# Seminars (GAN)

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Event When Speaker Title Presentation Material
GAN 9th June 2014
14:30 to 15:30
M Aka Points on spheres and their orthogonal lattices
It is a classical question to understand the distribution (when projected to the unit sphere) of the solutions of x^2+y^2+z^2=D as D grows. To each such solution v we further attach the lattice obtained by intersecting the hyperplane orthogonal to v with the set of integral vectors. This way, we obtain, for any D that can be written as a sum of three squares, a finite set of pairs consisting of a point on the unit sphere and a lattice. In the talk I will discuss a joint work with Manfred Einsiedler and Uri Shapira which considers the joint distribution of these pairs in the appropriate spaces. I will outline a general approach to such problems and discuss dynamical input needed to establish that these pairs distribute uniformly.
GAN 10th June 2014
14:30 to 15:30
On continued fraction expansion of potential counterexamples to mixed Littlewood conjecture.
Mixed Littlewood conjecture proposed by de Mathan and Teulie in 2004 states that for every real number $x$ one has $$\liminf_{q\to\infty} q\cdot |q|_D\cdot ||qx|| = 0.$$ where $|*|_D$ is a so called pseudo norm which generalises the standard $p$-adic norm. In the talk we'll consider the set $\mad$ of potential counterexamples to this conjecture. Thanks to the results of Einsiedler and Kleinbock we already know that the Haudorff dimension of $\mad$ is zero, so this set is very tiny. During the talk we'll see that the continued fraction expansion of every element in $\mad$ should satisfy some quite restrictive conditions. As one of them we'll see that for these expansions, considered as infinite words, the complexity function can neither grow too fast nor too slow.
GAN 11th June 2014
11:30 to 12:30
Simultaneous dense and nondense orbits for commuting maps
We show that, for two commuting automorphisms of the d-torus, many points have drastically different orbit structures for the two maps. Specifically, using measure rigidity, the Ledrappier-Young formula, and the Marstrand slicing theorem, we show that the set of points that have dense orbit under one map and nondense orbit under the other has full Hausdorff dimension. This mixed case, dense orbit under one map and nondense orbit under the other, is much more delicate than the other two possible cases. Our technique can also be applied to other settings. For example, we show the analogous result for two elements of the Cartan action on compact higher rank homogeneous spaces. This is joint work with V. Bergelson and M. Einsiedler.
GAN 12th June 2014
11:30 to 12:30
Distribution of values of linear maps on quadratic surfaces.
We discuss the distribution of values of linear maps on quadratic surfaces. The problem can be set up in the framework of unipotent dynamics and then Ratner's Orbit Closure Theorem can be used to establish conditions sufficient to ensure that the set of values is dense. We also indicate how a quantitative version of this result follows from equidistribution properties of unipotent flows on homogeneous spaces. A crucial ingredient in the latter is a non-divergence result for certain spherical averages. In order to establish this result we show how one can use ideas of Y. Benoist and J.F. Quint that were developed in the context of random walks on homogeneous spaces.
GAN 12th June 2014
14:30 to 15:30
T Ward Rigid and flexible properties of group automorphisms
This is an overview of dynamical properties of group automorphisms, with a particular emphasis on dynamical attributes that can vary smoothly as opposed to those that exhibit rigidity.
GAN 13th June 2014
11:30 to 12:30
On continued fraction expansion of potential counterexamples II
GAN 13th June 2014
14:30 to 15:30
Some applications of Mahler's method
We will discuss Mahler's method in transcendence theory and recent advances in it, as well as some applications of emerging results. If time allows, we will discuss a problem in algebraic dynamics interwinded with the involved technics.
GAN 16th June 2014
16:00 to 17:00
Y Benoist Spectral gap in simple Lie groups : Clay Mathematics Institute Senior Scholar Lecture.
Consider two matrices a , b in the orthogonal group SO(d) that span a dense subgroup. We will first recall that, in dimension at least 3, for n large, the set of words of length n in a and b become equidistributed in SO(d). We will then see that, when the matrices a, b have algebraic coefficients, the precision of this equidistribution is exponentially small in n. This joint work with N. de Saxce extends previous results of Bourgain and Gamburd for the unitary groups SU(d).
GAN 17th June 2014
11:30 to 12:30
V Gadre Partial sums of excursions along random geodesics.
In the theory of continued fractions, Diamond and Vaaler showed the following strong law: for almost every expansion, the partial sum of first n coefficients minus the largest coefficient divided by n \log n tends to a limit. We will explain how this generalizes to non-uniform lattices in SL(2, R) with cusp excursions in the quotient hyperbolic surface generalizing continued fraction coefficients. The general theorem relies on the exponential mixing of geodesic flow, in particular on the fast decay of correlations due to Ratner. Analogously, similar theorems are true for the moduli space of Riemann surfaces.
GAN 17th June 2014
14:30 to 15:30
Periodicity, complexity and shifts.
A beautiful example of a global property being determined by a local one is the Morse-Hedlund Theorem; it gives the relation between the global property of periodicity for an infinite word in a finite alphabet and local information on the complexity of the word. I will discuss higher dimensional versions of this problem, and the use of local conditions on complexity to determine global properties of the configuration, and applications of this to the automorphism group of shifts. This is joint work with Van Cyr.
GAN 18th June 2014
11:30 to 12:30
Lower bounds for entropy of random walks.
GAN 18th June 2014
14:30 to 15:30
Effective Ratner Theorem for ASL(2, R) and the gaps of the sequence \sqrt n modulo 1
Let G=SL(2,\R)\ltimes R^2 and Gamma=SL(2,Z)\ltimes Z^2. Building on recent work of Strombergsson we prove a rate of equidistribution for the orbits of a certain 1-dimensional unipotent flow of Gamma\G, which projects to a closed horocycle in the unit tangent bundle to the modular surface. We use this to answer a question of Elkies and McMullen by making effective the convergence of the gap distribution of sqrt n mod 1.
GAN 19th June 2014
14:30 to 15:30
B Host Fourier analysis of multiplicative functions and a combinatorial application.
GAN 20th June 2014
11:30 to 12:30
Y Benoist Spectral gap in simple Lie groups II
GAN 20th June 2014
14:30 to 15:30
Asymptotical behavior of piecewise contractions of the interval.
A map $f:[0,1)\to [0,1)$ is a piecewise contraction of $n$ intervals, if there exists a partition of $[0,1)$ into intervals $I_1, \ldots, I_n$ and every restriction $f\vert_{I_i}$ is an injective Lipschitz contraction. Among other results we will show that a typical piecewise contraction of $n$ intervals has at least one and at most $n$ periodic orbits. Moreover, for every point $x$, the $\omega$-limit set of $x$ equals a periodic orbit.

The talk is based in a joint work with B. Pires and R. Rosales.

GAN 23rd June 2014
11:00 to 12:00
F Ramirez Khintchine types of translated coordinate hyperplanes
This talk is about the problem of simultaneously approximating a tuple of real numbers by rationals, when one of the real numbers has been prescribed. This corresponds to describing the set of rationally approximable points on translates of coordinate hyperplanes in Euclidean space. It is expected that under a divergence condition on the desired rate of approximation, we should be able to assert that almost every point on the hyperplane is rationally approximable at that rate, like in Khintchine's Theorem. We will discuss some positive results in this direction. These can be seen as living in the degenerate counterpart to work of Beresnevich and Beresnevich--Dickinson--Velani, where similar results were achieved for non-degenerate submanifolds of Euclidean space.
GAN 23rd June 2014
13:30 to 14:30
Hall rays for Lagrange spectra of Veech surfaces
The Lagrange spectrum is a subset of the real line which can be described in terms of badly approximable numbers. In joint work with Hubert and Marchese, we defined a generalization of Lagrange spectra for translation surfaces and badly approximable interval exchange transformations. Both in the classical case and in the case of Veech translation surfaces, Lagrange spectra can be also described in terms of asymptotic depths penetration of hyperbolic geodesics in the cusps of the associated hyperbolic surface (the Teichmueller curve). In joint work with Mauro Artigiani and Luca Marchese, we show that high Lagrange spectra of Veech surfaces can be calculated using a symbolic coding in the sense of Bowen and Series. We then prove that the Lagrange spectrum of any Veech translation surface contains a semi-line, i.e. a Hall's ray.
GAN 24th June 2014
14:30 to 15:30
Diophantine approximation and the geometry of limit sets in Gromov hyperbolic metric spaces.
Let $(X,d)$ be a Gromov hyperbolic metric space, and let $\partial X$ be the Gromov boundary of $X$. Fix a group $G\leq\operatorname{Isom}(X)$ and a point $\xi\in\partial X$. We consider the Diophantine approximation of a point $\eta\in\partial X$ by points in the set $G(\xi)$. Our results generalize the work of many authors, in particular Patterson ('76) who proved most of our results in the case that $G$ is a geometrically finite Fuchsian group of the first kind and $\xi$ is a parabolic fixed point of $G$.
GAN 24th June 2014
15:45 to 16:45
Y Benoist Stationary Measures and Invariant Subsets of Homogeneous Spaces: Part I
GAN 26th June 2014
13:30 to 14:30
Effective Ratner equidistribution for SL$(2,\mathbb R)\ltimes(\mathbb R^2)^{\oplus k}$ and applications to quadratic forms
Let $G=\text{SL}(2,\mathbb R)\ltimes(\mathbb R^2)^{\oplus k}$ and let $\Gamma$ be a congruence subgroup of SL$(2,\mathbb Z)\ltimes(\mathbb Z^2)^{\oplus k}$. I will present a result giving effective equidistribution of 1-dimensional unipotent orbits in the homogeneous space $\Gamma\backslash G$. The proof involves spectral analysis and use of Weil's bound on Kloosterman sums. I will also discuss applications to effective results for variants of the Oppenheim conjecture on the density of $Q(\mathbb Z^n)$ on the real line, where $Q$ is an irrational indefinite quadratic form. (Based on joint work with Pankaj Vishe.)
GAN 27th June 2014
11:30 to 12:30
Improvement of Dirichlet's Theorem and homogeneous dynamics
We consider improvement of Dirichlet's Theorem for vectors in two dimensional Euclidean space. It is proved by Shah in 2009 that for alomost every point of an analytic nonplanar curve Dirichlet's theorem is nonimprovable. In this talk we show that similar results hold for certain straight lines and fractals.
GAN 27th June 2014
14:30 to 15:30
Let ||·|| denote the distance to the nearest integer and, for a prime number p, let |·|_p denote the p-adic absolute value. In 2004, de Mathan and Teulié asked whether $inf_{q?1} q·||qx||·|q|_p = 0$ holds for every badly approximable real number x and every prime number p. When x is quadratic, the equality holds and moreover, de Mathan and Teullié proved that $lim inf_{q?1} q·log(q)·||qx||·|q|_p$ is finite and asked whether this limit is positive. We give a new proof of de Mathan and Teullié's result by exploring the continued fraction expansion of the multiplication of x by p with the help of a recent work of Aka and Shapira. We will also discuss the positivity of the limit.
GAN 27th June 2014
15:45 to 16:45
Y Benoist Stationary Measures and Invariant Subsets of Homogeneous Spaces: Part II
GANW01 30th June 2014
09:00 to 09:50
Gap Distributions, Homogeneous Dynamics, and Random Lattices
We discuss the connections between cusp excursions of the horocycle flow, random lattices, and gap distributions between Farey fractions, and show how some aspects of this picture can be generalized to the setting of translation surfaces.
GANW01 30th June 2014
10:00 to 10:50
Divisibility properties in higher rank lattices
In a joint work with Manfred Einsiedler we discuss a relationship between the dynamical properties of a maximal diagonalizable group $A$ on certain arithmetic quotients and arithmetic properties of the lattice. In particular, we consider the semigroup of all integer quaternions under multiplication. For this semigroup we use measure rigidity theorems to prove that the set of elements that are not divisible by a given reduced quaternion is very small: We show that any quaternion that has a sufficiently divisible norm is also divisible by the given quaternion. Restricting to the quaternions that have norm equal to products of powers of primes from a given list (containing at least two) we show that the set of exceptions has subexponential growth.
GANW01 30th June 2014
11:30 to 12:20
Quasirandom Groups, Minimally Almost Periodic Groups and Ergodic Ramsey Theory
According to the definition introduced by T. Gowers in 2008, a finite group G is called D-quasirandom for some parameter D, if all non-trivial unitary representations of G have dimension greater or equal to D. For example, the group SL(2, F_p) is (p-1)/2 quasirandom for any prime p. Informally, a finite group is quasirandom if it is D-quasirandom for a large value of D. Answering a question posed by L. Babai and V. Sos, Gowers have shown that, in contrast with the more familiar "abelian" situation, qusirandom groups can not have large product-free subsets. The goal of this lecture is to discuss the connection between the combinatorial phenomena observed in quasirandom groups and the ergodic properties of the minimally almost periodic groups (these were introduced in 1934 by J. von Neumann as groups which do not admit non-constant almost periodic functions). This connection will allow us to give a simple explanation the dynamical underpinnings of some of the Gowers' results as well as of the more recent results obtained in joint work with T. Tao and in the work of T. Austin.
GANW01 30th June 2014
13:30 to 14:20
A Salehi Golsefidy Expansion properties of linear groups
Starting with finitely many matrices S in GL(n,Q), we will discuss when the Cayley graphs of congruence quotients of the group generated by S modulo a sequence of integers can form a family of expanders. Then we will focus on the case of powers of primes and show that such a property is dictated by the Zariski-topology.
GANW01 30th June 2014
15:00 to 15:50
Counting and equidistribution of common perpendiculars: arithmetic applications
I will survey several arithmetic applications, obtained in joint works with Parkkonen, of our counting (and simultaneous equidistribution of initial and terminal tangent vectors) of the common perpendiculars between two closed locally convex subsets of a negatively curved manifold: generalisation of Mertens' formula for imaginary quadratic number fields, counting quadratic irrationals with bounded crossratios, equidistribution of rational points in the Heisenberg group, counting of arithmetic chains in the Heisenberg group with Cygan diameter bounded from below.
GANW01 30th June 2014
16:00 to 16:50
Every flat surface is Birkhoff and Osceledets generic in almost every direction: Clay Mathematics Institute Senior Scholar Lecture
This is a simple argument which also works in the homogeneous setting.
GANW01 1st July 2014
09:00 to 09:50
J Chaika The Hausdorff dimension of not uniquely ergodic 4-IETs has codimension 1/2.
The main results of this talk are: a) The Hausdorff dimension of not-uniquely 4-IETs is 2 1/2 as a subset of the 3 dimensional simplex b) The Hausdorff dimension of flat surfaces in H(2) whose vertical flow is not uniquely ergodic is 7 1/2 as a subset of an 8 dimensional space c) For almost every flat surface in H(2) the set of directions where the flow is not uniquely ergodic has Hausdorff dimension 1/2. These results all say that the Hausdorff codimension of these exceptional sets is 1/2. Masur-Smillie showed that the Hausdorff codimension was less than 1. It follows from work of Masur that the Hausdorff codimension is at least 1/2. This is joint work with J. Athreya.
GANW01 1st July 2014
10:00 to 10:50
M Einsiedler Adelic equidistribution and property (tau)
We give a uniform equidistribution statement for maximal subgroups on adelic quotients. The result is effective and applies to cases where the results of Mozes and Shah may not be applicable. The argument can also be used to give an alternative proof of property (tau) (but with weaker exponents than established by Clozel). We will also discuss an extension/application concerning integer points on spheres and their orthogonal lattice.
GANW01 1st July 2014
11:30 to 12:20
Inhomogeneous multiplicative Littlewood conjecture and logarithmic savings
We use the dynamics on SL(3,R)/SL(3,Z) to get logarithmic savings in the inhomogeneous multiplicative Littlewood setting. This is a joint work with Alex Gorodnik.
GANW01 1st July 2014
13:30 to 14:20
L Flaminio On the Greenfield-Wallach and Katok conjectures
In the early 70's, Greenfield and Wallach studied fields globally hypo-elliptic vectors fields on compact manifolds and made the following conjecture : Let $G$ be a Lie group and let $H$ be a closed subgroup Such That $G/H$ is compact. Let $X ^ *$ be the vector field on $G / H$ determined by some element $X$ in the Lie algebra of $G$. Given by If $X ^ *$ is globally hypo-elliptic, then $G / H$ is a torus.'' In a work in collaboration with F.~Rodriguez-Hertz and G. Forni, we gave a positive solution to this problem. In this talk I will recall thistory of the problem, wxplain its relation with Katok's conjecture on cohomologically free vector fields and give some idea of the proof.
GANW01 1st July 2014
15:00 to 15:50
Localized representation theory
The talk will concern an aspect of the representation theory of Lie groups motivated by applications to bounds for period integrals. I will introduce a notion of a vector in an irreducible unitary representation being "localized". I will then plead the case that this definition is interesting. For example, recall that Schur's lemma says that the equivariant operators on an irreducible unitary representation are constant multiples of the identity. I will describe an asymptotic classification of "approximately equivariant" operators when restricted to localized vectors. I will indicate some applications of this simple-sounding result, particularly to establishing the invariance of certain limiting measures that arise in semiclassical analysis and number theory.
GANW01 1st July 2014
16:00 to 16:50
G Forni Effective equidistribution for twisted averages of the horocycle flow
We describe an approach, based on scaling, to bounds on twisted ergodic averages of the classical horocycle flow. As far as we know the best estimates available so far are due to A. Venkatesh and are based on effective equidistribution and rate of mixing for the horocycle flow itself. Our bounds are significantly better, in particular they are independent of the spectral gap of the representation. The main motivations come from the general question of establishing effective rate of equidistribution for unipotent flows and from the particular case of effective equidistribution for horocycle maps (or for the horocycle flow along polynomial sequences). We present joint work with L. Flaminio (Lille) and J. Tanis (Rice).
GANW01 2nd July 2014
09:00 to 09:50
Attaching shortest vectors to lattice points and applications
We highlight a simple construction, appeared in the work of D. Badziahin, A. Pollington and S. Velani where they proved Schmidt's conjecture, which attaches to a lattice point an integral vector that is shortest in a certain sense. Such a construction turns out to be useful in studying badly approximable vectors and bounded orbits of unimodular lattices. It can be used to prove: (1) The set $\mathrm{Bad}(i,j)$ of two-dimensional badly approximable vectors is winning for Schmidt's game; (2) $\mathrm{Bad}(i,j)$ is also winning on non-degenerate curves and certain straight lines; (3) Three-dimensional unimodular lattices with bounded orbits under a diagonalizable one-parameter subgroup form a winning set (at least in a local sense).
GANW01 2nd July 2014
10:00 to 10:50
Dimension of Self-similar Measures and Additive Combinatorics
I will discuss recent progress on the problem of computing the dimension of a self-similar set or measure in $\mathbb{R}$ in the presence of non-trivial overlaps. It is thought that unless the overlaps are "exact" (an essentially algebraic condition), the dimension achieves the trivial upper bound. I will present a weakened version of this that confirms the conjecture in some special cases. A key ingredient is a theorem in additive combinatorics that describes in a statistical sense the structure of measures whose convolution has roughly the same entropy at small scales as the original measure. As time permits, I will also discuss the situation in $\mathbb{R}^d$.
GANW01 2nd July 2014
11:30 to 12:20
A Kontorovich Applications of Thin Orbits
We will discuss various natural problems in arithmetic and dynamics, which at their core are questions about thin groups and orbits.
GANW01 2nd July 2014
13:30 to 14:20
Lattice points counting and Diophantine approximable on manifolds
It has been known for at least about a century that there exist algebraic badly approximable points confined by algebraic relations. On the other hand, until very recently nothing was known regarding badly approximable points subject to generic smooth (non-algebraic) relations. In this talk I will describe a technique in metric Diophantine approximation that has recently enabled to shed some light on this problem. Time permitting I will mention some other applications of the technique.
GANW01 2nd July 2014
15:00 to 15:50
U Shapira Escape of mass in positive characteristics
I will describe some peculiar features of the geodesic flow in positive characteristics which occur in sharp contrast to analogue discussions in characteristics zero. In particular, situations where a natural limiting procedure leads to escape of mass and accumulation on singular measures. This is joint work with Frederic Paulin and Alex Kemarsky.
GANW01 2nd July 2014
16:00 to 16:50
Oppenheim conjecture and related problems
Oppenheim conjecture which I proved in mid-eighties says that the set of values at integral points of an irrational indefinite quadratic form in 3 or more variables is dense in R (this is a slightly stronger than the original formulation). I will discuss various approaches to this conjecture and its quantitative generalizations.
GANW01 3rd July 2014
09:00 to 09:50
Perturbations of Weyl sums, and uniform distribution mod 1
Significant advances in our understanding of mean values of exponential sums have resulted from recent progress in Vinogradov's mean value theorem, in particular as a consequence of efficient congruencing methods. These advances lead to improved pointwise estimates for Weyl sums. In this talk we will explore the consequences of such ideas for perturbed exponential sums in which lower degree coefficients are permitted to vary over a set of large measure. The conclusions we obtain are related to recent work of Livio Flaminio and Giovanni Forni.
GANW01 3rd July 2014
10:00 to 10:50
Gauss maps for simultaneous approximation
Levy's constant measures the exponential growth rate for the sequence of denominators of the convergents of a real number. Khintchine proved existence for almost every real number and Levy computed the constant to be $\pi^2/12\ln2$. This result is a standard exercise in modern textbooks on ergodic theory. In this talk, we generalize it to higher dimensions with Levy's constant defined using the sequence of best approximation denominators. The main ingredient of the proof is constructing the analog of the Gauss map for continued fractions. This work is joint with Nicolas Chevallier.
GANW01 3rd July 2014
11:30 to 12:20
Metric Diophantine approximation: the well approximable theory on manifolds
I will give an overview of recent development regarding the metric theory of well approximable sets restricted to manifold. In particular, I will focus on the strengthening of the fundamental theorems of Khintchine and Gallagher, and demonstrate a basic principle that enables one to establish inhomogeneous extremality from (homogeneous) extermality. The end result of the latter is the inhomogeneous strengthening of the Kleinbock-Margulis theorem that validates the Baker-Sprindzuk Conjecture.
GANW01 3rd July 2014
13:30 to 14:20
Hausdorff dimension estimates for bounded orbits on homogeneous spaces of Lie groups
This work is motivated by studying badly approximable vectors, that is, ${\bf x}\in {\bf R}^n$ such that $\|q{\bf x} - {\bf p}\| \ge cq^{-1/n}$ for all ${\bf p}\in {\bf Z}^n$, $q\in {\bf N}$. Computing the Hausdorff dimension of the set of such ${\bf x}$ for fixed $c$ is an open problem. I will present some estimates, based on the interpretation of a badly approximable vector via a trajectory on the space of lattices, and then use exponential mixing to estimate from above the dimension of points whose trajectories stay in a fixed compact set. Joint work with Ryan Broderick.
GANW01 3rd July 2014
15:00 to 15:50
Limiting distributions of translates of curves on homogeneous spaces
We show that in various cases, under natural conditions, limits of translates of pieces of curves on homogeneous spaces get equidistributed in the whole space. These type of results have applications to diophantine approximation.
GANW01 3rd July 2014
16:00 to 16:50
Homogeneous dynamics, unitary representations, and Diophantine exponents
We will describe an explicit quantitative form of the duality principle in homogeneous dynamics. This allows the reduction of a diverse set of quantitative equidistribution problems on homogeneous spaces G/H to the problem of giving explicit spectral bounds for the restriction of automorphic representations of G to the stability subgroup H. We will demonstrate this approach by deriving bounds for Diophantine approximation exponent on homogeneous varieties, a problem raised by Serge Lang already in 1965, but which have seen little progress since then. The Diophantine exponents we derive are in many cases best possible, a remarkable fact that follows from an important and useful representation-theoretic phenomenon which we will highlight. Based on Joint work with Alex Gorodnik (Bristol University) and on joint work with Anish Ghosh (TIFR Mumbai) and Alex Gorodnik.
GANW01 3rd July 2014
17:00 to 18:00
The SL(2,R) action on Moduli space: Mordell Lecture
We prove some ergodic-theoretic rigidity properties of the action of SL(2; R) on the moduli space of compact Riemann surfaces. In particular, we show that any ergodic measure invariant under the action of the upper triangular subgroup of SL(2; R) is supported on an invariant a ffine submanifold. The main theorems are inspired by the results of several authors on unipotent flows on homogeneous spaces, and in particular by Ratner’s seminal work. This is joint work with Maryam Mirzakhani and Amir Mohammadi.
GANW01 4th July 2014
09:00 to 09:50
M Pollicott Higher Teichmuller Theory and Thermodynamical Formalism
One can consider cocompact Fuchsian groups from the point of view of representations of a surface group into SL(2, R). There has been recent interest in representations into SL(d, R) for d > 2 which we might call Higher Teichmuller'', particular with the recent work of Bridgeman, Canary, Labourie and Sambarino. This talk will describe the usefulness of classical ideas in thermodynamic formalism in the modern study of such representations.
GANW01 4th July 2014
10:00 to 10:50
Diophantine property in groups
Let x, y be two real numbers and consider all numbers that can be expressed from them using addition and subtraction. Denote by B_l the set of numbers that can be obtained by using x and y at most l times. A simple Borel-Cantelliargument shows that the smallest positive element of B_l is at least cl^(-1-e) for almost all pairs x,y. In the lecture we will investigate the analogue of this property in non-commutative Lie groups.
GANW01 4th July 2014
11:30 to 12:20
Y Guivarch Spectral gap properties for random walks on homogeneous spaces: examples and consequences
Co-authors: J.-P Conze, B. Bekka, E .LePage

Let E be a homogeneous space of a Lie group G, p a finitely supported probability measure on G, such that supp(p) generates topologically G. We show that, in various situations, convolution by p has a spectral gap on some suitable functional space on E . We consider in particular Hilbert spaces and Holder spaces on E and actions by affine transformations. If G is the motion group of Euclidean space V ,we get equidistribution of the random walk on V. If G is the affine group of V,p has a stationary probability, and the projection of p on GL(V) satisfies "generic" conditions we get that the random walk satisfies Frechet's extreme law , and Sullivan's Logarithm law.

GANW01 4th July 2014
13:30 to 14:20
S G Dani Multi-dimensional metric approximation by primitive points
We consider Diophantine inequalities of the form $| \Theta {\bf q} + {\bf p} - {\bf y} |\leq \psi(| {\bf q} |)$, with $\Theta$ is a $n\times m$ matrix with real entries, ${\bf y} \in \mathbb R^n$, $m,n\in {\bf N}$, and $\psi$ is a function on ${\bf N}$ with positive real values, and seek integral solutions ${\bf v} =({\bf q}, {\bf p})^t$ for which the restriction of ${\bf v}$ to the components of a given partition $\pi$ are primitive integer points. In this setting, we shall discuss metrical results in the style of the Khintchine-Groshev Theorem. Solutions for analogous doubly metrical inequalities will also be discussed.
GANW01 4th July 2014
15:00 to 15:50
Unipotent flows on infinite volume manifolds
Unipotent flows on homogeneous spaces obtained as quotients of a Lie group by a lattice have been extensively studied over the past 40 years or so, with several fundamental applications. Starting from the late 80's there have been several results aiming at extending some of these results to certain infinite volume settings. However, our understanding of the subject, beyond the horospherical foliation in the rank one case, remains rather poor. In this talk we will discuss some, ongoing, attempts in this direction. This is a joint project with Hee Oh.
GANW01 4th July 2014
16:00 to 16:50
Y Benoist Random walk on p-adic flag varieties
According to a theorem of Furstenberg, a Zariski dense probability measure on a real semisimple Lie group admits a unique stationary probability measure on the flag variety. In this talk we will see that a Zariski dense probability measure on a p-adic semisimple Lie group admits finitely many stationary probability measures on the flag variety and we will classify these measures. This is a joint work with JF. Quint.