Torus action topology 1000-1M12TDT
The lecture will be devoted to study differential manifolds with torus(product of circles) actions. A lot of explicit examples of such actions comes from complex algebraic geometry or symplectic geometry. We will study an important topological invariant of an action: the equivariant cohomology. Often a torus action has only a finite number of fixed points. Then the localization formula allows in a miraculous way to relate the topology of the manifold to a certain combinatorial object: the GKM-graph. A number of interesting results is obtained by studying the Grassmannian case. The laws of cohomology calculus on Grassmannians can be translated into identities involving rational functions of several variables.
I will discuss:
- Examples of manifolds with torus actions (Grassmannians, flag varieties, toric varieties)
- Equivariant cohomology for torus actions
- Atiyah-Bott (or Berline-Vergne) localization theorem
- GKM-graphs (Goresky-Kottwitz-MacPherson)
- Application of localization theorem to Schubert calculus
Type of course
Requirements
Prerequisites
Prerequisites (description)
Learning outcomes
Knowledge about main methods of equivariant topology, basic examples of torus actions on manifolds, ability to compute discussed invariants in concrete examples.
Assessment criteria
Zaliczenie ćwiczeń na podstawie zadań rozwiązywanych na zajęciach (50%) i zadań domowych (50%)
Egzamin ustny: problemy do samodzielnego rozwiązania (50%),
sprawdzenie znajomości materiału z wykładu (50%)
Bibliography
W. Fulton, Equivariant cohomology in algebraic geometry, lectures at Columbia University, notes by Dave Anderson, 2007.
V. Guillemin, S. Sternberg,Supersymmetry and equivariant de Rham theory. Springer-Verlag, Berlin, 1999
Additional information
Additional information (registration calendar, class conductors, localization and schedules of classes), might be available in the USOSweb system: