Ten challenging riddles/puzzles/problems of varying difficulty, take as much time as needed...

Students entering

- Background from Analysis
- Hahn-Banach, Banach limit
- Banach-Alaoglu, weak-* topology on P(X), Kantorovich metric
- Basic Hilbert space techniques
- Spectral theorem for compact operators
- Spectral theorem for normal operators: Halmos' note
- Examples of Lie groups
- SL(2,R) as Isom(H
^{2}), SO(n,1) as Isom(H^{n}) - KAK, KP, KMAN decompositions
- Heisenberg group
- Some discrete groups
- Free groups and graphs
- Bass-Serre theory (...)
- Gromov hyperbolic groups (...)
- Amenable groups (...)
- Some non-discrete groups
- Topological groups - basics and examples (Lie, loc compact, Polish)
- Haar measure on compact groups
- Modular functions, lattices
- Linear groups
- Selberg's lemma
- Furstenberg's lemma and quasi-projective transformations
- Borel's density theorem
- Growth of linear groups
- Tits alternative (...)
- Unitary Representations
- Peter-Weyl theorem
- Howe-Moore's theorem
- Induced representations
- Amenable groups vs. property (T)
- Rigidity
- Teichmuller space
- Mostow rigidity (rank one)
- Boundary maps (...)

[Alex Furman] [Papers] [Talks] [CV]