(August 30) Tomasz Downarowicz: Entropy Properties in Non-uniquely Ergodic Systems - In a non-uniquely ergodic system (action of a single homeomorphism T) the seemingly most complete information about entropy is in the entropy function (assigning to each invariant measure the entropy of T with respect to this measure). However, for many aspects like asymptotic h-expansiveness, existence of a subshift cover, etc., we need a more subtle tool, namely we need to know how the entropy function is approximated by certain relative entropy functions. We will present some recent results concerning the entropy function, relative entropy functions, relative variational principles, existence of subshift covers, criteria for asymptotic h-expansiveness, and the like. We will propose a new topological invariant (which we call "entropy structure") carrying complete information about the above discussed entropy properties of the system.

(September 6) Hillel Furstenberg: Invariant Graphs, Chromatic Numbers, and Recurrence Properties of Sets of Integers - Suppose (V,E) is a graph with V = Z = the integers, and suppose the edge set is determined by a subset Q of Z, with (n,m) in E iff n-m is in Q. What is the condition on Q so that the chromatic number of this graph be finite? We'll see that this depends on recurrence properties of the set Q relative to dynamical systems on compact spaces. This will lead to a natural open question in topological dynamics and we'll discuss aspects of this.

(September 13) Chris Hoffman: Which Toral Endomorphisms are Isomorphic? - Any integer valued matrix M with nonzero determinant generates a det(M) to one map of the unit cube onto itself. Katznelson proved that if |det(M)|=1 and M has no eigenvalues which are roots of unity then the map generated by M is (measurably) isomorphic to a Bernoulli shift. Thus two such matrices generate isomorphic actions if and only if the actions have the same entropy. We will discuss this problem in the case that |det(M)|>1.

(September 27) Misha Brin: The integrability of the central distribution - We show that the central distribution of a partially hyperbolic diffeomorphism $f\colon M\to M$ is uniquely integrable if the stable and unstable foliations are quasi-isometric in the universal cover $\widetilde{M}$, i.e., if two points lie in the same leaf, then the distance between them in $\widetilde{M}$ is bounded from below by the linear function of the distance along the leaf.

(November 1) Dmitri Dolgopyat: Transversely hyperbolic systems with symmetries - We describe mixing rates of transversely hyperbolic systems with symmetries such as Axiom A flows and compact group extensions of Axiom A diffeomorphisms which are related to accessibility properties of their stable and unstable laminations. We then describe some applications to statistical properties of such systems and their small perturbations.

(February 19) Mike Keane: Once reinforced random walks - In this lecture, I would like to discuss a recent idea of mine, possibly effective in proving recurrence of once reinforced random walks on ladders and in two dimensions. After an elementary presentation of the problem and results up until now, the simplest case of the calculation in mind will be discussed. It is not obvious that this will lead to success, but as of now it looks promising.

(March 07) Tomasz Downarowicz: Attainability of symbolic extension entropy - Symbolic extension entropy of a topological dynamical system (X,T) is the infimum of topological entropies of all symbolic extensions (Y,S) of (X,T). We will provide an explicit example for which this infimum is not attained. On the other hand we indicate a condition, concerning the simplex of invariant measures, which guarantees that an even stronger attainability holds.

(April 4) Sebastien Ferenczi: Joinings of three-interval exchange transformations - We show that among 3-interval exchange transformations there exists a dichotomy: T has minimal self-joinings whenever the associated subshift is linearly recurrent, and is rigid otherwise. We build also a family of simple rigid $3$-interval exchange transformations, which is a step towards an old question of Veech, and a family of rigid 3-interval exchange transformations which includes Katok's rank one map.

(April 11) Oh: Equidistribution of integer points and unipotent flows - Let $f$ be a homogeneous polynomial with integer coefficients and let $V_m$ be the variety defined by $f=m$. In the early sixties Linnik raised a problem of undertstanding the distribution of integer points $V_m(Z)$ as $m$ tends to infinity. In complete generality it seems hopeless to attack this question, except when the number of variables of $f$ is much bigger than the degree of $f$ in which case the Hardy-Littlewood circle method can be applied.

In this talk, we will discuss the Linnik problem when $f$ arises in invariant theory, explaining how the well developed theory of unipotent flows, culminating in Ratner's measure classfication theorem, becomes relevant.

This is a joint work with A. Eskin.

(April 18) Chris Hoffman: Mixing Time for Biased Card Shuffling - Consider a deck of n cards labeled 1 through n. We employ the following biased shuffling. At each stage we pick a pair of adjacent cards uniformly at random. Then with probability p>.5 we replace the cards with the lower numbered card before the higher numbered card. With probability 1-p we replace the cards with the higher numbered card first. We prove that the mixing time for this system is O(n^2). This proves a conjecture of Persi Diaconis. This is joint work with Itai Benjamini, Noam Berger, and Elchanan Mossel.

(May 2) Anthony Quas: Voronoi Percolation - We shall discuss a percolation problem that arose in the search for high entropy Bernoulli measures on higher-dimensional shifts of finite type. A random lattice is formed in d-dimensional space by dividing the space into "Voronoi Cells". These cells are then declared independently to be open with probability p, or closed with probability 1-p. We discuss a critical probability: a value p_c such that for p>p_c, there are almost surely infinite open components, but for p