Winter 2014/15: Homogeneous and Symmetric Spaces
What 


When 
Oct 22, 2014 02:00 PM
to Feb 11, 2015 04:00 PM 
Where  SR404, Freiburg 
Contact Name  Konrad Voelkel 
Add event to calendar 
vCal iCal 
The programme
The first third is about the differentialgeometric picture around (mostly compact) Lie groups and Riemannian symmetric spaces. The example of Grassmannians will give us strong motivation because of their parameter space interpretation. Riemannian symmetric spaces will be very important for the outlook, when we study those Riemannian symmetric spaces which have a compatible complex structure  Hermitian symmetric domains. We end the section on Lie groups with an overview of the adjoint action and a discussion of root systems and Dynkin diagrams, an important organising principle for compact Lie groups, linear algebraic groups, their homogeneous spaces and completions.
The second third is about the algebrogeometric picture around (mostly linear) algebraic groups. We will see which place semisimple linear groups (the nice ones) have in the world of algebraic groups, and how to construct the quotient varieties we want to study. The theory of compactifications will be introduced, which will be used in the last part.
The last part is an outlook. First we discuss higherdimensional uniformization, where homogeneous spaces show up as universal covers. Then we look at period domains in Hodge theory, which is another parameterspace story. After learning a bit about Hermitian symmetric spaces in general, we come to quotients of them, Shimura varieties, which are useful for arithmetic geometry. Finally, there are the affine Grassmannians, which are infinite dimensional homogeneous spaces.
Schedule:
 22.10.  Lie Groups  Martin Schwald
 29.10.  Quotients after Lie Group Actions  Anja Wittmann and Felix Grimm
 05.11.  Grassmannians  YiSheng Wang
 12.11.  Riemannian Symmetric Spaces  Giulia Battiston
 19.11.  The Adjoint Action  Wolfgang Soergel
 26.11.  Algebraic Groups  Max Schmidtke and Jens Eberhardt
 03.12.  Reductive Linear Algebraic Groups  Max Schmidtke and Jens Eberhardt
 10.12.  Quotients after Algebraic Group Actions  Helene Sigloch and ThiamSun Pang
 17.12.  Compactifications  Michael Rottmeier and Sergei Kovalenko
 07.01.  Dirac Operators on Homogeneous Spaces  Pavle Pandzic
 14.01.  FREE
 21.01.  Period Domains  Natalie Peternell
 28.01.  Hermitian Symmetric Spaces  Florian Beck
 04.02.  Shimura Varieties  Annette HuberKlawitter
 11.02.  Affine Grassmannians  Oliver Bräunling