Meetings: 3106 MSB from 9 to 10 on Thursdays.
Credit: The variable unit request form can be found here
https://www.math.ucdavis.edu/grad/gr
ad-student-handbook and should be filled out if you want course credit.
References:
[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 | | |
October 12 references:
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.