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Monday, 23 September 2019
Time Speaker Title Resources
09:30 to 10:15 Stefano Luzzatto Lecture I - Hyperbolicity and Physical Measures

A main goal in the ergodic theory of differentiable dynamical systems is the study of the existence of physical measures. One main strategy is to prove that a system is hyperbolic and then to use this hyperbolicity to construct physical measures. In this course I discuss methods and techniques to verify hyperbolicity in a family of geometric models for the Lorenz equations.

10:45 to 11:30 Sylvain Crovisier Homoclinic classes and equilibrium measures

The dynamics of uniformly hyperbolic diffeomorphisms is well understood (after works by Anosov, Smale, Sinaï, Ruelle, Bowen,…). In this mini-course we will be interested by the non-uniformly hyperbolic part of the dynamics of arbitrary diffeomorphisms. From classical tools developed by Pesin and Katok, this dynamics naturally decomposes into pieces, called homoclinic classes, which generalize Smale’s uniformly hyperbolic basic pieces.

We will present recent results that allow to associate a good symbolic coding for each homoclinic class (due to Sarig, Ben Ovadia, Buzzi-Crovisier-Sarig). Combined with Yomdin theory, this can be used for studying the equilibrium measures on each class. For instance
one will prove that a class can support at most one measure which maximizes the entropy and deduce some properties of that measure.

Reference: arXiv:1811.02240.

11:45 to 12:30 Alex Blumenthal Lagrangian chaos and mixing for models in fluid mechanics

Lagrangian flow describes the motion of a passive tracer particle (e.g., a mote of dust) in a fluid, giving rise to a flow of diffeomorphisms on the fluid domain which is volume preserving when the fluid is incompressible. When the fluid is subjected to forcing, it is anticipated that this flow is chaotic in terms of sensitivity with respect to initial conditions (i.e., a positive Lyapunov exponent) and exponentially fast decay of correlations (equivalently, fast mixing of passively adverted scalars, such as chemical concentration). Unfortunately, rigorously proving these chaotic properties for most types of deterministic forcing seems hopelessly out of reach.

On the other hand, the presence of stochastic driving often renders tractable the problem of rigorously verifying chaotic regimes. In a joint work with Jacob Bedrossian (UMD) and Sam Punshon-Smith (U Brown), we prove these chaotic properties for the Lagrangian flow for an incompressible fluid evolving according to the Navier-Stokes equations on the torus subjected to Sobolev-regular, white-in-time forcing. As a consequence, for the models we consider, we resolve an open question in the study of passive scalar turbulence in the Batchelor regime (i.e., fixed viscosity with molecular diffusivity going to zero).

14:30 to 14:50 Sergey Kryzhevich Invariant Measures for Non-autonomous systems and ergodic inverse shadowing

This is a joint work with Prof. Sergey Pilyugin. We look how numerical methods could be applied to find sets of invariant measures for continuous maps of compact sets. First of all, we naturally expand the concept of invariant measures for so-called methods that is 'non-autonomous' discrete systems. Then, we consider sequences of methods, converging to a given map. Calculating distances between measures in Kantorovich - Waserstein metrics, we consider distances between sets of all probability invariant measures of maps or methods in the corresponding Hausdorff metrics. We show that the set of invariant measures is always lower semicontinuous. We introduce the so-called Ergodic Inverse Shadowing (EIS) property that implies upper semicontinuity. We study properties of the set of all diffeomorphsms with EIS. Particularly, we demonstrate that the C^1 interior of this set is just the set of all Omega - stable diffeomorphisms.

14:50 to 15:10 Davide Ravotti Mixing time-changes of nilflows

Nilflows on nilmanifolds are classical examples of parabolic homogeneous flows. Their ergodic properties are well-understood: although almost every nilflow is uniquely ergodic, they are never weak mixing. The absence of mixing is the result of an algebraic obstruction, namely a factor isomorphic to a linear flow on a torus. On the contrary, we show that for generic smooth time-changes of any uniquely ergodic nilflow, mixing holds unless the time-change function is measurably trivial. This result confirms some heuristic principle on the ergodic properties of smooth parabolic flows.

This is a joint work with Artur Avila, Giovanni Forni, and Corinna Ulcigrai.

15:10 to 15:30 Adriana Cristina Sanchez Chavarria Lyapuniv exponents of probability distributions with non compact support

A recent result of Bocker–Viana asserts that the Lyapunov exponents of compactly supported probability distributions in GL(2, R) depend continuously on the distribution. We investigate the general, possibly non-compact case. We prove that the Lyapunov exponents are semi-continuous with respect to the Wasserstein topology, but not with respect to the weak* topology. Moreover, they are not continuous with respect to the Wasserstein topology

16:00 to 16:45 Khadim War TBA
Tuesday, 24 September 2019
Time Speaker Title Resources
09:30 to 10:15 Sylvain Crovisier Lecture II - Homoclinic classes and equilibrium measures

The dynamics of uniformly hyperbolic diffeomorphisms is well understood (after works by Anosov, Smale, Sinaï, Ruelle, Bowen,…). In this mini-course we will be interested by the non-uniformly hyperbolic part of the dynamics of arbitrary diffeomorphisms. From classical tools developed by Pesin and Katok, this dynamics naturally decomposes into pieces, called homoclinic classes, which generalize Smale’s uniformly hyperbolic basic pieces.

We will present recent results that allow to associate a good symbolic coding for each homoclinic class (due to Sarig, Ben Ovadia, Buzzi-Crovisier-Sarig). Combined with Yomdin theory, this can be used for studying the equilibrium measures on each class. For instance one will prove that a class can support at most one measure which maximizes the entropy and deduce some properties of that measure.

Reference: arXiv:1811.02240.

10:45 to 11:30 Stefano Luzzatto Lecture II - Hyperbolicity and Physical Measures

A main goal in the ergodic theory of differentiable dynamical systems is the study of the existence of physical measures. One main strategy is to prove that a system is hyperbolic and then to use this hyperbolicity to construct physical measures. In this course I discuss methods and techniques to verify hyperbolicity in a family of geometric models for the Lorenz equations.

11:45 to 12:30 Dalia Terhesiu Rates of mixing for the periodic Z^d billiard map (infinite horizon)

Optimal rates of mixing (of arbitrarily high order) for the periodic Z^d billiard map with finite horizon have been obtained recently by F. Pene. First order mixing for the infinite horizon case is also known. Rates of mixing (higher order mixing) in the infinite horizon case poses new challenges. In joint work with F. Pene, we obtain second order mixing.

14:30 to 14:50 Peyman Eslami Inducing schemes for piecewise expanding maps of R^n

For piecewise expanding maps of Rn Rn I will show how to construct an inducing scheme where the base map is Gibbs-Markov and the return times have exponential tails. The existence of such a structure has many consequences in regards to the statistical properties of systems with discontinuities and non-uniform expansion.

14:50 to 15:10 Gabriella Keszthelyi Dynamical properties of biparametric skew tent maps (joint work with Zoltan Buczolich)

We consider skew tent maps Tα,β(x) with Tα,β(x) = βαx for 0 ≤ x ≤ αand Tα,β(x) = β1−α(1 − x) for α < x ≤ 1. With this choice of parametersTα,β maps [0, 1] into [0, 1] and (α, β) is the vertex of Tα,β. The dynamics ofTα,β for (α, β) ∈ [0, 1]2
is interesting when (α, β) ∈ U = {(α, β) : 0.5 < β ≤1, 1 − β < α < β}. Denote by h(α, β) the topological entropy of Tα,β. It is well-known that h(α, β) is strictly monotone increasing along vertical line segments in U. It is natural to ask what happens if we move in the horizontal direction, that is for fixed β we consider h(α) = h(α, β), for (α, β) ∈ U. Turned out that h(α) is strictly monotone increasing on (1 − β, β). To deal with this question one needs to consider equi-topological entropy curves in the square. We denote by M = K(α, β) the kneading sequence of Tα,β and by Λ = Λα,β its Lyapunov exponent. For a given kneading squence M we consider isentropes (or equi-topological entropy, or equi-kneading curves), (α, ΨM(α)) such that K(α, ΨM(α)) = M. On these curves the topological
entropy h(α, ΨM(α)) is constant. We show that Ψ0 M(α) exists and the Lya-punov exponent Λα,β can be expressed by using the slope of the tangent to the isentrope. Since this latter can be computed by considering partial derivatives of an auxiliary function ΘM a series depending on the kneading sequence which converges at an exponential rate, this provides an efficient new method of finding the value of the Lyapunov exponent of these maps.

15:10 to 15:30 Luciana Silva Salgado Weak Hyperbolicity for Singular Flows

Dominated splittings perhaps is the first notion of weak hyperbolicity that have emerged after the uniform hyperbolic theory.

In the case of singular flows, the study of Lorenz attractors have inspired the definition of the so called Singular Hyperbolicity. Nowadays, there are several other notions of weak hyperbolicity for
flows beyond the classical singular hyperbolicity one.

In this talk, I am interested in weak and uniform hyperbolic structures in a broad sense and its characterizations, mainly by using the notion of infinitesimal Lyapunov functions.

References:
[1] L. S. Salgado. Singular hyperbolicity and sectional Lyapunov exponents of various orders. Proc. of the Amer. Math. Soc., Vol. 147, doi.org/10.1090/proc/14254. 2019.
[2] L. S. Salgado. Star flows: a characterization via Lyapunov functions. preprint ArXiv-1704.01987. 2017.
[3] V. Araujo, L. S. Salgado. Infinitesimal Lyapunov functions for singular flows. Math. Z., 275, 3-4, 863–897. 2013.
[4] A. Arbieto, L. S. Salgado. On critical orbits and sectional hyperbolicity of the nonwandering set for flows. J. of diff. equations 250 (2011) 2927–2939.

16:00 to 17:30 Marcelo Viana Lyapunov exponents, from the 1960's to the 2020's

The ergodic theory of Lyapunov exponents, initiated by the work of Furstenberg and Kesten at the dawn of the 1960s, has been a remarkably active field of mathematics over all these decades and, through the works of Oseledets, Pesin and many, many others, found extremely important applications in the realm of smooth dynamics.  The speaker will survey some of the classical results and more recently developments, with a view to the future of the field.

Wednesday, 25 September 2019
Time Speaker Title Resources
09:30 to 10:15 Stefano Luzzatto Lecture III - Hyperbolicity and Physical Measures

A main goal in the ergodic theory of differentiable dynamical systems is the study of the existence of physical measures. One main strategy is to prove that a system is hyperbolic and then to use this hyperbolicity to construct physical measures. In this course I discuss methods and techniques to verify hyperbolicity in a family of geometric models for the Lorenz equations.

10:45 to 11:30 Sylvain Crovisier Lecture III - Homoclinic classes and equilibrium measures

The dynamics of uniformly hyperbolic diffeomorphisms is well understood (after works by Anosov, Smale, Sinaï, Ruelle, Bowen,…). In this mini-course we will be interested by the non-uniformly hyperbolic part of the dynamics of arbitrary diffeomorphisms. From classical tools developed by Pesin and Katok, this dynamics naturally decomposes into pieces, called homoclinic classes, which generalize Smale’s uniformly hyperbolic basic pieces.

We will present recent results that allow to associate a good symbolic coding for each homoclinic class (due to Sarig, Ben Ovadia, Buzzi-Crovisier-Sarig). Combined with Yomdin theory, this can be used for studying the equilibrium measures on each class. For instance one will prove that a class can support at most one measure which maximizes the entropy and deduce some properties of that measure.

Reference: arXiv:1811.02240.

11:45 to 12:30 Jana Rodriguez-Hertz Minimality and stable ergodicity

Generically in Diff1m(M3), the existence of a minimal expanding invariant foliation implies stable Bernoulliness. Also, generically in Diff1m (M3), if an expanding f-invariant foliation of dimension u is minimal and there is a periodic point of unstable index u,then the foliation is stably minimal.For higher dimensions, we prove that generically in Diff1m(M), if an expanding f-invariant foliation of dimension u is minimal, and there is a periodic point of unstable index u, then the foliation is stably minimal. Also there is a hyperbolic ergodic component that is essentially dense.

This is joint work with Gabriel N´u˜nez.

Thursday, 26 September 2019
Time Speaker Title Resources
09:30 to 10:15 Sylvain Crovisier Lecture IV- Homoclinic classes and equilibrium measures

The dynamics of uniformly hyperbolic diffeomorphisms is well understood (after works by Anosov, Smale, Sinaï, Ruelle, Bowen,…). In this mini-course we will be interested by the non-uniformly hyperbolic part of the dynamics of arbitrary diffeomorphisms. From classical tools developed by Pesin and Katok, this dynamics naturally decomposes into pieces, called homoclinic classes, which generalize Smale’s uniformly hyperbolic basic pieces.

We will present recent results that allow to associate a good symbolic coding for each homoclinic class (due to Sarig, Ben Ovadia, Buzzi-Crovisier-Sarig). Combined with Yomdin theory, this can be used for studying the equilibrium measures on each class. For instance one will prove that a class can support at most one measure which maximizes the entropy and deduce some properties of that measure.

Reference: arXiv:1811.02240.

10:45 to 11:30 Stefano Luzzatto Lecture IV - Hyperbolicity and Physical Measures

A main goal in the ergodic theory of differentiable dynamical systems is the study of the existence of physical measures. One main strategy is to prove that a system is hyperbolic and then to use this hyperbolicity to construct physical measures. In this course I discuss methods and techniques to verify hyperbolicity in a family of geometric models for the Lorenz equations.

11:45 to 12:30 Andy Scott Hammerlindl Partially hyperbolic surface endomorphisms

Partially hyperbolic surface endomorphisms are a family of not necessarily invertible surface maps which are associated with interesting dynamics. The dynamical behaviour of these maps is less understood than their invertible counterparts, and existing results show that they can exhibit properties not possible in the invertible setting. In this talk, I will discuss recent results regarding the classification of partially hyperbolic surface endomorphisms. We shall see that either the dynamics of such a map is in some sense similar to a linear map, or that the map falls into a special class of interesting examples. This is joint work with Layne Hall.

14:30 to 14:50 Catalina Freijo Continuity of Lyapunov exponents for linear cocycles with a single holonomy

We consider a fixed hyperbolic dynamic in the base and study how the Lyapunov exponents vary as functions of the cocycle. The continuity of the Lyapunov exponents has been proved by Backes, Brown and Butler for cocycles that admit stable
and unstable uniform holonomies. In this talk, we present a partial result of a conjecture of Marcelo Viana that states that a single uniform holonomy is sufficient to guarantee the continuity. This is a joint work with Karina Marin (UFMG).

14:50 to 15:10 Christian S. Rodrigues On the representation of measures

Consider a family of probability measures $\{\mu_{x}\}_{x\in X}$ on a topological space $M$ parametrised by a given set $X$. Representing $\{\mu_{x}\}_{x\in X}$ consists in finding a map $F: X \times \Omega \to M$ such that, for each $x \in X$, we have, $\mu_{x} = F(x,\cdot)_{*}\mathbb{P}$, where $(\Omega, \mathbb{P})$ is an auxiliary probability space. Such questions arise in Probability, Geometry and several other areas. In particular, in Dynamical Systems, it appears in the context of disintegration of measures, and in random perturbation of dynamics, where one is interested in finding a probability on the spaces of maps which mimics a given Markov chain. In this talk, we provide sufficient conditions such that family of probabilities on manifolds can be represented by regular maps. The talk is based on a joint work with Jost, Matveev, and Portegies.

15:10 to 15:30 Nuno Luzia On Bowen's equation for non-conformal repellers

We give a version of Bowen's equation for the Hausdorff dimension of non-conformal repellers, with application to Non-linear Lalley-Gatzouras carpets. A main difference with respect to the conformal case
is that the equation might have several solutions, meaning there might be several measures of full dimension.

16:00 to 16:45 Ian Melbourne Uniformly hyperbolic flows: Rapid mixing for Holder observables

Decay of correlations for Axiom A flows remains an open and notoriously difficult problem. A reasonable conjecture is that an open and dense set of Axiom A flows have exponential decay of correlations for all nontrivial hyperbolic basic sets, but despite some isolated positive results this conjecture seems beyond modern technology.

A more tractable problem is to prove rapid mixing (mixing at all polynomial orders). This property is known to be open and dense, but only for observables that are sufficiently smooth. In this talk, we explain how to remove this restriction, proving that for an open and dense set of Axiom A flows the nontrivial hyperbolic basic sets are rapid mixing for Holder observables. Consequences and extensions will be mentioned. (Joint work with Caroline Wormell.)

Friday, 27 September 2019
Time Speaker Title Resources
10:45 to 11:30 Zhiyuan Zhang On the set of pseudo-rotations on annulus

A homeomorphism of the annulus without any periodic point is called a pseudo-rotation. Each pseudo-rotation has a unique rotation number in \mathbb R / \mathbb Z. We show that for a Baire generic rotation number \alpha \in \mathbb{R} / \mathbb{Z}, the set of area preserving C^\infty pseudo-rotation of the annulus \mathbb{A} with rotation number \alpha equals to the closure of the set of area preserving C^\infty pseudo-rotations which are smoothly conjugate to the rotation R_{\alpha} . As a corollary, a C^\infty generic area preserving pseudo-rotation of the annulus with a Baire generic rotation number \alpha is weakly mixing. This is a joint work with Barney Bramham.

11:45 to 12:30 Carlangelo Liverani Parabolic dynamics and distributions

I will illustrate in a simple example the relation between the distributional obstructions to the growth of the ergodic integral in a parabolic flow and the eigendistributions of the hyperbolic dynamics that renormalizes the flow. Such a relation is expected to hold in greater generality whereby providing a new approach to the study of parabolic flow.

Monday, 30 September 2019
Time Speaker Title Resources
10:00 to 11:00 S G Dani Diophantine approximation with nonsingular integral transformations

We consider the component-wise action of the multiplicative semigroup of nonsingular $n \times n$ matrices with integer entries, on the Cartesian product of $p \leq n-1$ copies of the Euclidean space $\mathbb R^n$, and discuss the effectiveness of the approximation of any target point by the orbit of a given initial point under the action. The exponent of approximation is proved to be $(n-p)/p$ for all initial points outside a null set that can be described by a certain Diophantine condition.

11:30 to 12:30 Debanjan Nandi Diphantine approximation, large intersections and geodesics in negative curvature

Let Γ be a group acting geometrically on a proper, geodesic, hyperbolic metric space X. We prove that the Γ action on the visual boundary of X has the large intersection property of Falconer. We discuss some geometric consequences. This is joint work with Anish Ghosh.

14:30 to 15:30 Shahar Mozes Surface subgroups in uniform lattices of some semisimple Lie groups

In a joint work with Jeremy Kahn and Francois Labourie we prove that any uniform lattice in a simple complex Lie group G contains a surface group. (I.e. the the fundamental group of an orientable surface of genus at least 2).
This theorem is a generalization of the celebrated Kahn–Markovic Theorem which deals with the case of G=PSL(2, C).

16:00 to 16:30 Shreyasi Datta Diophantine Inheritance and dichotomy for $p$-adic measures

I will talk about inheritance of exponents for affine subspaces and a Diophantine dichotomy for analytic $p$-adic manifold answering a conjecture of Kleinbock-Tomanov.

16:30 to 17:00 Andreas Wieser Planes in four space and four associated CM points

In the spirit of H. Maass and W. Schmidt we consider two-dimensional rational subspaces of the four-dimensional Euclidean space. To any such subspace one can attach two CM points originating from the subspace and
its orthogonal complement. The accidental (local) isomorphism between SO(4) and SO(3) x SO(3) allows us to define two further CM points naturally attached to the subspace. We show simultaneous
equidistribution of subspaces of a fixed discriminant together with all their CM points under two splitting conditions. A crucial input is the classification of joinings of higher-rank diagonalizable actions on
homogeneous spaces by Einsiedler and Lindenstrauss. This is joint work with Menny Aka and Manfred Einsiedler.

Tuesday, 01 October 2019
Time Speaker Title Resources
09:00 to 10:00 Jens Marklof Kinetic theory for the low-density Lorentz gas (Joint work with Andreas Strombergsson)

The Lorentz gas is one of the simplest and most widely-studied models for particle transport in matter. It describes a cloud of non-interacting gas particles in an infinitely extended array of identical spherical scatterers, whose radii are small compared to their mean separation. The model was introduced by Lorentz in 1905 who, following the pioneering ideas of Maxwell and Boltzmann, postulated that its macroscopic transport properties should be governed by a linear Boltzmann equation. A rigorous derivation of the linear Boltzmann equation from the underlying particle dynamics was given, for random scatterer configurations, in three seminal papers by Gallavotti, Spohn and Boldrighini-Bunimovich-Sinai. The objective of this lecture is to develop an approach for a large class of deterministic scatterer configurations, including various types of quasicrystals. We prove the convergence of the particle dynamics to transport processes that are in general (depending on the scatterer configuration) not described by the linear Boltzmann equation. This was previously understood only in the case of the periodic Lorentz gas through work of Caglioti-Golse and Marklof-Strombergsson. Our results extend beyond the classical Lorentz gas with hard sphere scatterers, and in particular hold for general classes of spherically symmetric finite-range potentials. We employ a rescaling technique that randomises the point configuration given by the scatterers' centers. The limiting transport process is then expressed in terms of a point process that arises as the limit of the randomised point configuration under a certain volume-preserving one-parameter linear group action.

10:00 to 11:00 Amos Nevo Lecture I: Intrinsic Diophantine approximation

An important goal of classical Diophantine approximation is quantitative analysis of the denseness of the set of rational vectors in their ambient Euclidean space. A challenging and substantial extension of the classical theory is to establish quantitative aspects of the denseness of rational points (and rational points with constrained denominators) in general homogeneous algebraic varieties. Here the denseness in question refers to the rational points on the variety itself, and so it is natural to call this subject "intrinsic Diophantine approximation". This problem was raised by Serge Lang more than half a century ago, but until recently material progress towards it was achieved only in the case of Abelian algebraic groups.

A systematic approach to this problem for homogeneous algebraic varieties associated with non-amenable algebraic groups was initiated and developed in joint work with A. Ghosh and A. Gorodnik over the last decade. This approach is based on dynamical arguments and effective ergodic theory, and some of its results apply even more generally, to actions of arbitrary lattice subgroups on homogeneous spaces. It leads to the derivation of uniform and almost sure Diophantine exponents for rational points and lattice orbits, as well as to an analog of Khinchin's dichotomy, and an analog of Schmidt's solution counting theorem. Furthermore, some of the results established using this approach are in fact best possible. We will explain some of the main results obtained and some of the ingredients in their proof, and present a variety of examples to demonstrate them, as time permits.

11:30 to 12:30 M.S. Raghunathan Ratner's Thorem - A representation theoretic approach

In this talk I will outline a proof of Ratner's theorem on unipotent flows on quotients of semisimple groups by co-compact lattices.

The proof makes use of results from representation theory, in particular Casselman's theorem of imbedding unitary modules in principal series.

Wednesday, 02 October 2019
Time Speaker Title Resources
09:00 to 10:00 Mahan M J Commensurators of thin groups

A celebrated theorem of Margulis characterizes arithmetic lattices in terms of density of their commensurators. A question going back to Shalom asks the analogous question for thin subgroups. We shall report on work during the last decade or so and conclude with a recent development. In recent work with Thomas Koberda, we were able to show that for a large class of normal subgroups of rank one arithmetic lattices, the commensurator is discrete.

10:00 to 11:00 Dmitry Kleinbock Lecture I; Khintchine-type theorems for values of homogeneous polynomials at integer points

Recently there has been a surge of activity in quantifying the density of values of generic quadratic forms and other polynomials at integer points. I will survey what is known (the work of Ghosh-Gorodnik-Nevo, Athreya-Margulis, Kelmer-Yu and others) and present some new and quite general results in this direction obtained via the so-called Siegel-Rogers method, which has recently been utilized by Athreya and Margulis, and later by Kelmer and Yu. This work is joint with Mishel Skenderi. If time allows I'll comment on other related problems.

11:30 to 12:30 Amos Nevo Lecture II: Intrinsic Diophantine approximation

An important goal of classical Diophantine approximation is quantitative analysis of the denseness of the set of rational vectors in their ambient Euclidean space. A challenging and substantial extension of the classical theory is to establish quantitative aspects of the denseness of rational points (and rational points with constrained denominators) in general homogeneous algebraic varieties. Here the denseness in question refers to the rational points on the variety itself, and so it is natural to call this subject "intrinsic Diophantine approximation". This problem was raised by Serge Lang more than half a century ago, but until recently material progress towards it was achieved only in the case of Abelian algebraic groups.

A systematic approach to this problem for homogeneous algebraic varieties associated with non-amenable algebraic groups was initiated and developed in joint work with A. Ghosh and A. Gorodnik over the last decade. This approach is based on dynamical arguments and effective ergodic theory, and some of its results apply even more generally, to actions of arbitrary lattice subgroups on homogeneous spaces. It leads to the derivation of uniform and almost sure Diophantine exponents for rational points and lattice orbits, as well as to an analog of Khinchin's dichotomy, and an analog of Schmidt's solution counting theorem. Furthermore, some of the results established using this approach are in fact best possible. We will explain some of the main results obtained and some of the ingredients in their proof, and present a variety of examples to demonstrate them, as time permits.

14:30 to 15:00 Sneha Chaubey The dynamics of Apollonian circle packings

In this talk, we define a dynamical system on the complex plane and view it in three different perspectives. First, as a system of reduction on the Descartes quadruples of all Apollonian circle packings, second, as a dynamical system on the tree of Lorentz quadruples and, third as a version of Gaussian complex continued fractions.

15:00 to 15:30 Manuel Luethi Equidistribution of simultaneous reductions of complex multiplication elliptic curves"

Under certain congruence conditions, the elliptic curves defined over the complex numbers with complex multiplication (CM) by a given order can be reduced to supersingular curves (SSC) defined over a finite field of prime characteristic. The (finite) set of isomorphism classes of SSC curves carries a natural probability measure. It was shown by Philippe Michel via progress on the subconvexity problem that the reductions of CM curves equidistribute among the SSC curves when the discriminant of the order diverges along the congruence conditions. We will describe a proof of equidistribution in the product of the simultaneous reductions with respect to several distinct primes of CM curves of a given order using a recent classification of joinings for certain diagonalizable actions by Einsiedler and Lindenstrauss. This is joint work with Menny Aka, Philippe Michel, and Andreas Wieser.

16:00 to 16:30 Arijit Ganguly Random walks on Tori and normal numbers in self similar sets.

We say a number x in [0,1] is normal if for any positive integer D, all finite words of same length with letters from the alphabet {0, 1, ... , D-1} occur with the same asymptotic frequency in the representation
of x in base D, or in simple words, its digital expansion is uniformly random in any base. Now the question is if the number x is chosen  randomly then how normal it is for x to be a normal one?'. This is
answered by the famous Normal number theorem of E. Borel which says that almost every number possesses this phenomenon.

It is generally believed that some naturally defined subsets of $\mathbb{R}$ also inherit the above property unless the set under consideration displays an obvious obstruction. This talk is about the study of Borel's theorem on fractals; cantor type sets for instance. We show that for certain fractals how the property of being normal can be related to the behaviour of trajectories under some random walk on tori, and consequently can be settled studying measures which are stationary' with respect to the random walk.

The talk is based on an ongoing joint work with Yiftach Dayan and Barak Weiss.

16:30 to 17:00 Runlin Zhang Translates of some homogeneous measures in arithmetic quotients of linear algebraic groups.

Let G be a linear algebraic group over Q, H be an observable Q-subgroup and \Gamma be an arithmetic lattice in G. Given a sequence of elements g_n in G(R), we study the limiting behavior of g_nH(R)\Gamma in G/\Gamma. For a bounded open set O in H(R), we give a criterion on when g_nO\Gamma diverges topologically to infinity in G/\Gamma using the work of Kleinbock--Margulis. And when the full orbit does not diverge, we classify all possible limiting measures. This analysis is based on the early work of Eskin--Mozes--Shah and Shapira--Zheng which ultimately relies on the Ratner's theorem on unipotent rigidity and the linearization technique developed by Dani--Margulis.

Thursday, 03 October 2019
Time Speaker Title Resources
10:00 to 11:00 Dmitry Kleinbock Lecture II: Khintchine-type theorems for values of homogeneous polynomials at integer points

Recently there has been a surge of activity in quantifying the density of values of generic quadratic forms and other polynomials at integer points. I will survey what is known (the work of Ghosh-Gorodnik-Nevo, Athreya-Margulis, Kelmer-Yu and others) and present some new and quite general results in this direction obtained via the so-called Siegel-Rogers method, which has recently been utilized by Athreya and Margulis, and later by Kelmer and Yu. This work is joint with Mishel Skenderi. If time allows I'll comment on other related problems.

11:30 to 12:30 Amos Nevo Lecture III: Intrinsic Diophantine approximation

An important goal of classical Diophantine approximation is quantitative analysis of the denseness of the set of rational vectors in their ambient Euclidean space. A challenging and substantial extension of the classical theory is to establish quantitative aspects of the denseness of rational points (and rational points with constrained denominators) in general homogeneous algebraic varieties. Here the denseness in question refers to the rational points on the variety itself, and so it is natural to call this subject "intrinsic Diophantine approximation". This problem was raised by Serge Lang more than half a century ago, but until recently material progress towards it was achieved only in the case of Abelian algebraic groups.

A systematic approach to this problem for homogeneous algebraic varieties associated with non-amenable algebraic groups was initiated and developed in joint work with A. Ghosh and A. Gorodnik over the last decade. This approach is based on dynamical arguments and effective ergodic theory, and some of its results apply even more generally, to actions of arbitrary lattice subgroups on homogeneous spaces. It leads to the derivation of uniform and almost sure Diophantine exponents for rational points and lattice orbits, as well as to an analog of Khinchin's dichotomy, and an analog of Schmidt's solution counting theorem. Furthermore, some of the results established using this approach are in fact best possible. We will explain some of the main results obtained and some of the ingredients in their proof, and present a variety of examples to demonstrate them, as time permits.

14:30 to 15:30 Federico Rodriguez-Hertz Rigidity for Anosov higher rank lattice actions.

In this talk I will discuss some simple consequence of various rates of mixing and propose some problems related to it.

16:00 to 17:00 Seonhee Lim Dimension bound for doubly badly approximable affine forms

We show that for all b and e>0, the Hausdorff dimension of the set of matrices e-badly approximable for the target b is not full. The doubly metric case follows.

It was known that for almost every matrix A, the Hausdorff dimension of the set of e-badly approximable target is not full. We show that if it is of full dimension, then A is singular on average. We will further discuss the singular on average case. (This is joint work with Wooyeon Kim and with Taehyung Kim.)

Friday, 04 October 2019
Time Speaker Title Resources
09:00 to 10:00 Dmitry Kleinbock Lecture III: Khintchine-type theorems for values of homogeneous polynomials at integer points

Recently there has been a surge of activity in quantifying the density of values of generic quadratic forms and other polynomials at integer points. I will survey what is known (the work of Ghosh-Gorodnik-Nevo, Athreya-Margulis, Kelmer-Yu and others) and present some new and quite general results in this direction obtained via the so-called Siegel-Rogers method, which has recently been utilized by Athreya and Margulis, and later by Kelmer and Yu. This work is joint with Mishel Skenderi. If time allows I'll comment on other related problems.

10:00 to 11:00 Cagri Sert "Measure classification and (non)-escape of mass for horospherical actions on regular trees"

Let T be a d-regular tree (d > 3), Γ < Aut(T) be a lattice and U < Aut(T) the stabilizer of some ξ ∈ ∂T, that do not contain hyperbolic elements (horo-spherical subgroup). In a first part, we study U-invariant ergodic probability
measures on Aut(T)/Γ and prove an Hedlund theorem when Γ is geometrically finite. In a second part, given a closed transitive subgroup G < Aut(T) and lattice Γ < G, we study non-escape of mass phenomenon for the U-action on
G/Γ, give examples of Γ with escape of mass for the U-action. Finally, we make connections between the geometric diophantine behaviour of ends and the speed of equidistribution of dense U-orbits in the aforementioned Hedlund
theorem. Joint work with Corina Ciobotaru and Vladimir Finkelshtein.

11:30 to 12:30 Rene Rühr Cut-And-Project sets and their moduli spaces

A cut-and-project set is constructed by restricting a lattice L in (d+m)-space to a domain bounded in the last m coordinates, and projecting it to the the space spanned by its d-dimensional orthogonal
complement. These point sets constitute an important example of so-called quasi-crystals.

During the talk, we shall present and give some classification results of the moduli spaces of cut-and-project sets, which were introduced by Marklof-Strömbergsson. These are obtained by considering the orbit
closure of the special linear group in d-space acting on the lattice L inside the space of unimodular lattices of rank d+m. Theorems of Ratner imply that these are meaningful objects.

This is a joint endeavour with Yotam Smilansky and Barak Weiss.