Syllabus and Reading List for Eskin-kleinbock Course

1. General introduction, Birkhoff’s Ergodic Theorem vs. Ratner’s Theorems on unipotent flows; measure classification implies classification of orbit closures; uniform convergence and the theorem of Dani-Margulis; the statement of the Oppenheim Conjecture. 2. The case of SL(2, R) (the mixing argument). We will be loosely following Ratner’s paper [18]. 3. The classification of invariant measures for the action of a unipotent subgroup of SL(2, R) n R (modulo non-divergence). 4. Quantitative non-divergence of unipotent flows and quasi-polynomial maps: the case of SL(2, R). 5. Quantitative non-divergence in higher rank. 6. Oppenheim and Quantitative Oppenheim (lower bounds). The work of Dani-Margulis [2]. 7. Quantitative Oppenheim (upper bounds) [4]. Systems of inequalities. 8. The work of Mozes-Shah [17] (limits of ergodic measures are ergodic). 9. Equidistribution of translates and applications to Diophantine equations. We will follow parts of [6] and [7]. 10. Equidistribution of translates and applications to improvement of Dirichlet’s Theorem [12, 14]. 11-12. Applications of non-divergence to metric Diophantine approximation: Baker-Sprindžuk Conjectures and their generalizations, Diophantine exponents of affine subspaces [9, 10, 11]. 13. Counting problems with billiards (slightly off-topic). 14. The work of Margulis-Goetze on quadratic forms. 15. If time allows: an overflow lecture. Possibly Oppenheim in signature (2, 2) and applications to quantum chaos.