Math 201a, Topics in Algebra: Moduli Theory of Representations

Course Information

First meeting:	Monday, January 13, 12:00PM-12:50PM Goldsmith Hall, Room 116	The weekly schedule will be discussed and set according to the preferences of people interested in the course, and then posted here.
Instructor:	Carl Wang Erickson	Office: Goldsmith 206
	cwe@brandeis.edu	Office Hours: TBD

Course Summary

The overall theme of the course is that homological data exert control over the geometry of a moduli space; we will focus on moduli spaces of representations. Each of these topics -- representations, moduli spaces, and homological algebra -- will be introduced at a pace calibrated to the background and interests of people in the course, which will be surveyed as we begin. I expect the first half-or-so of the course to consist of the introduction of these topics and the exploration some common first examples of homological control of geometry that will appear along the way. The rest of the course will explore concepts introduced in these two papers:

The deformation theory of representations of fundamental groups of compact Kähler manifolds by Goldman and Millson
Introduction to A_∞-algebras and modules by Bernhard Keller

The first will serve to explain how the local (infinitesimal) structure of a moduli space of representations is controlled by homological data, namely, a differential graded Lie algebra. The second will be used to show how some of the global structure of a moduli space of representations is controlled by homological data, namely, the A_∞-algebra structure of a certain Yoneda algebra.

Each student is asked to give a talk on a topic of their choice. Details and options for topics will be posted once student interests have been surveyed.

Syllabus

Full course information and a more detailed course description are available in the syllabus.

Resources

Representation Theory
- Elements of the Representation Theory of Associative Algebras by Assem, Simson, and Skowronski. The appendices contain a summary of category-theoretic terms we will use.
- Representations of quivers by Michael Brion.
Scheme theory, group actions, moduli theory
- Algebraic Geometry I by Ulrich Görtz and Torsten Wedhorn.
- Lectures on Algebraic Geometry I by Gunter Härder.
- Rational Points on Varieties by Bjorn Poonen. Chapters 2, 3, and 5 are helpful in supplementing a basic text on scheme theory.
- Geometric Invariant Theory by Mumford, Fogary, and Kirwan.
Homological algebra
- An Introduction to Group Cohomology by Romyar Sharifi
- See Appendix A.4 of the book of Assem, Simson, and Skowronski above.
- See Chapter 2 in the book of Gunter Härder above.
Local structure of moduli spaces of representations
- Deforming Galois Representations by Barry Mazur.
- The deformation theory of representations of fundamental groups of compact Kähler manifolds by Goldman and Millson
Global structure of moduli spaces of representations
- Moduli of representations of finite-dimensional algebras by Alastair King
- Introduction to A_∞-algebras and modules by Bernhard Keller

Return to Carl's webpage.