
Time and place  Thursday 1113 Schreiber 210 




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). In the fall semester we studied mixing and effective mixing, and in this semester, we will study applications of these results.  



Schedule 
Counting lattice orbits on the upper half plane and counting closed geodesics We will follow ideas originating from Margulis' thesis, relating mixing and the count of points in an orbit of lattice orbits on the hyperbolic plane. Some of the ideas can be found in Margulis' thesis ( On some aspects of the theory of Anosov systems ) and some can be found in a survey of Einsiedler and Ward, see section 5.2 of the Durham lecture notes. Room 210, 14:10 Counting lattice orbits on affine symmetric space (after EskinMcMullen) The talk will be based on the paper of Eskin and McMullen, "Mixing, counting, and equidistribution in Lie groups", Duke Math. J. Volume 71, (1993), 181209. Counting lattice orbits on affine symmetric space (continued) This will be a continuation of Rene's talk from last week. Rene will complete his proof of the equidistribution result (theorem 1.2 of the EskinMcMullen paper) and Arijit will explain how to deduce counting results. Here are the notes for Rene's two talks.
Counting lattice orbits on affine symmetric space (continued) Arijit will explain the proof of Theorem 1.4 of the EskinMcMullen paper.
Asymptotic distribution of Frobenius numbers, following J. Marklof In the paper The asymptotic distribution of Frobenius numbers, Inventiones Mathematicae 181 (2010) 179207, Jens Marklof made a connection between a classical problem involving Frobenius numbers, and homogeneous dynamics. Further work on this topic was done by Han Li (see this paper, Compositio Math. 2014 ) and Andreas Strombergsson (see this paper, Acta Arith. 2012 ). In this talk the results will be introduced and the connection to geometry of numbers and homogeneous dynamics will be made. Click here for the notes for Yotam's lecture.
Daniel ElBaz Asymptotic distribution of Frobenius numbers (continued) PLEASE NOTE SPECIAL DAY AND TIME. The talk will discuss the dynamical ideas behind the proofs of the results of Marklof and Li discussed in Yotam's talk.
Yiftach Dayan Khintchine's theorem and its analogues in homogeneous dynamics NOTE SPECIAL TIME AND PLACE The talk will state Khintchine's theorem in diophantine approximations, discuss the background and the Borel Cantelli lemma which is the basic tool for proving such results. Then the dynamical interpretation, following Dani, will be presented. The talk will follow the papers "Logarithm laws for flows on homogeneous spaces", by Kleinbock and argulis, Inv. Math. 138 (1999), 451494, and "Disjoint spheres, approximation by imaginary quadratic numbers, and the logarithm law for geodesics ", by Sullivan, Acta Math. 149 (1982), 215237. Also lecture notes of Gorodnik will be used.
Khintchine's theorem and its analogues in homogeneous dynamics (continued) A continuation of last week's talk with proofs of the divergence case and the connection to quantitative mixing.




Previous years 
Fall 2014
