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Winter 2014/15: Homogeneous and Symmetric Spaces

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What
  • Seminar
When Oct 22, 2014 02:00 PM to
Feb 11, 2015 04:00 PM
Where SR404, Freiburg
Contact Name Konrad Voelkel
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The programme

The first third is about the differential-geometric 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 algebro-geometric 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 higher-dimensional uniformization, where homogeneous spaces show up as universal covers. Then we look at period domains in Hodge theory, which is another parameter-space 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 - Yi-Sheng 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 Thiam-Sun 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 Huber-Klawitter
  • 11.02. - Affine Grassmannians - Oliver Bräunling

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