This is a page of information about the reading group for the Atiyah and Bott paper "The Yang-Mills Equations over Riemann Surfaces".

The reading group grew out of the PhD reading groups run through the Heriot-Watt Mathematical Physics group which most of the participants were members of.

We will decide on a reasonable number of pages to read each week, probably content dependent but likely to be under 10.

The participants include: Calum Ross (me) UCC, Lukas Müller MPI Bonn, Lennart Schmidt NUS, Vincenzo Marotta Heriot-Watt, Grigorios Giotopoulos HW, Bruno Barton-Singer HW, Juan Carlos Morales Parra HW, Corina Keller University of Monpellier, Luuk Stehouwer MPI Bonn, and others to be added.

The page here, is for a course which covered some of the same details and may be a useful resource and source of further references.

Date | Pages of [1] that we read | Presenter | Summary of what we discussed |
---|---|---|---|

22/10/20 | N/A | N/A | We had a brief chat about how we would proceed with the reading group. It was a housekeeping session where we discussed how to proceed. We have decided to aim for 1 hour sessions, 12:00-13:00 UK time (BST this week and GMT in future weeks), with ~30 minutes for a presentation and ~30 minutes for a discussion. The plan is for the presenter to change each week though there is no obligation to volunteer. I (Calum) will send round a PDF of the paper so that we all have a copy where the page numbers match up. |

29/10/20 | Introduction: pg 524 to 528 (pg 3 to pg 7 of the PDF) | Calum | We discussed the introduction and tried to get a feel for the spaces that we will study in the paper. The notes that I presented are uploaded here. |

5/11/20 | Section 1: Morse Theory: pg 528 to 533 (pg 7 to 12 of the PDF) | Bruno | Bruno gave a nice introduction to Morse theory and we spent a bit of time trying to understand the case of critical surfaces. We touched on equivariant Morse theory but decided to come back to that next week. |

12/11/20 | Section 1: Morse Theory: pg 528 to 534 (pg 7 to 13 of the PDF) | Bruno | We started by discussing the completion principle for critical submanifolds before moving on to equivariant Morse theory. Bruno presented the two examples: The height function on a sphere with a U(1) action, and the "mod" height function on the sphere with a U(1) action. |

19/11/20 | Section 13: Equivariant cohomology: pg 604 to 606 (pg 83 to 85 of the PDF) | Lukas | Lukas gave an intrduction to equivaraint cohomology following chapter 12 of [1]. The notes that he covered are uploaded here. |

26/11/20 | Section 1: Equivariant Morse theory: pg 534 to 537 (pg 13 to 16 of the PDF) | Calum | I attempted to discuss the Morse stratification and the construction of the stable and unstable manifolds of a critical point. The notes that I presented are uploaded here . |

3/12/20 | Section 1: Equivariant Morse theory pg 537 to pg 539 (pg 16 to pg 18) | Calum | I finished presenting the material on the Morse strata. The notes that I used have been updated here. |

10/12/20 | Section 2: Topology of gauge group pg 539 to 542 (pg 18 to 21 of the PDF) | Luuk | Luuk nicely summarised the materal from the paper and gave some useful background material on the tools being used behined the scenes to establish the results. |

17/12/20 | Section 2: Topology of gauge group pg 543 to 545 (pg 22 to 24 of the PDF) | Luuk | Luuk gave a nice introduction to complex K-theory. For those of us with less experience with K-theory this was a chance to see the prerequisites for understanding this section of the paper. I think that Luuk is going to send his notes round via email. |

7/1/21 | Section 2: The Yang-MillsTopology of gauge group pg 543 to 545 (pg 22 to 24 of the PDF) | Luuk | After a break for Christmas and New Year we will reconvene to finish up reading section 2. Luuk will return to finish the dicussion of Section 2. |

14/1/21 | Section 3:The Yang-Mills Functional | Grigorios | TBC |

21/1/21 | Section 4:The Yang-Mills equations | Grigorios | TBC |

28/1/21 | Section 5: Yang-Mills over a Riemann surface | Lennart | TBC |

4/2/21 | Section 5: Yang-Mills over a Riemann surface cont: | Lennart | TBC |

11/2/21 | Section 6: Representations of the Fundamental group cont: | Lukas and Luuk | TBC |

18/2/21 | Section 6: Representations of the Fundamental group cont: | N/A general discussion | TBC |

25/2/21 | Section 7: Holomorphic Vector bundles | Lennart | TBC |

4/3/21 | Section 7: Holomorphic Vector bundles cont: | Lennart | TBC |

11/3/21 | Section 8: Relation with Yang-Mills: | TBC | TBC |

- [1]: The Yang-Mills Equations over Riemann Surfaces, M. Atiyah and R. Bott ” Phil. Trans. R. Soc. Lond. A. 308 (1983), 523–615.