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

- (f) Kirwan, The cohomology rings of moduli spaces of bundles over Riemann surfaces

- (g) Thaddeus, Conformal field theory and the cohomology of the moduli space of stable bundles

- (h) Narasimhan-Seshadri, Holomorphic vector bundles on a compact Riemann surface

## 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 | Stable bundles I | (a) sections 7&8 |

October 30th | Semon Rezchikov | Stable bundles II | (a) sections 7&8 |

November 6th | Yash Deshmukh | Donaldson's proof of the Narasimhan-Seshadri Theorem | (b) |

November 13th | Juan Muñoz | Cohomology of the moduli space of stable bundles over a Riemann surface | (a) section 9 |

November 20th | Francesco Lin | Holonomy and stable bundles | |

November 27th | Thanksgiving week! | No talk | |

December 4th | TBA | ||

December 11th | TBA | ||

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.

Stable Bundles and Yang Mills Connections

This is based on sections 7 and 8 of Atiyah-Bott. I will

explain what the Harder-Narasimhan filtration is, and use the HN

filtration on the space of connections prove some things about

semi-stable bundles, for example that the moduli space of semistable

bundles on a riemann surfaces of coprime rank and degree has no

torsion in its cohomology. I will then outline 'infinite dimensional

GIT' idea which is guiding Atiyah-Bott's computations.

We describe how to view the correspondence between stable holomorphic structures and unitary connections with constant central curvature on a unitary bundle over a riemann surface as the analogue of the Kempf-Ness theorem. We will then provide the details of Donaldson's proof of this correspondence. Finally we relate this to the original statement proved by Narasimhan and Seshadri, and mention some generalizations of the result if time permits.

This is based on section 9 of the Atiyah-Bott paper. The space of holomorphic structures on a complex vector bundle over a Riemann surface has an 'equivariantly perfect' stratification. We use this to obtain results about the integral cohomology of the moduli space of stable bundles over a Riemann surface. Namely, we will prove it is torsion-free, show how to compute the Betti numbers, and describe a set of multiplicative generators.

We'll discuss Thaddeus' approach to the computation of the cohomology of N_0(2,1) using the trace of holonomy around a fixed loop as a perfect Morse-Bott function.