[GKM]Equivariant cohomology, Koszul duality, and the localization theorem by M Goresky, R Kottwitz2 and R MacPherson

[AB]The moment map and equivariant cohomology by M Atiyah and R Bott

[Pro]Symplectic geometry notes by N Proudfoot

[GZ]Equivariant de Rham Theory and Graphs by V Guillemin and C Zara

[TW] The cohomology rings of Abelian symplectic quotients by S Tolman and J Weitsman

[Tym]An introduction to equivariant cohomology and homology, following Goresky, Kottwitz, and MacPherson by J Tymoczko

Software for computations

[Bro][Lectures on the Cohomology of Groups by K Brown

[GS]Convexity properties of the momentum mapping by V Guillemin and S Sternberg

[H] Topological properties of subanalytic sets by Robert Hardt

[E]Notes by J Erickson

**Course Outline:** This is a rough outline and (apart from exam dates) subject to change.

Day | Speaker | Topic |

October 5 | Yuze Luan | Introduction |

October 12 | Timothy Paczinski | Examples |

October 19 | Timothy Paczinski | Equivariant Chains |

October 26 | Jon Erickson | Symplectic Actions |

November 3 | Jon Erickson | Symplectic Actions |

November 10 | Jon Erickson | Symplectic Actions |

November 17 | Yuze Luan | Intersection Homology |

November 24 | holiday | |

December 1 | Raymond Chan | Equivariant Formality |

December 8 | Koszul Duality | |

January 11 | Jon Erickson | Examples: Notes [E] |

January 18 | Jon Erickson | Examples: Notes [E] |

January 25 | Timothy Paczinski | Subanalytic Sets [H] |

February 1 | Timothy Paczinski | Subanalytic Sets [H] |

February 8 | Timothy Paczinski | Equivariant Chains |

February 15 | No Meeting | |

February 22 | Jon Erickson | Derived Sheaf Categories |

February 29 | ||

March 7 | ||

March 14 |

In [Tym sec 4] there are three nice examples including the case of the projective plane as [Tym ex 4.1]. The formula in [Tym thm 3.4] used there agrees with [GKM thm 1.2.2] and is equivalent to the different looking one in [GKM thm 7.2]. These examples might also enable you to work out a couple for yourself just starting from a graph such as a polygon in the plane. The definition [GZ def 2.1.1] gives a notion of such a graph. In their notation the cohomology computation [GKM thm 1.2.2] becomes the combination of [GZ thm 1.7.1] and [GZ thm 1.7.3]. The phrasing is algebraic rather than symplectic. The definition of equivariantly formal involves a spectral sequence on which Ken Brown's book Cohomology of Groups has a very nice exposition and [Bro sec 5.1] has a shortened version.

Jon Erickson's notes for symplectic actions.