References
1. Metric spaces of non-positive curvature, Bridson-Haefliger, Springer-1991 (week 1-8)
2. Ratner Theorems on Unipotent
flows, Dave Witte Morris, Chicago press (week 5-8)
3. Ergodic theory and Topological
dynamics of group actions on homogeneous spaces Bekka-Mayer,
LMS lecture note series 269. Cambridge Univ. Press. (week 1-8)
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 10-15)
5. Dynamical systems and diophantine
approximation, Bugeaud, Dalbo,
Drutu, eds.SMF-2009 (week 13-15)
6. Homogeneous spaces, moduli spaces and arithmetic, Einsiedler et al. (Clay Math Proceedings) AMS-2007 (week
10-15)
Syllabus
PART
I : Geometry of homogeneous spaces
Week 1-2 (Sept 5-12).
Basic properties of metric spaces of non-positive curvature (hyperbolic plane
on Sept 5-7, more general spaces Sept 10-12)
Week
3-4 (Sept 19-28). Lie groups and Lie algebra, Cartan
decomposition, Iwasawa decomposition, homogeneous
spaces, etc. (SL_2 on Sept 19, SL_n on Sept 21-28)
Week
5-6 (Oct 5-12). Isometries, group actions, geodesic flow, unipotent
flow. Ratner theorems I.
Week
7-8 (Oct 17-26) Boundary theory : Tits boundary, Gromov boundary, Busemann
boundary, and Martin boundary. Ratner theorems II.
Week
9 (Oct. 31- Nov. 2). : Discussion session. Midterm
presentation.
PART II : Homogeneous
dynamics and number theory
Week 10-11 (Nov.
7-16). Introduction to Diophantine approximation.
Week 12 (Nov. 21-23).
Quadratic forms, Oppenheim conjecture
Week
13 (Dec 28-30). Littlewood conjecture
Week 14-15
(Dec. 5-12). Survey on badly approximable
sets, Hausdorff dimension, etc.
Week
15 (Dec. 12-14). Discussion session. Final presentation and/or oral exam.