| Bergelson | The multifarious van der Corput trick. | We will discuss diverse applications of the so called van der Corput trick which range from the classical Weyl's theorem on the equidistribution of polynomials to modern results such as the polynomial Szemeredi theorem and its ramifications. We will also discuss some natural open problems and conjectures which focus on various forms of uniform distribution and connections with ergodic Ramsey theory. |
| Bourgain | On random walks and expansion in SL_d(q). | We review recent developments around expansion in SL_d(q) with a brief description of the underlying results and techniques. |
| Breuillard | Non commutative diophantine approximation and the Tits alternative. | Using tools from diophantine geometry, we prove a strong uniform version of the Tits alternative, improving earlier results of Eskin-Mozes-Oh and Breuillard-Gelander. We derive from it a uniform upper bound for the number of words that fall in a shrinking neighborhood of 1 in a given Lie group, in the spirit of an earlier work of Kaloshin-Rodnianski. Such non commutative diophantine properties are key to several spectral gap / equidistribution conjectures concerning dense subgroups of Lie groups, as emphasized in particular by works of Gamburd-Jacobson-Sarnak and Bourgain-Gamburd. |
| Einsiedler | Spectral gap and effective equidistribution of semisimple orbits. | The dynamics on homogeneous spaces has many interesting connections to number theory. One of the main problems here is to understand the distribution of closed orbits for subgroups H of the ambient Lie group G. In joint work with G.Margulis and A.Venkatesh we prove an error rate in the equidistribution for semisimple subgroups H acting on congruence quotients of G. This makes use of spectral gap in the form of property (tau). We will discuss the relationship between spectral gap, effective decay of matrix coefficients, and effective equidistribution. |
| Furman | Stiffness of large groups of toral automorphisms. | I will report on a joint work with Jean Bourgain, Elon Lindenstrauss and Shahar Mozes. For the action of a Zariski dense subgroup G in SL(d,Z) on the torus T^d it is shown that the only G-invariant and, more generally, stationary probability measures on T^d are the obvious ones: convex combinations of the Lebesgue measure and atomic measures at rational points. The proof employs a projection theorem of Bourgain related to the sum-product estimates. |
| A. Katok | Measure rigidity for actions of higher-rank abelian groups from topology and dynamics. | I will survey recent results joint with Boris Kalinin and Federico Rodriguez Hertz (in various combinations) where existence and properties of an absolutely continuous invariant measure for actions of higher rank abelian groups is deduced from homological or general dynamical properties of the action. The method can be viewed as a development of the first method used to prove measure rigidity for positive entropy measures for actions by toral automorphisms in the 1996 paper joint with Ralf Spatzier. I will also discuss open problems including the potential of using more recent methods developed in measure rigidity of homogeneous actions in the general non-homogeneous situations. |
| Kelmer | Spectral gap for quotients of products of PSL_2(R). | In this talk I will discuss the notion of a strong spectral gap for quotients of semi-simple Lie groups. I will go over what is known with regard to this property for general Gamma\G, and present new results establishing an effective gap when G is a product of copies of PSL(2,R). This is joint work with Peter Sarnak. |
| Kleinbock | An "almost all versus no" dichotomy for orbits on homogeneous spaces. | Several years ago I proved (in an involved and somewhat mysterious way) the following theorem: suppose M is an analytic submanifold of R^n which contains a not very well approximable (NVWA) vector; then almost all its vectors are NVWA. Recently, jointly with Barak Weiss, we found a simple argument establishing this and other similar results. Both old and new proofs use quantitative nondivergence on the space of lattices. |
| Michel | The subconvexity problem for GL_2 L-functions. |
In this talk we will describe the general subconvexity problem for central value of GL_n automorphic
L-functions. We will also explain the resolution of this problem in the case of GL_1 and GL_2 automorphic forms over a general (fixed) number field, and this uniformly in all parameters (the spectral, level and s-aspects). The main ingredient of the proof are - suitable representations of the central values in terms of automorphic periods which factor over local period integrals of matrix coefficients, -the spectral decomposition of such periods -the spectral gap property for GL_2 matrix coefficients -and the amplification method of Iwaniec We will also explain an application -explained to us by Andre Reznikov- of the uniformity of our bounds to the study of the restriction of Maass forms of large Laplace eigenvalue along a fixed closed geodesic. This is joint work with Akshay Venkatesh. |
| Mirzakhani | Dynamics over moduli spaces of Riemann surfaces. | In this talk we will describe the ergodic properties of different geometric flows defined on bundles over moduli spaces of Riemann surfaces. We will discuss some applications and related open problems motivated by the analogy between the moduli space of surfaces and homogeneous spaces of Lie groups. |
| Mohammadi | Unipotent flows in positive characteristic. | Margulis' celebrated proof of the Oppenheim's conjecture and Ratner's seminal work on the proof of Raghunathan's conjectures are considered as pioneer works in the theory of unipotent flows. However the question in the case of positive characteristic local fields is left wide open. In this talk we will address the recent improvements in this issue. In particular we will focus on the recent joint work with M. Einsiedler on classifications of joinings for flows of certain unipotent groups. We will also draw the attention to an application in quasi-isometry rigidity. |
| Oh | Rational points and unipotent flows. | Understanding the rational points of a projective variety is a fundamental subject
in arithmetic geometry.
Main motivation of this talk is a conjecture of Manin in 1987 on the asymptotic
number of rational points of height at most T as T tends to infinity. By studying the rigid properties of flows on homogeneous spaces of adelic groups, we can approach this problem for the projective varieties which are compactifications of affine homogeneous varieties. I will state a general equidistribution theorem of a sequence of semisimple adelic periods, and explain several theorems on rational points, including some new cases of Manin's conjecture, as applications. This talk is based on a joint work with Alex Gorodnik. |
| Sarnak | Arithmetic and classical variance | TBA, joint with Luo and Rudnick. |
| Shah | Equidistribution of expanding translates of curves and Diophantine approximation. | We show that for almost all points on any analytic curve on R^{k} which is not contained in a proper affine subspace, the Dirichlet's theorem on simultaneous approximation, as well as its multiplicative variation, cannot be improved. The result is obtained by proving asymptotic equidistribution of evolution of an analytic curve segment on a strongly unstable leaf under certain partially hyperbolic flow on the space of unimodular lattices in R^{k+1}. |
| Tao | The quantitative distribution of polynomial sequences on nilmanifolds. | The asymptotic equidistribution of linear and polynomial sequences on nilmanifolds has been studied by many authors (e.g. Green, Parry, Shah, Leibman). In particular Leibman showed the following Ratner-type theorem: any polynomial sequence ( g^{p(n)} x )_{n=1}^\infty on a nilmanifold is always equidistributed in a subnilmanifold, after one partitions the integers into finitely many residue classes. I will discuss a quantitative analogue of this result, in joint work with Ben Green, which establishes a similar equidistribution statement for finite orbits ( g^{p(n)} x )_{n=1}^N . This result is of relevance to a certain number-theoretic conjecture of Ben and myself, which we call the Mobius-nilsequences conjecture. |
| Templier | On the equidistribution of roots of quadratic congruences. | Motivated by a question by Erd\"os on large prime factors of polynomial sequences, Hooley discovered the equidistribution of roots of quadratic congruences. I will review his ingenious proof and its geometric significance (spectral gap). Another important result related to quadratic forms is Duke's Theorem. I will then present a new shifted convolution problem and some ideas leading to its resolution. It has application to moments of Rankin-Selberg L-functions and to independence of Heegner points on rational elliptic curves. |
| Ullmo | Equidistribution+Galois=André-Oort. | We will explain how to combine ideas from ergodic theory and Galois theory to obtain a proof of the André-Oort conjecture under the Riemann Hypothesis for CM-fields. |
| Venkatesh | Torus orbits on the space of rank 3 lattices. | I will discuss compact orbits of the diagonal torus on the space of rank 3 lattices, and explain a uniform distribution result in this context that is joint work with Einsiedler, Michel and Lindenstrauss. I'll then discuss the parts of the proof that relate to harmonic analysis. For instance, one question that arises naturally concerns decay of eigenfunctions on homogeneous spaces; as time permits I'll try to discuss it in a more general context. |
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