Prof. Paul Gunnells, LGRT 1115L, 545–6009, gunnells at math dot umass dot edu.
Tuesdays and Thursdays, 12–1, in LGRT 1115L.
Singular spaces arise naturally in many contexts, including algebraic geometry and representation theory. Any singular space has a decomposition into manifolds (a “stratification”), and so the study of them is a mixture of topology, geometry, and combinatorics.
The goal of this course is to present the basics of stratified spaces and to illustrate the general theory with some important examples. The main topics (time permitting) will be the following:
General material, including stratifications, Whitney conditions, local structures (tubes, normal slices, links), basic examples of naturally occuring singular spaces.
Isolated singularities of complex hypersurfaces (the Milnor fibration and related topics).
Intersection homology.
Compactifications of locally symmetric spaces.
Any one of these parts could form the basis of an arbitrarily long course. Hence we will not be able to go into details. My goal instead is to expose you to as much interesting geometry as I can, and to provide pointers to the literature.
There is no book that is really suitable for a textbook, although there are many useful references. The following is a list of materials I think could be interesting/helpful.
My TWIGS talk on stratified spaces.
John Mather’s notes. This is considered by some to be the definitive source for the details of stratified spaces.
Mark Goresky’s overview of the work of Thom and Mather.
Matthias Kreck’s Stratifolds. See especially the notes on Differential Algebraic Topology. I haven’t read this, but it looks interesting.
A Geometric Proof of Existence of Whitney Stratifications by V. Y. Kaloshin.
Stratified Morse Theory, by M. Goresky and R. MacPherson. This is a long book with a lot of stuff in it.
It includes a description of the basics of stratified spaces. Some of the results of this book have been
summarized in this.
Singular points of complex hypersurfaces, by John Milnor.
Singularities and Topology of Hypersurfaces, by Alexandru Dimca.
Walter Neumann’s survey on the topology of complex hypersurface singularities.
Anatomy of a singularity, by Liviu I. Nicolaescu. This summarizes most of the geometry of an isolated hypersurface singularity into a few pages.
Plane algebraic curves, by Egbert Brieskorn and Horst Knorrer.
Three-dimensional link theory and invariants of plane curve singularities, by David Eisenbud and Walter Neumann.
The story of the 120-cell, by John Stillwell. Explains many of the manifestations of the 120-cell and its relation to the Poincare homology sphere.
Intersection cohomology, edited by A. Borel.
An Introduction to Intersection Homology Theory, by F. Kirwan.
Intersection Homology & Perverse Sheaves with Applications to Singularities, by L. Maxim.
Intersection Homology Theory, by Goresky and MacPherson.
Intersection Homology II, by Goresky and MacPherson.
A course by Mark Goresky on derived category of sheaves, intersection homology, etc. Lecture 7, by the way, discusses Whitney stratifications.
Geometry of compactifications of locally symmetric spaces, by Lizhen Ji and Robert MacPherson.
Mark Goresky’s notes on the geometry of compactifications of modular varieties (a special class of locally symmetric spaces).
The grades for this course will be based on a final paper. This will be an expository article of no less than five and no more than ten pages that you will prepare on a topic related to material in the course and your own interests. My goal is to simulate as accurately as possible the experience of writing an original research paper. Here’s how it will work:
We will discuss possible topics individually, probably starting right after spring break. The goal is to let students write based on their own interests (e.g. analysis, geometry, topology, number theory, etc.).
You will research these topics outside of class and start to prepare a draft. You can ask me for advice if you need help. As an example of what kind of writing I have in mind, take a look at the singular spaces TWIGS talk. This is on the short side, since the talk was only an hour. It’s also somewhat informal. You could pretend that you have two hours to give a lecture, and write up notes from that.
A draft of your paper should be completed within three weeks (a deadline will be given later), which you will submit to me. I will read these drafts and make comments.
You’ll get the drafts back as soon as possible with comments, which you should not take personally. You will address the comments and prepare a new draft. I will give you even more nitpicky and annoying comments, and so forth. Experience shows that we will require a few iterations of this before everyone is happy. Then you will prepare and submit a final version.
Any kind of writing is challenging, and writing mathematics poses its own challenges. It’s rare for a graduate student to get detailed feedback on writing before the thesis, but I feel that such feedback would have been extremely helpful to me. This should be a good opportunity for you to get some.