Seminar on Equivariant Localization

This is a working group on the Atiyah-Bott Localization Theorem. We start by recalling the basics on equivariant cohomology and then we move to the statement and proof of the theorem. The seminar sessions take place on Thursday - 6pm at room 241 building A of IME. Would you like to give a talk? please contact the organizers Alejandro (labarbosat@gmail.com) or Cristian (cortiz@ime.usp.br). Below you can find our schedule. The full schedule can be found here.

Last talk - November 10

Speaker: Alejandro Barbosa (IME-USP)

Title: Atiyah-Bott Localization

Abstract: In this talk, we will present the notion of push-forward map in the context of equivariant cohomology and we will briefly discuss some of its properties. Finally, we will prove the Atiyah-Bott localization theorem.

References:

V. W. Guillemin, and S. Sternberg. Supersymmetry and equivariant de Rham theory. Springer Science & Business Media, 2013.

Atiyah, M. F., and R. Bott. “The moment map and equivariant cohomology.” Topology 23.1 (1984): 1-28  

J. H. Lee. Atiyah-Bott Localization in equivariant cohomology. 

A. Bapat. Equivariant Cohomology and the Localization Theorem

Past talks

October 27

Speaker: Eduardo de Carvalho Andrade (IME-USP)

Title: Equivariant characteristics classes

Abstract: Suppose S and G are two compact Lie groups, and P and M are manifolds. For an S-equivariant principal G-bundle P → M, the equivariant characteristic classes live in the equivariant cohomology ring H_{S}^{*}(M ). Since the equivariant cohomology of M is the cohomology of its Cartan model, it is natural to ask if equivariant characteristic classes can be constructed in the Cartan model out of the curvature of a connection on P. This is possible and in this seminar such construction will be shown.

References:

R. Bott, L. Tu, "Equivariant characteristic classes in the Cartan model"

October 13

Speaker: Alejandro Barbosa (IME-USP)

Title: Some properties of equivariant cohomology

Abstract: We will discuss the relation between the equivariant cohomology associated to an action by a compact connected Lie group G with the equivariant cohomology given by the induced action of the maximal torus of G. Also we will discuss some properties of long exact sequences associated to the equivariant cohomology that will be a key part in the proof of Atiyah-Bott localization.

References: 

1. A. Bapat, "Equivariant cohomology and the localization theorem"

2. V. Guillemin, S. Sternberg, "Supersymmetry and equivariant de Rham theory", Springer Verlag

3. J. Lee, "Atiyah-Bott localization in equivariant cohomology"

October 16

Speaker: Mateus de Melo (IME-USP)

Title: Cartan Theorem (Equivariant de Rham Theorem)

Abstract: The Cartan model for actions of compact Lie groups computes the real singular cohomology of Borel's construction. It is the statement of the Cartan Theorem (or Equivariant de Rham Theorem). We will present this result with examples and applications to classical cases. We will sketch a proof for free actions (principal bundles) and prove the theorem when the Lie group has a finite dimension smooth classifying space. If time allows, we will mention how to show the general case using the Bott-Schulmann complex for the associated action groupoid.

References: 

4. K. Behrend, "Cohomology of stacks. Intersection theory and moduli", ICTP Lect. Notes. 2004;19:249-94

5. O. Goertsches, L., Zoller, "Equivariant de Rham cohomology: theory and applications", São Paulo J. Math. Sci. 13, 539–596 (2019)

6. L. W. Tu, "Introductory Lectures on Equivariant Cohomology", Princeton University Press 204, 2020

September 29

Speaker: Fabricio Valencia (IME-USP)

Title: Cartan complex of equivariant differential forms

Abstract: In this talk, I will introduce the Cartan complex of equivariant differential forms which is associated to a compact Lie group action over a smooth manifold. Then, I will show how to recover it by means of Morse-Bott theoretical tools.

References: 

1. A. Kübel and A. Thom, "Equivariant differential cohomology"

2. D. M. Austin and P. J. Braam, "Morse-Bott theory and equivariant cohomology" 

September 22

Speaker: Alejandro Barbosa (IME-USP)

Title: Equivariant cohomology - The Borel model

Abstract: We present the concept of equivariant cohomology of a Lie group acting on a smooth manifold using Borel's approach and we give some examples and properties of this notion.

References: 

7. E. Meinreken, "Equivariant cohomology and the Cartan model"

8. L. W. Tu, "Introductory Lectures on Equivariant Cohomology", Princeton University Press 204, 2020

September 15

Speaker: Guilherme Vasconcelos (IME-USP)

Title: Universal bundles and classifying spaces

Abstract: In this talk we shall introduce the concepts of universal bundles and classifying spaces. Given a Lie group G, such bundles are special types of principal G-bundles, which satisfy certain criteria. In turn, the base space of this structure is what is called a classifying space for the group G. Moreover, a universal bundle allows one to pull back every principal G-bundle from its classifying space, which is one of the key features of this structure.

References: 

1- L. W. Tu, Introductory Lectures on Equivariant Cohomology, Princeton University Press 204, 2020.

2- N. Steenrod - The topology of fibre bundles, Princeton University Press, 1999.

3- S. A. Mitchell, Notes on principal bundles and classifying spaces, Lecture Notes. University of Washington

August 25

Speaker: Emma Cupitra (IME-USP)

Title: Principal G-bundles

Abstract: We will introduce principal G-bundles, their morphisms and main examples. We also recall the associated bundle construction given by a G-representation.

References: 

1. S. Kobayashi, K. Nomizu, "Foundations of Differential Geometry", vol.1, New York, 1963

2. S. A. Mitchell, "Notes on principal bundles and classifying spaces", Lecture Notes. University of Washington

3. L. W. Tu, "Differential Geometry connection, curvature and characteristic classes", Springer Verlag