Annual Lecture Series

We prove quantitative recurrence and large deviations results for the Teichmuller geodesic flow on the connected components of strata of the moduli space Q_g of holomorphic unit-area quadratic differentials on a compact genus g>1 surface. We also prove analogous results for certain symmetric random walks on Q_g.

The first talk will provide an introduction to quadratic differentials and strata, and the second will focus on my results.

We prove quantitative recurrence and large deviations results for the Teichmuller geodesic flow on the connected components of strata of the moduli space Q_g of holomorphic unit-area quadratic differentials on a compact genus g>1 surface. We also prove analogous results for certain symmetric random walks on Q_g.

The first talk will provide an introduction to quadratic differentials and strata, and the second will focus on my results.

We generalize some well-known limit theorems of classical probability theory in the context of nilpotent Lie groups. In particular, using a harmonic analysis approach, we prove the local limit theorem for products of centered i.i.d. random variables on the Heisenberg group $H$. We also get a precise estimate of the behavior of the walk on $H$ by comparing it to the associated gaussian walk. Using a different approach, weaker estimates for a smaller class of measures are also obtained on arbitrary nilpotent Lie groups. As an application, we show that symmetric unipotent random walks on homogeneous spaces &G/\Gamma$ are equidistributed, thus yielding a kind of probabilistic version of Ratner's equidistribution theorem.

Let $G$ be a connected Lie group of polynomial growth. We show that $G$ has strict polynomial growth and obtain a formula for the asymptotics of the volume of large balls. This is done via the study of the asymptotic shape of metric balls. We show that large balls, after a suitable renormalization, converge to a limiting compact set, which can be interpreted geometrically as the unit ball for some Carnot-Caratheodory metric on the associated graded nilshadow. The results hold for a large class of pseudometrics including left invariant Riemannian metrics or ``word metrics'' associated to a compact generated set. A similar description can be made in the greater generality of locally compact $G$. As an application, we derive new pointwise ergodic theorems on nilpotent Lie groups and Lie groups of polynomial growth.

The bulk of these talks is devoted to the development of basic ideas of the boundary control method in inverse problems. We consider the example of the inverse problem of the reconstruction of a Riemannian manifold with a boundary, i.e. its differentiable and metric structures, from the so-called Gel'fand boundary spectral data. These are the eigenvalues and restrictions to the boundary (or part of the boundary) of the eigenfunctions of the Laplace operator with Neumann boundary conditions.

We prove the uniqueness (up to an isometry) and describe a reconstruction algorithm. The method is based on elements of the PDE-control, spectral theory, properties of the hyperbolic PDE's, as well as a special representation of the Riemannian manifold. If time permits, the rest of talks will be devoted to more advanced aspects of the boundary control method including invers problems for the Maxwell and Dirac systems, equivalence of inverse problems of different types, stability in inverse problems, etc.

The bulk of these talks is devoted to the development of basic ideas of the boundary control method in inverse problems. We consider the example of the inverse problem of the reconstruction of a Riemannian manifold with a boundary, i.e. its differentiable and metric structures, from the so-called Gel'fand boundary spectral data. These are the eigenvalues and restrictions to the boundary (or part of the boundary) of the eigenfunctions of the Laplace operator with Neumann boundary conditions.

We prove the uniqueness (up to an isometry) and describe a reconstruction algorithm. The method is based on elements of the PDE-control, spectral theory, properties of the hyperbolic PDE's, as well as a special representation of the Riemannian manifold. If time permits, the rest of talks will be devoted to more advanced aspects of the boundary control method including invers problems for the Maxwell and Dirac systems, equivalence of inverse problems of different types, stability in inverse problems, etc.

The notion of an amenable group was introduced (under the name measurable) by von Neumann in 1929 with a purpose of understandings the roots of the phenomenon known as Banach-Tarskii Paradox. The notion was studied and used in different branches of mathematics by many prominent mathematicians (Tarskii, Bogolyubov). Staring from the 1950's it began to play an important role in the theory of dynamical systems and different versions of this notion are the subject of current research in Ergodic Theory. There are hundreds of equivalent definitions of the class of amenable groups , but we are still very far from having a complete picture which describes which groups are amenable and which are not.

In my talk I will focus on a series of results related to two Problems of M.Day from 1957, one of which is known as von Neumann Conjecture. I will explain how these problems were solved and also present the history of the solutions to some other problems in the same spirit. Some of the results are very recent.

The classes of groups called Branch Groups and Finite Automata Groups will play an essential role in the exposition. These classes are important also from dynamical point of view because interesting dynamical systems can be associated to them (the simplest one is the well known Adding Machine).

ssociated to this graph. The spectrum of a finitely generated group is the spectrum of the Cayley graph of the group. More generally, one can consider the spectrum of a Schreier graph associated to a group and a subgroup.

The spectral theory of graphs and groups is extremely interesting subject related to many other fascinating topics (Ramanujan graphs and expenders, Ihara zeta function, Poisson and Martin boundary, amenability and random walks, reduced C*-algebras and idempotents).

Many questions in operator theory, operator K-theory, theory of representations and in abstract harmonic analysis can be reduced to a questions about spectral properties of some associated graph or a group.

For instance, the famous criterion of H.Kesten states that a finitely generated group is amenable if and only if its spectral radius is 1.

In my talk I will give an overview of results about spectra of Cayley Graphs and spectra of Shreier graphs associated to Fractal groups. These groups will be generated by finite automata and will be of branch type.

We will describe a method of computation of a spectra based on use of self-similarity properties of a group and of related objects and on the idea of introducing extra parameters in the spectral problem. The last idea leeds unexpectedly to the invariance of the multidimensional spectrum with respect to some rational mapping $f$ of the same dimension.

Thus the spectrum becomes an $f$- invariant set. The rational mappings which arise in this way are very interesting and some of them remind of the Henon map. They posses a property that we call integrability and which we use for computation of von Neimann-Kesten-Serre spectral measure and of Ihara zeta function which can be defined for infinite graphs as well. Some interesting examples of computation of Ihara zeta function of infinite graphs and groups will be considered at the end of the talk.

In a jointwork with J. Palis, we consider smooth surface diffeomorphisms exhibiting a horseshoe with an outside homoclinic tangency. When the dimension of the horseshoe is smaller than 1, the maximal invariant set in a neighbourhood of the union of the horseshoe and the tangency orbit is still hyperbolic for most diffeomorphisms close to the bifurcation, according to previous work of PALIS and TAKENS. We assume that the dimension of the horseshoe is larger than one, but not much larger. We then show that for most diffeomorphisms after the bifurcation, the maximal invariant set is a "non uniformly hyperbolic horseshoe" : this object is the analogue, for saddle-like dynamics, of Henon-like attractors.

In a jointwork with J. Palis, we consider smooth surface diffeomorphisms exhibiting a horseshoe with an outside homoclinic tangency. When the dimension of the horseshoe is smaller than 1, the maximal invariant set in a neighbourhood of the union of the horseshoe and the tangency orbit is still hyperbolic for most diffeomorphisms close to the bifurcation, according to previous work of PALIS and TAKENS. We assume that the dimension of the horseshoe is larger than one, but not much larger. We then show that for most diffeomorphisms after the bifurcation, the maximal invariant set is a "non uniformly hyperbolic horseshoe" : this object is the analogue, for saddle-like dynamics, of Henon-like attractors.

We introduce bounded continuous cohomology of locally compact groups and give a functorial caracterization of it. We'll show how amenable actions in the sense of Zimmer play a fundamental role, in that they lead to resolutions allowing the compution of these cohomology groups in low degrees. We will then apply these techniques to get rigidity results on actions of lattices by diffeomorphisms on the circle, and deformation rigidity results concerning lattices in complex hyperbolic spaces.

We introduce bounded continuous cohomology of locally compact groups and give a functorial caracterization of it. We'll show how amenable actions in the sense of Zimmer play a fundamental role, in that they lead to resolutions allowing the compution of these cohomology groups in low degrees. We will then apply these techniques to get rigidity results on actions of lattices by diffeomorphisms on the circle, and deformation rigidity results concerning lattices in complex hyperbolic spaces.

We introduce bounded continuous cohomology of locally compact groups and give a functorial caracterization of it. We'll show how amenable actions in the sense of Zimmer play a fundamental role, in that they lead to resolutions allowing the compution of these cohomology groups in low degrees. We will then apply these techniques to get rigidity results on actions of lattices by diffeomorphisms on the circle, and deformation rigidity results concerning lattices in complex hyperbolic spaces.

Tube-log Riemann surfaces were discovered 10 years ago. These are Riemann surfaces build by cylinders and planes pasted in a similar way to the Riemann surface of the logarithm. They are the main tool in a series of constructions in holomorphic dynamics, providing examples or counter-examples answering a series of old questions. Most answers went against popular beliefs.

Tube-log Riemann surfaces were discovered 10 years ago. These are Riemann surfaces build by cylinders and planes pasted in a similar way to the Riemann surface of the logarithm. They are the main tool in a series of constructions in holomorphic dynamics, providing examples or counter-examples answering a series of old questions. Most answers went against popular beliefs.

Tube-log Riemann surfaces were discovered 10 years ago. These are Riemann surfaces build by cylinders and planes pasted in a similar way to the Riemann surface of the logarithm. They are the main tool in a series of constructions in holomorphic dynamics, providing examples or counter-examples answering a series of old questions. Most answers went against popular beliefs.

I will discuss new proofs of the two related classical facts: the geodesic flow on the ellipsoid and the billiard ball map inside it are completely integrable. It is well known that the geodesic flow and the billiard ball map preserve natural symplectic structures associated with the Euclidean geometry of the ambient space. The proofs are based on the observation that they also preserve symplectic structures associated with the projective model of the hyperbolic geometry inside the ellipsoid.

The Euler equations describe the motion of an ideal incompressible fluid. Every real fluid, which is not the liquid helium, is viscous and compressible. But if the viscosity and compressibility effects are small, we may anticipate that the fluid behaves as if it were inviscid and incompressible. But this belief is not confirmed by observation of the real motions of nearly ideal fluids, which are called turbulent. The most striking fact is that the rate of the energy dissipation in such flows does not tend to zero, as the viscosity tends to zero!

The Euler equations may be recast as some integral identities, expressing the local mass and momentum balance. Every solution of the Euler equations satisfies these identities ( which may be called the weak Euler equations); but there may a priori exist very nonregular (merely square integrable) velocity fields, which satisfy the weak Euler equations. Such velocity fields are called weak solutions of the Euler equations.

Our main conjecture is that these weak solutions have something to do with the turbulent flows; namely, that the flow field for a turbulent flow of a fluid with very small viscosity and compressibility is asymptotically close to some weak solution, as viscosity and compressibility tend to zero. The aim of my lectures is to present some rigorous arguments in favor of this hypothesis.

There is a few sparce results concerning the weak solutions. The first nontrivial example was presented in 1993 by V. Scheffer. It was a vector field u(x, t) in L^2(R^2 x R), which vanishes outside the ball |x|^2+|t|^2<1, and thus violently breaks uniqueness, energy conservation, and even energy monotonicity.

I am going to explain the simplified example of such situation. This is a weak solution (in fact, an unbounded and everywhere discontinuous vector field) on a 2-dim torus u(x, t), which vanishes for |t|>1. These examples show that the formal definition of a weak solution is not complete, and we need some further restrictions. But such solutions are interesting by their own, for they demonstrate other interesting fluiddynamical phenomenon, namely the "inverse cascade."

Next, I shall explain the construction of a more realistic 3-dim weak solution, whose kinetic energy monotonically decreases in time. This solution is also everywhere discontinuous and unbounded, while has some realistic features. The construction starts from a simple mechanical system having the property that the kinetic energy decreases, while there is no explicit friction; it requires Generalized Flows, introduced recently by Y. Brenier, and some harmonic analysis.

At last, I am going to discuss the ways to the construction and theory of true, physically reasonable weak solutions of the Euler equations.

The Euler equations describe the motion of an ideal incompressible fluid. Every real fluid, which is not the liquid helium, is viscous and compressible. But if the viscosity and compressibility effects are small, we may anticipate that the fluid behaves as if it were inviscid and incompressible. But this belief is not confirmed by observation of the real motions of nearly ideal fluids, which are called turbulent. The most striking fact is that the rate of the energy dissipation in such flows does not tend to zero, as the viscosity tends to zero!

The Euler equations may be recast as some integral identities, expressing the local mass and momentum balance. Every solution of the Euler equations satisfies these identities ( which may be called the weak Euler equations); but there may a priori exist very nonregular (merely square integrable) velocity fields, which satisfy the weak Euler equations. Such velocity fields are called weak solutions of the Euler equations.

Our main conjecture is that these weak solutions have something to do with the turbulent flows; namely, that the flow field for a turbulent flow of a fluid with very small viscosity and compressibility is asymptotically close to some weak solution, as viscosity and compressibility tend to zero. The aim of my lectures is to present some rigorous arguments in favor of this hypothesis.

There is a few sparce results concerning the weak solutions. The first nontrivial example was presented in 1993 by V. Scheffer. It was a vector field u(x, t) in L^2(R^2 x R), which vanishes outside the ball |x|^2+|t|^2<1, and thus violently breaks uniqueness, energy conservation, and even energy monotonicity.

I am going to explain the simplified example of such situation. This is a weak solution (in fact, an unbounded and everywhere discontinuous vector field) on a 2-dim torus u(x, t), which vanishes for |t|>1. These examples show that the formal definition of a weak solution is not complete, and we need some further restrictions. But such solutions are interesting by their own, for they demonstrate other interesting fluiddynamical phenomenon, namely the "inverse cascade."

Next, I shall explain the construction of a more realistic 3-dim weak solution, whose kinetic energy monotonically decreases in time. This solution is also everywhere discontinuous and unbounded, while has some realistic features. The construction starts from a simple mechanical system having the property that the kinetic energy decreases, while there is no explicit friction; it requires Generalized Flows, introduced recently by Y. Brenier, and some harmonic analysis.

At last, I am going to discuss the ways to the construction and theory of true, physically reasonable weak solutions of the Euler equations.

The Euler equations describe the motion of an ideal incompressible fluid. Every real fluid, which is not the liquid helium, is viscous and compressible. But if the viscosity and compressibility effects are small, we may anticipate that the fluid behaves as if it were inviscid and incompressible. But this belief is not confirmed by observation of the real motions of nearly ideal fluids, which are called turbulent. The most striking fact is that the rate of the energy dissipation in such flows does not tend to zero, as the viscosity tends to zero!

The Euler equations may be recast as some integral identities, expressing the local mass and momentum balance. Every solution of the Euler equations satisfies these identities ( which may be called the weak Euler equations); but there may a priori exist very nonregular (merely square integrable) velocity fields, which satisfy the weak Euler equations. Such velocity fields are called weak solutions of the Euler equations.

Our main conjecture is that these weak solutions have something to do with the turbulent flows; namely, that the flow field for a turbulent flow of a fluid with very small viscosity and compressibility is asymptotically close to some weak solution, as viscosity and compressibility tend to zero. The aim of my lectures is to present some rigorous arguments in favor of this hypothesis.

There is a few sparce results concerning the weak solutions. The first nontrivial example was presented in 1993 by V. Scheffer. It was a vector field u(x, t) in L^2(R^2 x R), which vanishes outside the ball |x|^2+|t|^2<1, and thus violently breaks uniqueness, energy conservation, and even energy monotonicity.

I am going to explain the simplified example of such situation. This is a weak solution (in fact, an unbounded and everywhere discontinuous vector field) on a 2-dim torus u(x, t), which vanishes for |t|>1. These examples show that the formal definition of a weak solution is not complete, and we need some further restrictions. But such solutions are interesting by their own, for they demonstrate other interesting fluiddynamical phenomenon, namely the "inverse cascade."

Next, I shall explain the construction of a more realistic 3-dim weak solution, whose kinetic energy monotonically decreases in time. This solution is also everywhere discontinuous and unbounded, while has some realistic features. The construction starts from a simple mechanical system having the property that the kinetic energy decreases, while there is no explicit friction; it requires Generalized Flows, introduced recently by Y. Brenier, and some harmonic analysis.

At last, I am going to discuss the ways to the construction and theory of true, physically reasonable weak solutions of the Euler equations.