The University of Auckland, 26–29 September 2017
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Overview
This will be a series of lectures on the geometry and topology of the Yang-Mills equations on a Riemann surface, aimed at advanced honours and graduate students.
Programme
- Lecture 1 (Tuesday 26 September, 3-4pm): Holomorphic vector bundles on a Riemann surface
- Abstract: This will be an introductory lecture covering the basic definitions concerning Riemann surfaces, holomorphic vector bundles, connections and gauge transformations.
- Lecture 2 (Wednesday 27 September, 3-4pm): The Narasimhan-Seshadri theorem
- Abstract: In this lecture I will discuss the celebrated theorem of Narasimhan-Seshadri identifying the moduli space of (projectively) flat unitary connections on a Riemann surface with the moduli space of semistable holomorphic vector bundles. I will also discuss Donaldson's gauge theoretic proof of this theorem.
- Lecture 3 (Thursday 28 September, 1-2pm): The Yang-Mills equations on a Riemann surface
- Abstract: In this and the following lecture I will discuss the famous paper of Atiyah and Bott "The Yang-Mills Equations over Riemann Surfaces", in which the cohomology of moduli spaces of holomorphic vector bundles are computed using ideas from physics (Yang-Mills theory) and infinite dimensional Morse theory.
- Lecture 4 (Friday 29 September, 1:30-2:30pm): The Atiyah-Bott recursive formula
- Abstract: Continuing from the previous lecture, I will aim to cover the recursive formula of Atiyah and Bott for the Betti numbers of the moduli spaces.
Background reading
1. M. F. Atiyah, R. Bott, The Yang-Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. London Ser. A 308 (1983), no. 1505, 523-615. PDF
2. S. K. Donaldson, A new proof of a theorem of Narasimhan and Seshadri, J. Differential Geom. 18 (1983), no. 2, 269-277. PDF