Review of Ratner's theorems

Ratner's result from the 1990's were a breakthrough that solved a longstanding open problem, have had many applications outside ergodic theory and are a basic tool in the work we will discuss. Yotam will present the main results and give a "horopherical" proof of for the space SL(2,R)/SL(2,Z). The presentation will follow a survey of Alex Eskin, Chapter 2 of Clay Institute, lecture notes on homogeneous flows (Pisa, 2007). .

Ratner's theorems (cont.) and conditional measures

Yotam will complete last week's lecture. Here are his notes and here is a question about them.

Uriya's lecture will be a review lecture about the construction and basic properties of conditional measures. For more information see the survey of Einsiedler and Lindenstrauss, pages 164-170. Here are the notes for Uriya's talk.

Leafwise measures, more on Ratner

Uriya completed his talk from last week and I made some remarks on the exercise left over from Yotam's talk, regarding the proof of the special case of Ratner's theorem, SL(2,R)/SL(2,Z).

Illumination in polygons

Amit will present his research on the illumination problem, this uses orbit-closure classification results of Eskin-Mirzakhani-Mohammadi to solve a problem in geometry. THE TALK WILL BE HELD AT ROOM SCHREIBER 8 AT 10:30.

Leafwise measures

Yiftach will present leafwise measure which is a generalization of the notion of conditional measures (and relies on it). The presentation is based on Clay notes, Einsiedler-Lindenstrauss survey, pages 170-185.

Random matrix products

Chen will give some information on the growth of vectors under a random matrix product, and on the distance of this vector from a `subspace of maximal growth'. Chen's presentation will rely on the discussion of the previous semester. Click here for Chen's notes (updated June 4!).

Outline of argument

In this talk we begin the proof of the main result (Theorem 1.3) of the paper by Eskin and Lindenstrauss. Itamar will give the outline of the proof and reduce the statement to the proof of Section 6.1.

Outline of argument (continued), and beginning of argument

Itamar will complete his outline of the proof of Theorem 1.3 and Tsviqa will begin the reduction of Proposition 6.1 to Proposition 6.2.

Proof of Proposition 6.1

Tsviqa will state Proposition 6.1 and explain how it follows from Proposition 6.2. The same topic was discussed in Alex Eskin's talks in Jerusalem, see this video.

Proof of Proposition 6.2

Rene will outline the proof of Proposition 6.2. This will bring together all the concepts we have discussed during the seminar.