Summer Graduate School

The study of locally symmetric manifolds, such as closed hyperbolic manifolds, involves geometry of the corresponding symmetric space, topology of towers of its finite covers, and number-theoretic aspects that are relevant to possible constructions.

The workshop will provide an introduction to these and closely related topics such as lattices, invariant random subgroups, and homological methods.

Suggested prerequisites:

It will be useful to be familar with basic notions on:

• Hyperbolic plane H2 and hyperbolic spaces Hn: disk/ball, upper-half plane/space models, geodesics, isometries.
• Elements of Lie groups, in particular SL(2;R)-action on H2, and SO(n;1)-action on Hn.
• Basic algebraic topology: fundamental group and covering spaces, homology, elements of cohomology, Poincare duality.
• Some notions of geometric group theory: Cayley graphs, quasi-isometries.
• Real Analysis: abstract measure theory and integration.
• Abstract Algebra: finite extensions of Q and their Galois groups.

Recommended background material:

• A. Hatcher, Algebraic Topology, Chapters 1,2,3.
• D.Witte Morris, Introduction to arithmetic groups, Chapters 1, 5, 6. (available at arXiv:0106063).
• C. Maclachlan, A. Reid, The arithmetic of hyperbolic 3-manifolds, GTM 219. Chapters 1, 2, and possibly 4.
• T. Gelander, A lecture on Invariant Random Subgroups. (available at arXiv:1503.08402).

Additional books

For eligibility and how to apply, see the Summer Graduate Schools homepage