||Time and place||Thursday 12:30-14 Schreiber 209
||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 semester we will study mixing, effective mixing, and applications.||
Mixing and weak-mixing -- the basics
Mautner phenomenon and Howe-Moore Theorem, I.
Based on chapters 2 and 3 of the book of Bekka and Mayer, Ergodic theory and topological dynamics of group actions on homogeneous Spaces.
Howe-Moore Theorem, II.
Completion of the proof. Based on chapters 2 and 3 of the book of Bekka and Mayer, Ergodic theory and topological dynamics of group actions on homogeneous Spaces.
Examples and effective mixing.
Yiftach will present some examples completing last week's discussion. Daniel will begin the proof of effective mixing, based on the survey of Manfred Einsiedler, Effective equidistribution and spectral gap, section 4.
Effective mixing (cont.)
Daniel will continue with the proof of effective mixing, again using this survey by Manfred Einsiedler, section 4. It will be useful to review the spectral theorem for commuting unitary operators.
Introduction to Property (T) of Kazhdan
Einsiedler's proof of effective mixing employs ideas related to Kazhdan's property (T) which is a property of central importance in representation theory of Lie groups, with many interesting applications. Sieye will review property (T) following Lubotzky's book Discrete groups, Expanding graphs, and invariant measures. She will also review the notion of amenability and explain the relation of these notions.
Introduction to Property (T) of Kazhdan (cont.)
The main result will be the proof that SL_3(R) has property T and the pair (ASL_2(R), R^2) has a relative property T (to be defined in the lecture). The texts followed will be Einsiedler and Ward, Functional analysis, spectral theory and applications, and Lubotzky, Discrete groups, Expanding graphs, and invariant measures.
Property (T) and finite generation
Following Kazhdan's original argument, it will be shown that a lattice in SL_3(R), or any other group with Property T, is finitely generated. The text followed will be Zimmer, Ergodic theory and semisimple groups.