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).

- (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

## 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 | |||

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

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.

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.