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  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 . 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|