We will cover the theory of unipotent flows, and its applications to number theory. We will start with quantitative recurrence results for unipotent flows. These results, on their own, lead to the solutions of many old problems in the matric theory of diophantine approximations. We will cover some of the applications (such as a proof of the Mahler conjecture), and subsequent developments.
We will then proceed to the measure classification results on unipotent flows (Ratner's theorem). We will give a proof of Ratner's theorem in a simple case, and then cover some of the applications. In particular we will give a proof of the Oppenheim conjecture (we may also present Margulis' original proof which does not rely on Ratner's theorem). We will then cover results on the quantitative Oppenheim conjecture, and some (both older and more recent) applications to diophantine equations. We will end by presenting very recent results of Margulis and Goetze on quadratic forms in five or more variables.
We will discuss the theory of diagonalizable group actions, and applications to number theory and other subjects, including:
Classification of measures on Gamma\G invariant under diagonalizable groups, particularly a maximal split torus (such as the full diagonal group in SL(n,Z)\SL(n,R)). The complete classification is an important open problem, we will discuss the state-of-the-art results toward this classification. We also discuss joinings and measurable factors of such actions with applications to equidistribution.
Classification of closed invariant sets under such actions: again the full classification is an important open problem, but there are substantial partial results which we will cover. We will give an application regarding values of products of n linear forms in n variables and the set of exceptions to Littlewood Conjecture.
Distribution of compact orbits of these actions, including Linnik's principle and a dynamical proof of Duke's theorem regarding equidistribution of certain collections of closed geodesics on the modular surface.
Will also investigate the connections to automorphic forms, and in particular to arithmetic quantum unique ergodicity.
We will possibly also cover quantitative aspects of equidistribution and connection to subconvexity and sparse trajectories; quantitative versions Furstenberg's theorem and possibly of other measure and orbit classification theorems.
The course will be on translation surfaces, interval exchange maps and Teichmuller flows. A list of contents may include (parts of) the following: Poligonal billiards and translation surfaces. Deformation of a translation surface by the action of GL(2,R). Dynamics of vertical vector fields and interval exchange maps. Suspension of interval exchange maps. Teichmuller flow. Moduli space of abelian differentials. Continued fraction algorithms for interval exchange maps. The cohomological equation for interval exchange maps. The cocycles of Zorich and of Kontsevich-Zorich and their spectral properties. Exponential mixing of the Teichmuller flow.
The first lecture introduces Heegner points and closed geodesics on the modular surface SL2(Z)\H and highlights some of their arithmetic significance. The second lecture discusses how subconvex bounds for certain automorphic L-functions yeild quantitative equidistribution results for Heegner points and closed geodesics.
Fuchsian groups, geodesic flows on surfaces of constant negative curvature and symbolic coding of geodesics (Svetlana Katok)
This will be an introductory course on geodesic flows on surfaces of constant negative curvature and symbolic coding of geodesics. Particular topics will include (parts of) the following:
*Hyperbolic geometry (geodesics, isometries, hyperbolic area)
*Fuchsian groups, their fundamental regions, connection with Riemann surfaces
*Examples: arithmetic Fuchsian groups: modular group and its subgroups; Fuchsian groups derived from quaternion algebras
*Geodesic flow on a Riemann surface as an example of a dynamical system with complicated "hyperbolic" behavior. Density of closed orbits, topological transitivity and ergodicity with respect to the Liouville measure. Anosov closing lemma and Livshitz theorem for the geodesic flow.
*Geodesic flow as a special flow over a cross-section. Symbolic coding of geodesics on surfaces of constant negative curvature via a fundamental region.
*A new approach to coding of geodesics on the modular surface via so-called (a,b)-continued fractions and its relation to Gauss reduction theory.
Advanced Mini Courses
This course is a follow-up to Yoccoz's course. It will center around applications of the "chaotic" aspect of the dynamics of the Teichmuller flow/Renormalization algorithm for interval exchange transformations.
The first application is the study of the Zorich phenomenon, regarding the asymptotic behavior in homology of trajectories of the vertical flow on a typical translation surface. To first approximation, trajectories grow linearly along the direction of the Schwartzmann asymptotic cycle. Closer inspection (first observed numerically) reveals a hierarchy of polynomial deviations organized in a Lagrangian flag: there is a main deviation concentrated along a 2-plane, followed by a secondary deviation concentrated along a 3-plane, etc. The work of Kontsevich and Zorich explained how this phenomenon is linked to the Lyapunov exponents of a certain linear skew-product over the Teichmuller flow, the Kontsevich-Zorich cocycle. The emergence of the Lagrangian flag was shown to be a consequence of the Zorich-Kontsevich conjecture, that the Lyapunov spectrum of this cocycle is simple. We will discuss an approach to this conjecture based on treating the Kontsevich-Zorich cocycle as a random matrix product. Simplicity of the spectrum is then a consequence of ''combinatorial richness'' of the cocycle, detected by hyperbolichyperbolicboundary behavior.
The second problem is the weak mixing property for typical interval exchange transformations which are not rotations. Here what one has to show is absence of measurable eigenfunctions. The renormalization algorithm ''smooths'' eigenfunctions, making them approximately constant (this is a common theme of renormalization theory). This leads to the Veech criterium for ruling out eigenvalues, in terms of the dynamics of a linear skew-product (the Kontsevich-Zorich cocycle), acting as automorphisms of tori. Chaoticity allows us to proceed by stochastic modelling. The analysis involves some information on the Lyapunov exponents and combinatorial richness (strictly contained in the discussion of the previous problem).
This minicourse deals with the large eigenvalue limit for eigenfunctions of the laplacian, on a compact riemannian manifold M. More precisely, we will be interested in the quantum (unique) ergodicity problem, which asks about the weak limits of the probability measures |phi(x)|^2 dx on M defined by the eigenfunctions phi, when the eigenvalue tends to infinity. Our approach will be purely focused on the dynamical, rather than arithmetic, aspects.
We will start with a brief introduction to microlocal analysis, which is a way to lift the problem to the cotangent bundle T^*M, and thus to relate it with a particular hamiltonian dynamical system, here the geodesic flow. We will discuss quantization procedures, and state the Egorov theorem, saying that the unitary flow induced by the laplacian converges, in some sense, to the geodesic flow, in the large eigenvalue limit (which is here equivalent to a "semiclassical" limit). This implies that the microlocal lifts of the measures |phi(x)|^2 dx on T^*M converge to invariant measures of the geodesic flow.
We will then proceed to show how the qualitative properties of the geodesic flow can lead to various properties for eigenfunctions. For instance, eigenfunctions in completely integrable systems exhibit a completely different behaviour from those in ergodic systems. We will prove the Snirelman theorem, which concerns the case when the geodesic flow is ergodic with respect to the Liouville measure. It says that a "density one" subsequence of the measures |phi(x)|^2 dx converges to the Liouvile measure. On a negatively curved manifold, Rudnick and Sarnak conjectured that the whole sequence actually converges, but this has remained an open problem so far.
The rest of the minicourse will be devoted to proving a recent result, joint with Stephane Nonnenmacher, giving a positive (and explicit) lower bound for the entropy of the limit measures. The result relies only on the Anosov property of the geodesic flow, it holds in variable negative curvature and arbitrary dimension. This bound implies, in particular, that less than half the mass of the measures |phi(x)|^2 dx can go, in the limit, to a finite union of closed geodesics. The proof combines some simple ideas of ergodic theory with technical semiclassical estimates obtained from WKB methods.
I will discuss the connection between effective equidistribution, the spectral gap, and questions in analytic number theory.
In the first lecture, I will try to give historical context by explaining Kloosterman's work from the 1920s, its connection to Selberg's 3/16 theorem, and the interpretation in terms of effective equidistribution of horocycle flows.
In the second and third lectures, I will discuss effective equidistribution of closed orbits of semisimple groups, acting on homogeneous spaces.This will be based on a joint paper with M. Einsiedler and G. Margulis. I will try to emphasize the connection to spectral gap in higher rank.
In a given algebraic variety X defined as the zero sets of polynomials with integral coefficients, understanding (1) the integral points, (2) the S-integral points and (3) the rational points of X is a classical topic in number theory and arithmetic geometry.
When the variety is a homogeneous space of a semisimple algebraic group, counting and equidistribution problems for these 3 different types of points can be approached in an unified way via either mixing or unipotent flows, and we are able give satisfactory answers in many important cases.
The goal of my 3 lectures is to explain these approach and illustrate concrete examples solved by these approach. When we are able to use mixing properties, our answers are given in effective forms.