Gauge theory learning seminar (Fall 2019)

The time and place for the seminar will be Wednesdays at 1:15-2:15 in room 622.

We'll start with the early paper by Atiyah and Bott on the Yang-Mills equations over Riemann surfaces, which will take us about 7 talks. Further topics for the remainder of the semester TBD.

If you want to be in the mailing list for this seminar then contact the organizer, Juan Muñoz (juanmunoz@math.columbia.edu).

 

References

- (a) Atiyah and Bott, Yang-Mills equations over Riemann surfaces

- (b) Donaldson, A new proof of a theorem of Narasimhan and Seshadri

- (c) Thaddeus, An introduction to the moduli space of stable bundles on  a Riemann surface

- (d) Thomas, Notes on GIT and symplectic reduction for bundles and varieties

- (e) Donaldson-Kronheimer, The geometry of four-manifolds

 

Schedule 

Date

Speaker

Title

References

September 25th  Organizational meeting     
October 2nd Aleksander Doan The Kempf-Ness theorem (e) Ch.6, (d)
October 9th  Mike Miller Equivariant Morse theory  (a) sections 1&2
October 16th Daniele Alessandrini Yang-Mills over a Riemann surface (a) sections 5&6
October 23rd Semon Rezchikov    
October 30th Yash Deshmukh    
November 6th Juan Muñoz    
November 13th Francesco Lin    
November 20th      
November 27th      
December 4th      
December 11th      
       

 

Abstracts

September 25th, Aleksander Doan

The Kempf-Ness theorem

An introduction to GIT and symplectic quotients, with a view towards the correspondence between Yang-Mills connections and stable bundles on Riemann surfaces
 
 

October 9th, Mike Miller

Equivariant Morse theory

Morse theory is (among many other things) a method to obtain constraints on the homology of a manifold based on the number of critical points and their indices of a smooth function, or more generally based on the topology of critical submanifolds. In some cases (when the Morse function is "complete"), the topology of the critical set describes the homology of the manifold without any additional work. In this talk, we will describe the equivariant analogue of this story, where complete Morse functions sometimes come for free based on group-cohomology considerations, and explain how Atiyah and Bott related these ideas to the study of the space of holomorphic bundles on a Riemann surface.

 

October 16th, Daniele Alessandrini 

Yang-Mills over a Riemann surface

When the Yang-Mills equations are considered on a Riemannian
manifold of dimension 2 (a surface), they become closely related with
the complex geometry of the conformal structure of the surface. We
will discuss the special properties of Yang-Mills connections on
surfaces, and how they are related with representations of a central
extension of the fundamental group of the surface. This is based on
section 5 and 6 of the Atiyah-Bott paper.