Some suggestions and links for grad students
I agree with what Terry Tao's career advice
Some tryout problems.
Ten challenging riddles/puzzles/problems of varying difficulty,
take as much time as needed...
Students entering the field might start by learning the topics listed below.
Some topics here are small and well defined, some are big subjects; many are
old classics, some are close to current action.
There is no good definition of what is meant by the field,
but it includes rigidity phenomena for group actions, geometry and dynamics.
- 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(H2), SO(n,1) as Isom(Hn)
- 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
- Unitary Representations
- Peter-Weyl theorem
- Howe-Moore's theorem
- Induced representations
- Amenable groups vs. property (T)
- Teichmuller space
- Mostow rigidity (rank one)
- Boundary maps (...)
Some notes for above topics and further reading