Hyperbolic geometry and dynamics:
talks and abstracts

May 18-22, 2015
Laboratory of Algebraic Geometry, Moscow

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Sasha Ananin (ICMC-USP Sao Carlos)
Spherical and hyperbolic 2-spheres with cone singularities

Slides (PDF)
We study a possible geometry on the space C(a_1,...,a_n) of spherical or hyperbolic 2-spheres with cone points of prescribed cone angles a_1,...,a_n and the corresponding completions of this space. The euclidean case, considered by W. Thurston, provides the famous 9 examples of nonarithmetic compact holomorphic 2-ball quotients constructed by Deligne-Mostow (when n=5). A geometry in question is not unique (even in the flat case considered by Thurston) and our main concern is to find a "good" hyperbolic geometry on C(a_1,...,a_n) that may conjecturally provide new examples of nonarithmetic compact holomorphic 2-ball quotients.

This is a joint work in progress with Carlos H. Grossi, Jaejeong Lee, and Joao dos Reis jr.

A geometry in question is not unique (even in the flat case considered by Thurston) and our main concern is to find a `good' hyperbolic geometry on $C(a_1,\dots,a_n)$ that may conjecturally provide new examples of nonarithmetic compact holomorphic 2-ball quotients.

This is a joint work in progress with Carlos H. Grossi and Jaejeong Lee.

Serge Cantat (Rennes)
Periodic points of automorphisms of projective surfaces (minicourse)

Let f be a holomorphic diffeomorphism of a complex projective surface X. Consider its periodic points of period n: there are examples for which the number of such points increases exponentially fast with the period n. What can be said on the number of such points and on their repartition in the surface X? I shall address this type of questions and explain how ideas of complex analysis, algebraic geometry, and dynamical systems can be combined in this context.

Simion Filip (University of Chicago)
Counting Techniques

It is classical (Gauss) that the number of integral lattice points in the ball on the plane is quadratic in the radius of the ball. One can address similar questions about lattices in hyperbolic or more general homogeneous spaces. The most effective tools involve a combination of ergodic theory and geometry of negatively curved spaces. I will explain some of the basic techniques that are used.

Simion Filip (University of Chicago)
The Multiplicative Ergodic Theorem (2 lectures)

The classical ergodic theorems of Birkhoff and von Neumann say that averaging a function over an orbit of a dynamical system converges, typically, to the average of the function on the entire space. When sampling a matrix-valued function along orbits, the asymptotic behavior is described by the Oseledets Multiplicative Ergodic Theorem. I will explain several proofs of this result, including a generalization to sampling isometries of non-positively curved spaces, due in various forms to Karlsson, Margulis, and Ledrappier.

Alex Furman (University of Illinois at Chicago)
1. Mostow rigidity
2. Locally symmetric metrics entropy minimizers after Besson-Courtois-Gallot
3. Quasi-isometric rigidity of lattices

Any compact surface of genus at least two admits a hyperbolic metric (i.e. is covered by the hyperbolic plane); in fact, one has a whole multidimensional family of such metrics - the Teichmuller space. However, in higher dimensions, if a compact manifold admits a hyperbolic (or other locally symmetric) structure, then it is unique. This remarkable phenomenon, discovered by Mostow in the sixties, is one of the cornerstones of modern hyperbolic geometry and geometric group theory. In the talks I plan to discuss Mostow's Strong Rigidity theorems, the work of Besson-Courtois-Gallot (that implies Mostow's rigidity), and survey some quasi-isometric rigidity results pertaining to lattices.

Vadim Kaimanovich (Ottawa)
Boundaries in topology and probability

Usually one deals with boundaries in the topological setup (for instance, the boundaries of various compactifications of topological spaces). I will describe the measure theory tools which allow one to define the so-called Poisson boundary of a Markov chain in the measure category. I will outline the relationship of this boundary with the topological boundaries for random walks on groups with hyperbolic properties and formulate the main problems in this area.

Olivier Guichard (Strasbourg)
Hyperbolic groups and subgroups of Lie groups

The aim of those lectures is first to give different characterizations of convex-compact subroups in terms of their action on the hyperbolic space or on the sphere at infinity and second to address the situation in higher rank exhibiting rigidity results as well as flexibilty phenomenon for the action on symmetric spaces and on flag varieties.

Dmitry Kleinbock (Brandeis)
Hyperbolic dynamics and intrinsic Diophantine approximation

Dynamics on homogeneous spaces of Lie groups has been a useful tool in solving many previously intractable Diophantine problems. In this talk I will describe some existing connections between homogeneous dynamics and Diophantine approximation, and then show how a similar approach can help quantify the density of rational points on quadric hypersurfaces (intrinsic approximation problems). The case of spheres is reduced to dynamics on hyperbolic manifolds. The new work is joint with Lior Fishman, Keith Merrill and David Simmons.

Andrei Malyutin (PDMI)
Random walks and hyperbolic spaces

Random walks on groups is a subject at the intersection of probability, group theory, geometry, etc. We are looking for new examples and approaches in this area by examining group actions on various classes of spaces. The core example here is the following case:

Dmitri Scheglov (UFF, Rio de Janeiro)
Logarithmic angular diffusion for rational right triangular billiards and high genus hyperelliptic limit(s)

Recently a group of physicists numerically observed peculiar ergodic properties of right triangular billiards( both rational and irrational), which do not hold for non-right triangles. In the joint project with G.Forni we established this numerical conjecture for rational case, which essentially amounts to splitting of KZ- cocyle on absolute and relative component over hyperellyptic moduli space.

I will also explain the current attempts to rationally approximate the general ( irrational) case and some partial progress.

Anatoly Vershik (PDMI)
How to classify the filtrations, --- e.g. the decreasing sequences of sigma- algebras or algebras

The Classification of the filtrations appears in the theory of dynamical systems, statistical physics, theory of stochastic processes and in classical analysis. This is the art of general problem: how to make the link between classification of finite faminlies of the pbjects and classification of infinite families. The main question: what one must add to the set of finite invariants and when we do not need the new invarinats. Classical example of the case of absense of new infinite invariants is Kolmogorov's (1933) zero-one law for Bernoulli scheme. But very often we need in the "highest" invariants.

I will tell about this problem in the framework of asymptotic studies of combinatorics and probability theory. No special knowledge is needed for understanding of this talk.

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Laboratory of Algebraic Geometry and its Applications