Renzo's Math 676

Math 676: Moduli Spaces

WHEN: MWF 12 – 12:50 pm

WHERE: ENGRG E 103 // in the oval with good weather

WHAT: Some good fun with moduli spaces

Moduli spaces are ubiquitous and overwhelmingly important objects in mathematics. Simply because when we have collections of mathematical objects that are similar, we just can’t content of looking at them one-by-one. We want to somehow see the complete collection at once with some sort of structure that tells us when the objects are more or less similar to each other. A moduli space achieves such a goal.

In this course, I would like to make a wide exploration of moduli spaces, with the goal of addressing these four main philosophical questions:

1. What are moduli spaces and how should we think of them?

2. How do you construct a moduli space?

3. How do you compactify a moduli space?

4. How do you work with moduli spaces?

The course will be driven by concrete examples, and weave in and out of these four points in such a way that hopefully, by the end of the semester, we will have a good feeling for what the answers are. Some of the characters that will appear (most of them somewhat superficially) are:

· Projective spaces.

· Elliptic (and hyperelliptic) curves.

· Grassmannians.

· Jacobians.

· Rational pointed curves.

· Riemann Surfaces.

· Toric Varieties.

· Quiver Representations.

· Hilbert Schemes.

While all moduli spaces are somewhat geometric (they are spaces after all!), they are not at all exclusive domain of algebraic geometry! They naturally feature in many other disciplines. Looking at the list above you might recognize some of your favorite analytic, arithmetic or combinatorial objects. I will try to keep a fairly ecumenical point of view, that seems to me to be the most likely to be beneficial to a somewhat diverse crowd as we’ll be.

Ideally the beginning of the course will be soft (I would like to assume very little previous knowledge), and the ending pretty steep (I would like to discuss some current research topics). Hopefully you will help me with adjusting the slope in between.

NOTES:

I will give a good effort at trying to keep my notes updated here. You are welcome to use them, and you are even more welcome to contribute to them, by adding/ editing/integrating/modifying them etc. Just let me know if you want to mess with them and we’ll set up a way to make that easy and natural.

Notes

Worksheet on psi classes

Also, here are some notes from a mini-course at University of Utah that might be useful:

Utah minicourse

GOOD BOOKS:

· J.Kock, I. Vainsencher. An Invitation to Quantum Cohomology.

· J.Harris, I.Morrison. The moduli space of Curves.

· Kentaro Hori, Sheldon Katz, Albrecht Klemm, Rahul Pandharipande, Richard Thomas, Ravi Vakil, Eric Zaslow. Mirror Symetry.

GOOD NOTES:

· I. Coskun. Birational Geometry of Moduli Spaces.

· A. Craw. Quiver Representations in Toric Geometry

· R. Thomas. Notes on GIT and Symplectic Reduction for Bundles and Varieties.