Following the organizational meeting, the topic chosen for the first 6-7 weeks was Veech surfaces (option 2 from the list).

Here is a list of exercises.

We will continuously update these as we go along, and add some solutions or hints to exercises we have already discussed.

Definitions of translation surfaces, relation to billiards and interval exchanges, Veech group, definition of Veech surface.

Arithmetic and non-arithmetic examples, statement of the Veech dichotomy and beginning of proof .

Some Spectral Methods in Dynamical Systems

We will explore some common methods in dynamics for proving the existence of absolutely continuous measures in order to illustrate the reliance on exponential decay of correlations. Then we will discuss the difficulties and some methods available for systems that exhibit only polynomial decay of correlations.

Veech surfaces III

Proof of the Veech dichotomy, periodic directions.

Veech surfaces IV

Masur ergodicity criterion, conclusion of the proof of the Veech dichotomy.

Time permitting, Barak Weiss will discuss non-lattice translation surfaces for which the Veech dichotomy holds, and related open problems.

Veech surfaces V

No small triangles condition

Veech surfaces VI

Smillie's theorem on lattice surfaces and closed orbits, statement of quantitative nondivergence for the horocycle flow on the moduli space of translation surfaces.

Veech surfaces VII

No small triangles condition (continued)

Veech surfaces VIII

No small triangles condition (continued). THIS MEETING WILL TAKE PLACE IN SCHREIBER BUILDING ROOM 209