References
1. Metric spaces of non-positive curvature, Bridson-Haefliger, Springer-1991 (week 1-4)
2. Ratner Theorems on Unipotent
flows, Dave Witte Morris, Chicago press (week 1-4)
3. Ergodic theory and Topological
dynamics of group actions on homogeneous spaces Bekka-Mayer,
LMS lecture note series 269. Cambridge Univ. Press. (week 1-4)
3. Lie groups beyond introduction, Knapp, Progress in Math.
140, Birkhauser (week 3-4, for advanced grad
students)
4. Ergodic Theory with a view
towards Number Theory, Einsiedler-Ward, Springer (GTM
259) (week 1-4)
5. Dynamical systems and diophantine
approximation, Bugeaud, Dalbo,
Drutu, eds.SMF-2009 (week 5-10)
6. Homogeneous spaces, moduli spaces and arithmetic, Einsiedler et al. (Clay Math Proceedings) AMS-2007 (week
12)
Syllabus
PART
I : Geometry of homogeneous spaces
Week 1-2 (March 4-15).
Basic properties of metric spaces of non-positive curvature (hyperbolic plane
on week 1, more general spaces week 2)
Week
3-4 (March 18-29). Lie groups and Lie algebra, Cartan
decomposition, Iwasawa decomposition, homogeneous
spaces, etc. (SL_2 on week 3, SL_n on week 4)
PART II : Homogeneous
dynamics and number theory
Week 5-6 (April 1-12). Introduction to Diophantine approximation.
Week 7-8 (April 15-26).
7-16). Diophantine approximation: homogeneous DA and works of Kleinbock-Margulis
Week 9-10 (April 29- May 10). Diophantine approximation: non-divergence on average and Hausdorff dimension
Week 11-12 (May 13-24). Diophantine approximation: inhomogeneous DA and Littlewood conjecture
Week 13 (May 27-31).
Quadratic forms, Oppenheim conjecture
Week
14 (June 3-7). Subconvexity of L-functions (survey)
Week
15 (June 10-14). Discussion session. Final presentation and/or oral exam.