** **

**Speaker: Theresa (Tess) Anderson – UW-Madison-USA**

** Title: A Spherical Maximal Function along the Primes.**

Abstract: Many problems at the interface of analysis and number theory involve showing that the primes, though deterministic, exhibit random behavior. The Green-Tao theorem stating that the primes contain infinitely long arithmetic progressions is one such example. In this talk, we show that prime vectors equidistribute on the sphere in the same manner as a random set of integer vectors would be expected to. We further quantify this via ergodic theorems, which connects classical tools from harmonic analysis with analytic number theory. This is joint work with Cook, Hughes, and Kumchev.

** **** ****Main Speaker : Boris Kalinin- Penn State-USA**

** ****Title: Lyapunov exponents of cocycles over non-uniformly hyperbolic**

**Systems**

** **Abstract:

We consider a linear cocycle A over a non-uniformly hyperbolic dynamical system,i.e. a diffeomorphism f preserving a hyperbolic measure \mu. The cocycletakes values in the group of invertible bounded operators on a Banach space V. For finite-dimensional V, we show that the Lyapunov exponents of A with respect to \mu can be approximated by the exponents at hyperbolic periodic orbits of f. For infinite-dimensional V, we show that the upper and lower Lyapunov exponents of A can be approximated in terms of the norms of the periodic return values of A, but not necessarily by the exponents at periodic orbits.

This is joint work with V. Sadovskaya..** **

**Speaker : Victoria Sadovskaya- Penn State-USA**

**Title: Boundedness and invariant norms for cocycles over hyperbolic systems**

Abstract:

We consider group-valued cocycles over dynamical systems with hyperbolic behavior, such as hyperbolic diffeomorphisms or subshifts of finite type. The cocycle takes values in GL(d,R) or in the group of invertible bounded linear operators on a Banach space. We will discuss how uniform boundedness and existence of invariant norms for the cocycle can be obtained from its periodic data, i.e. the set of its return values along the periodic orbits in the base. This is joint work with B. Kalinin.

**Speaker: Giulio Tiozzo- University of Toronto-Canada.**

**Title: On the local Hoelder exponent of the entropy function**

Abstract:

exponent at any point equals the *value* of the function?

It turns out that such functions arise quite naturally from dynamical systems, and we will see

an example which comes from real unimodal maps.

The proof is elementary and relies on the symbolic dynamics.

**Speaker :**

**Behrang Forghani- University of Connecticut – USA**

**Tittle: Poisson boundary of random walks on free semigroups**

**Speaker: Kasun Fernando- UMD- College Park**

**Title:**EDGEWORTH EXPANSIONS FOR WEAKLY DEPENDENT RANDOM VARIABLES

(Kasun Fernando and Carlangelo Liverani)

**Speaker: Mrinal Roychowdhury**

**Title: Quantization **

Abstract. Quantization for probability distributions refers to the idea of estimating a given probability by a discrete probability supported by a set with no more than n points. It has broad application in signal processing and data compression. Quantization dimension gives the speed how fast the specified measure of the error goes to zero as n approaches to infinity. Quantization dimension is also connected with other dimensions of dynamical systems such as Hausdorff, packing and box counting dimensions. I will talk about it.

**Speaker: Michael Lin – Ben Gourion University-**

**Speaker: Guy Cohen – Ben Gourion University**

**Speaker: Maryam Hosseini- IPM**

**Speaker**:

**Jason Atnip – UNT-**

**Title: ****An Almost Sure Invariance Principle for Random Dynamical Systems**

Abstract: In this talk we deal with a large class of dynamical systems having a version of the spectral gap property. Our primary class of systems comes from random dynamics, but we also deal with the deterministic case. We show that if a random dynamical system has a fiberwise spectral gap property as well as an exponential decay of correlations in the base, then, developing on Gou\”{e}zel’s approach, the system satisfies the almost sure invariance principle. The result is then applied to uniformly expanding random systems like those studied by Denker and Gordin and Mayer, Skorulski, and Urba\’nski.

**Speaker: Tushar Das- University of Wisconsin-La Cross**

**Title: Singular systems of linear forms with a prescribed uniform irrationality exponent**

Abstract:

The singularity (in the sense of Diophantine approximation) of a system of m linear forms in n variables, has been studied in dynamical terms since Dani’s seminal correspondence principle. At the workshop last year, I spoke about our resolution of the Kadyrov-Kleinbock-Lindenstrauss-Margulis conjecture regarding the Hausdorff dimension of the set of singular forms. This involved creating a flexible Schmidt-type game that allows precise dimension computations, as well as extending the Schmidt-Summerer-Roy parametric geometry of numbers. I will present a progress report on what we have been able to accomplish since my last talk (for instance, a precise formula for the packing dimension of the tau-singular mxn matrices whenever n\geq 2), and end with further open problems that have arisen in the wake of our latest results (for instance, we conjecture that when n=1 and m\geq 3, the packing dimension is discontinuous as a function of tau). This is joint work with Lior Fishman (UNT), David Simmons (York) and Mariusz Urbanski (UNT).

**Main Speaker: Albert Fathi – Georgia-Tech**

**Title:** **Recurrence on abelian cover. Application to closed geodesics in manifolds of negative curvature**

If h is a homeomorphism on a compact manifold which is chain-recurrent, we will try to understand when the lift of h to an abelian cover (i.e. the covering whose Galois group is the first homology group of the manifold) is also chain-recurrent.

This is related to the proof by John Franks of the Poincaré-Birkhoff theorem.

It has new consequences on density of classes of closed geodesics in a manifold of negative curvature.