(F_n), which recently re attract the interest of many people, and describe some of the main problems, and some recent developments.
    
    
  • Nov 6,2008 : Fanny Kassel, Orsay University, France
    
    Clifford-Klein forms of semisimple rank-one group manifolds
    
    Abstract: Let G be a connected semisimple rank-one Lie group or a Zariski-connected semisimple rank-one algebraic group over a local field. I will describe all torsion-free discrete subgroups Gamma of G\times G acting properly discontinuously on G by left and right multiplication. I will also give a criterion on Gamma for the quotient to be compact, and discuss the deformation of such compact quotients.
    
    
  • Nov 18, 2008 (Joint with Geometry/Topology Seminar): Alan Reid, University of Texas at Austin/IAS
    
    LERF and the Lubotkzy-Sarnak Conjecture
    
    Abstract: The Lubotzky-Sarnak Conjecture asserts that the fundamental group of a finite volume hyperbolic manifold does not have Property $\tau$. Put in a geometric context, this conjecture predicts a tower of finite sheeted covers for which the Cheeger constant goes to zero. This Conjecture has attracted a lot of attention recently because of its connections to the topology of finite sheeted covers of closed hyperbolic 3-manifolds. This talk will discuss these connections, together with recent work that connects these circle of ideas with the group theoretic property LERF (a far reaching generalization of residually finite).
    
    
  • Nov 20, 2008: Jean Francois Quint, University of Paris 13, France
    
    Exponential drift and stationary measures on tori.
    
    Abstract:in a recent joint work with Yves Benoist, we developed a new method for the analysis of orbit structure for action of non-connected subgroups of Lie groups on homogeneous spaces. This method allowed us to prove that, if $\Gamma$ is a Zariski dense subgroup of $SL_d(\mathbb R)$, every $\Gamma$-orbit in $SL_d(\mathbb R)/SL_d(\mathbb Z)$, is either finite or dense. In this talk, I will show how this method can be used to recover the following result of Bourgain, Furman, Lindenstrauss and Mozes : if $\mu$ is a finitely supported probability measure on $SL_2(\mathbb Z)$ which support spans a Zariski dense subgroup of $SL_2(\mathbb R)$, then every extremal $\mu$-stationary Borel probability measure on the torus $T^2$ either is finitely supported or equals Lebesgue measure.
    
    
  • Nov 25,2008: Alex Gamburd, UCSC and Northwestern university
    (Alex will also be giving an Algebra seminar "Expanders: from arithmetic to combinatorics and back" on Nov 24)
    
    Title: Uniform spectral gap bounds
    
    Abstract:
    
    
    Past Schedule for 2007
    
    
  • 9/27 /2007
    Alex Kontorovich, Brown university
    The Hyperbolic Lattice Point Count in Infinite Volume with Applications to Sieves
    
    abstract: We develop novel techniques using only abstract operator theory to obtain asymptotic formulae for lattice counting problems on infinite-volume hyperbolic manifolds, with error terms which are uniform as the lattice moves through ``congruence'' subgroups. These methods have their origins in the work of Selberg, Lax-Phillips and Duke-Rudnick-Sarnak, and the uniformity relies on the spectral gap established in Bourgain-Gamburd and Bourgain-Gamburd-Sarnak. We give an application to the theory of affine linear sieves.
    
    
  • 10/11 /2007
    Amir Mohammadi, Yale university
    Discrete Transitive Actions on Bruhat-Tits Building
    
    abstract: There are many discrete transitive actions on an $r$-regular tree, i.e. rank one Bruhat-Tits tree. However it is rare to have such action on a higher rank Bruhat-Tits Building. The objective of this joint on going work with "A. Salehi Golsefidy" is to classify all discrete transitive actions on Bruhat-Tits building of higher rank algebraic groups over characteristic zero local fields. In this talk we will see how arithmetic structure of such lattices and volume formula of Prasad for arithmetic lattices will give a non-existence result for such a lattice in $\rm {PGL_n} (F)$ for $n>8$ and give a list of 17 potential such examples for $4
    
    
  • 10/25 /2007
    Alex Furman, UIC (also see his G-T talk on 10/24)
    Invariant and stationary measures for groups of toral automorphisms
    
    abstract: Joint work with J. Bourgain, E. Lindenstrauss, S. Mozes. Given a Zariski dense group G of toral automorphisms we prove that the only invariant or, more generally, stationary measures on the torus are combinations of Lebesgue and atomic measures.
    
    
  • 11/15 /2007
    Emmanuel Breuillard, IAS and Polytechnique
    On the growth of Linear groups and Arithmetic heights
    
    abstract: We introduce a notion of minimal height for a finite subset of matrices with coefficients in an algebraic closure of Q and show an analog of the "Margulis Lemma" in this situation, which asserts that sets of small height must generate virtually solvable subgroups. This result allows to prove a uniform lower bound for the growth of finitely generated subgroups of GL_d(C), thus improving earlier results of Eskin-Mozes-Oh. We also make a connection with the Lehmer conjecture.
    
    
  • 11/29 /2007
    Alireza Salehi Golsefidy, Princeton University
    Translates of horospherical measures and rational points on G/U
    
    abstract: (Joint with A. Mohammadi) There are three, more or less related, methods to understand distribution of integral or rational points on a homogeneous variety: Automorphic forms, mixing, and ergodic theory. In this talk, I will explain the general frame work of the second and the third methods. Then for the purpose of studying the distribution of rational points on a toric bundle on the flag variety, i.e. SL (n,R)/U where U is the set of real matrices with diagonal entries equal to 1, I will consider translates of the probability measure on SL(n,R)/SL(n,Z) which is induced by the Haar measure on U and supported on U SL(n,Z)/SL(n,Z). I will explain the possible scenarios for the limit of the translates, and show how it can help to study the distribution of the rational points.