MAT 520: Introduction to Geometry: Symmetric Spaces
Time: M F 9:30 a.m. - 11:00 a.m.
Classroom: Fine 1001
Start Date: Monday, February 9, 2009
Instructor: Anna Wienhard, email
Office: Fine 1007
Office Hours: TBA
Symmetric spaces (and locally symmetric spaces) play a crucial role in Algebraic Geometry, Differential Geometry, Mathematical Physics, Number Theory, and Representation Theory. They arise as moduli spaces (parameter spaces) for variations of geometric and arithmetic objects, e.g. the space of all k-dimensional subspaces of an n-dimensional vector spaces is a symmetric space, and the moduli space of elliptic curves is a locally symmetric space.
Any symmetric space can be decomposed into irreducible symmetric spaces. There are three types of irreducible symmetric spaces: Euclidean, compact and noncompact, where the latter two are related by an interesting duality. An example of such dual spaces are the two-sphere and the hyperbolic plane.
Due to their rich symmetry group symmetric spaces can be described both differential geometrically as well as algebraically. We will start from the differential geometric definition of (locally) symmetric spaces
and establish their relation with certain real Lie algebras with involution. We study the structure theory of these Lie algebras and of the corresponding Lie groups as well as the geometric properties of compact and noncompact symmetric spaces. If time permits we will also discuss the special class of Hermitian symmetric spaces which are related to bounded symmetric domains.
Throughout the course we will see many examples of symmetric spaces and various geometric models for them.
Symmetric Space Contest
Problem Set 1
Problem Set 2
Problem Set 3