Monday, 11 March 2019

In the first set of lectures, the basic objects will be introduced and their behavior will be analyzed emphasizing the role of hyperbolicity and preparing the second set of lectures given by Sebastien Gouëzel.

These lectures should culminate with the following theorem: the fundamental inequality is an equality only if the group is virtually free.

We will discuss the following topics:

- asymptotic entropy, drift, the Guivarc'h fundamental inequality

- harmonic measure and asymptotic behavior of the walk on a hyperbolic group

- the Martin boundary and the Green distance

- dimension of the harmonic measure and the fundamental inequality

Tuesday, 12 March 2019

In the first set of lectures, the basic objects will be introduced and their behavior will be analyzed emphasizing the role of hyperbolicity and preparing the second set of lectures given by Sebastien Gouëzel.

These lectures should culminate with the following theorem: the fundamental inequality is an equality only if the group is virtually free.

We will discuss the following topics:

- asymptotic entropy, drift, the Guivarc'h fundamental inequality

- harmonic measure and asymptotic behavior of the walk on a hyperbolic group

- the Martin boundary and the Green distance

- dimension of the harmonic measure and the fundamental inequality

Wednesday, 13 March 2019

In the first set of lectures, the basic objects will be introduced and their behavior will be analyzed emphasizing the role of hyperbolicity and preparing the second set of lectures given by Sebastien Gouëzel.

These lectures should culminate with the following theorem: the fundamental inequality is an equality only if the group is virtually free.

We will discuss the following topics:

- asymptotic entropy, drift, the Guivarc'h fundamental inequality

- harmonic measure and asymptotic behavior of the walk on a hyperbolic group

- the Martin boundary and the Green distance

- dimension of the harmonic measure and the fundamental inequality

TBA

Thursday, 14 March 2019

We will discuss the following topics:

- harmonic measure and asymptotic behavior of the walk on a hyperbolic group

- the Martin boundary and the Green distance

- dimension of the harmonic measure and the fundamental inequality

We define the expanding cone of a horospherical subgroup in a simple noncompact Lie group. The cone generalizes the usual cone in the special linear group used to study weighted Diophantine approximation. The trajectories of points on a horospherical orbit with respect to the action of semigroups in the expanding cone have nice dynamical properties. We discuss effective multiple correlations, effective multiple pointwise ergodic theorem and central limit theorem. We also give a characterization of nonzero variance in the central limit theorem using a Livsic type theorem on homogeneous space.

TBA

Friday, 15 March 2019

For first passage percolation (FPP), a classical model of putting a random metric on a graph, we shall describe an old question going back to Furstenberg about the existence of bigeodesics on Euclidean lattices. We shall describe recent partial developments in related questions and discuss why the answer to question in Euclidean lattices is expected to be different from that in hyperbolic graphs.

We define the expanding cone of a horospherical subgroup in a simple noncompact Lie group. The cone generalizes the usual cone in the special linear group used to study weighted Diophantine approximation. The trajectories of points on a horospherical orbit with respect to the action of semigroups in the expanding cone have nice dynamical properties. We discuss effective multiple correlations, effective multiple pointwise ergodic theorem and central limit theorem. We also give a characterization of nonzero variance in the central limit theorem using a Livsic type theorem on homogeneous space.

TBA

Monday, 18 March 2019

In this mini-course I will focus on the qualitative aspects of the relationship between the Poisson boundary of random walks and the structural properties of the underlying groups.

We establish a connection between extreme values of stable random fields arising in probability and groups G acting geometrically on spaces X exhibiting negative curvature properties: CAT(-1) spaces, Tehcmuller space, higher rank symmetric spaces. The connection is mediated by the action of the group on its limit set equipped with a conformal or quasiconformal measure. Based on motivation from extreme value theory, we introduce an invariant of the action called extremal cocycle growth and show that its non-vanishing is equivalent to finiteness of the Bowen-Margulis measure for the associated unit tangent bundle U(X/G) provided X is not a tree whose edges are (up to scale) integers. We also establish an analogous statement for normal subgroups of free groups.This is joint work with Jayadev Athreya.

We establish a connection between extreme values of stable random fields arising in probability and groups G acting geometrically on spaces X exhibiting negative curvature properties: CAT(-1) spaces, Tehcmuller space, higher rank symmetric spaces. The connection is mediated by the action of the group on its limit set equipped with a conformal or quasiconformal measure. Based on motivation from extreme value theory, we introduce an invariant of the action called extremal cocycle growth and show that its non-vanishing is equivalent to finiteness of the Bowen-Margulis measure for the associated unit tangent bundle U(X/G) provided X is not a tree whose edges are (up to scale) integers. We also establish an analogous statement for normal subgroups of free groups.This is joint work with Jayadev Athreya.

Tuesday, 19 March 2019

In this mini-course I will focus on the qualitative aspects of the relationship between the Poisson boundary of random walks and the structural properties of the underlying groups.

Building on the minicourse by Peter Haïssinsky, we will continue the study of random walks on hyperbolic groups, a setting in which the interaction between the geometry of the group and the randomness of the walk is most clearly visible. We will focus on the problem of the local limit theorem, i.e., the asymptotics of the return probability to the origin in large time. While the local limit question is easy to answer on Z^d thanks to Fourier series arguments, the problem is much more delicate in hyperbolic groups, and ultimately rests on geometric arguments. We will touch the following topics:

- automatic structure on hyperbolic groups,

- symbolic dynamics,

- Anonca inequalities, saying that the random walk tends to follow geodesics, up to the spectral radius,

- Asymptotics of the Green function at its spectral radius thanks to approximate differential inequalities

- Tauberian theory

Wednesday, 20 March 2019

In this mini-course I will focus on the qualitative aspects of the relationship between the Poisson boundary of random walks and the structural properties of the underlying groups.

Thursday, 21 March 2019

Friday, 22 March 2019

Building on the minicourse by Peter Haïssinsky, we will continue the study of random walks on hyperbolic groups, a setting in which the interaction between the geometry of the group and the randomness of the walk is most clearly visible. We will focus on the problem of the local limit theorem, i.e., the asymptotics of the return probability to the origin in large time. While the local limit question is easy to answer on Z^d thanks to Fourier series arguments, the problem is much more delicate in hyperbolic groups, and ultimately rests on geometric arguments. We will touch the following topics:

- automatic structure on hyperbolic groups,

- symbolic dynamics,

- Anonca inequalities, saying that the random walk tends to follow geodesics, up to the spectral radius,

- Asymptotics of the Green function at its spectral radius thanks to approximate differential inequalities

- Tauberian theory

Consider a probability measure preserving system $(X,\mu,T)$ and a sequence of subsets $B_n\subset X$ whose measure shrinks with $n$ and satisfy $B_m \supset B_{m+1}$. A common pastime among ergodic theorists is to understand various properties of the points whose $n$'th iterate under $T$ hits the set $B_n$ for infinitely many $n$. That is, the set $\mathcal{A}_{\text{i.o.}}:=\{x\in X: T^n(x)\in B_n \text{ for infinitely many } n\}$. Typical questions in this direction, known as \emph{shrinking target problems}, concern how the measure of $\mathcal{A}_{\text{i.o.}}$ depends on the rate with which the $B_n$'s shrink. For ergodic systems the measure of $\mathcal{A}_{\text{i.o.}}$ is always zero or one and in many interesting cases this jump occurs when $\sum_n \mu(B_n)$ goes from being finite to infinite. When the measure is zero we are also interested in the Hausdorff dimension of the set and its dependence on the shrinking rate of $\mu(B_n)$.

In this talk we study an interesting subset (aside from a set of measure zero) of $\mathcal{A}_{\text{i.o.}}$ known as the set of \emph{eventually always hitting} points which was recently introduced by Kelmer. These are the points whose orbit up to time $n$ will never have empty intersection with $B_n$ for all sufficiently large $n$. That is, the set $\mathcal{E}_{\text{ah}}:=\{ x\in X : \exists\, n_0(x) \;\forall\, n\geq n_0(x)\; \{ T^k(x)\}_{k=0}^n \cap B_n\neq \emptyset \}$. We will discuss recent results giving necessary and sufficient conditions for $\mathcal{E}_{\text{ah}}$ to be of full measure as well as dimension estimates for the case when $\mathcal{E}_{\text{ah}}$ is of zero measure. In particular we will focus on eventually always hitting points for interval maps.

This is joint work with Philipp Kunde (Hamburg/Indiana), Tomas Persson (Lund).