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
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
(and hyperelliptic) curves.
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.
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.
Worksheet on psi classes
Also, here are
some notes from a mini-course at University of Utah that might be useful:
J.Kock, I. Vainsencher. An Invitation to Quantum Cohomology.
J.Harris, I.Morrison. The
moduli space of Curves.
Sheldon Katz, Albrecht Klemm, Rahul Pandharipande, Richard Thomas, Ravi Vakil, Eric Zaslow. Mirror Symetry.
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.
N. Proudfoot. Geometric Invariant Theory and Projective Toric Varieties.
S. Keel. Intersection Theory of Moduli Space of Stable N-Pointed Curves
of Genus Zero.