Geometry, arithmetic, topology, and dynamics of discrete groups

Seminars

1. Arithmetic & quasi-arithmetic hyperbolic reflection groups. General plans. Discrete subgroups of Lie groups. Seminar 1

Educational Research Part

1. Smooth manifolds. Manifolds with boundary. Riemannian manifolds. Examples. Complex manifolds. Riemann surfaces. Fundamental groups. Coverings.
2. Algebraic varieties. Examples.
3. Algebraic groups. Examples. Lie groups. Algebraic groups over R and C as Lie groups.
4. Complex algebraic groups. Lie algebras. Simple, semi-simple, reductive groups.
5. Simple roots. Classification of simple and semi-simple algebraic groups over C.
6. Algebraic groups over non-closed fields (R, Q, Q[sqrt d], etc.).
7. Algebraic varieties and Riemannian manifolds as homogeneous spaces (G/H). Symmetric homogeneous spaces. Classical spaces E^n, S^n, H^n.
8. Discrete subgroups of Lie groups. Discrete groups of isometries. Biebarbach’s theorems. Fundamental domains. Fuchsian groups.
9. Properly discontinuous actions. Affine transformations. Flat affine manifolds. The Auslander conjecture (1964).
10. Cayley graphs. Ramanujan graphs. Expanders. Graphs of groups. Actions of groups on trees.
11. Arithmetic discrete groups. Rigidity theorems.
12. Hyperbolic manifolds.

Discussion of Papers

1. Alex Kontorovich & Kei Nakamura, “Geometry and arithmetic of crystallographic sphere packings”.
2. Olivier Mila, “Nonarithmetic hyperbolic manifolds and trace rings”.
3. Vincent Emery & Olivier Mila, “Hyperbolic manifolds and pseudo-arithmeticity”.

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