# Moduli Spaces in Representation Theory - Math 797RM

**Instructors: Chris Elliott and Owen Gwilliam**

### Course Description

This course will be a guided tour of moduli spaces that have played a central role in topology, differential geometry, and representation theory. The emphasis will be on explicit examples rather than theory.

The approach will be a hybrid between a lecture class and a seminar. For the first few weeks, we will give lectures laying foundations and introducing terminology. But we will also have you form small groups, focused around a common interest, in which you will read a key text (e.g., a paper by Atiyah) and work together to master it. In the latter part of the course, each group will give presentations, so you get a chance to practice speaking and so that we can all benefit from what your group has learned. We expect the class to be highly interactive and dynamic, and we will create a welcoming atmosphere to discuss confusions and to grow as mathematicians.

We will start out with the following:

- A gentle introduction to the “functor of points” approach to geometry
- Basic examples of moduli spaces: Grassmannians, flag varieties, configuration spaces of points.

Later topics may include:

- Moduli of vector bundles and G-bundles, the Harder-Narasimhan filtration.
- Quivers, representations of quivers, the path algebra.

### Prerequisites

Basic notions of differential geometry and algebra.

### Possible Topics for Exploration

The following list of topics is only a suggestion, other topics that the participants are interested in are also welcome.

- Stability of holomorphic vector bundles, including some famous comparison theorems (e.g. Donaldson-Uhlenbeck-Yau).
- Symplectic reduction and moment maps.
- Geometric invariant theory.
- Instantons and twistors in Yang-Mills theory.
- Higgs bundles and the Hitchin integrable system.
- Quiver varieties.
- Hilbert schemes of points.
- Applications to physics, e.g. Seiberg-Witten theory.

### Contact

Chris Elliott: celliott@math.umass.eduOwen Gwilliam: gwilliam@math.umass.edu

Last modified Wednesday October 28th, 2020.