||Time and place||Thursday 10:10-12 Kaplun 324 (NOTE CHANGE OF ROOM)
||About||We will amuse ourselves by reading some papers about applications of homogeneous dynamics in number theory (particularly diophantine approximation and geometry of numbers) and geometry of discrete sets (Delone sets). This year we will try a mini-course format, with sequences of talks about a particular topic (one or two topics per semester).||
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