Math 585, Topics in Mathematical Physics, Fall 2006

Instructor: Shannon Starr

Title: The Mathematics of Mean Field Spin Glasses and the Replica Method

Time/Place: HYLAN 206, 12:30--1:45 Tue, Thu

Office Hours: 11:05--12:20 (and by appointment) HYLAN 1017

Preliminary Description: Math585.txt

Grading policies: The grading will be relaxed. If you attend regularly, you will receive an "A". Any student who wants to may present a talk at the end of the semester, but it will not be mandatory. If you decide (around the middle of the semester) that you would like to present a talk, then I will be very happy to discuss topics with you and/or help you with finding and reading a paper.

Goals: The topic of mean-field spin glasses is less specialized than it seems. One aspect of the mathematical theory is that it gives us an opportunity to learn fundamental ideas in probability and analysis, and then use them on a real example. Some of these ideas are: large deviation principle; exchangeable random variables and de Finetti's theorem; concentration of measure; extreme value statistics.

Lecture Notes: Lecture2.pdf, Lecture3b.pdf, Add-Entropy.pdf, Lecture4.pdf, Lecture5.pdf, Lecture6.pdf, Lecture7.pdf, Lecture8.pdf, Lecture9.pdf, Lecture10.pdf, Lecture11.pdf, Lecture12b.pdf, Lecture13.pdf.

Other online references: These should only be viewed as supplemental reading. The lecture notes will be self-contained. The lecture notes will contain a fuller list of citations, for example to papers.

- See the notes/reviews prepared by Michel Talagrand prior to his book: http://www.proba.jussieu.fr/users/talagran/expository/index.html
- See the notes prepared by Anton Bovier prior to his book: http://www.wias-berlin.de/people/bovier/files/erlangen.pdf
- Here is a copy of David Aldous and J. Michael Steele's recent review of their ``Objective Method'': Aldous-Steele.pdf. This file was freely available on the internet at some time. I will remove it if asked by Aldous, Steele or a representative of Springer-Verlag, where it is now published in ``Encyclopedia of Mathematical Sciences, Volume 110. Probability Theorem I.'' A.-S. Sznitman, S.R.S. Varadhan subseries editors. ``Probability on Discrete Structures'', Pages 1--72 (2004).

Other offline references. I own the following books. Please feel free to borrow them from me whenever you like:

- Talagrand's book for sale: http://www.springer.com/west/home/generic/order?SGWID=4-40110-22-2302936-0. Incidentally, there is a nice book review (especially for graduate students) by David Aldous here: http://www.stat.berkeley.edu/~aldous/Misc/talagrand.pdf#search=%22Michel%20Talagrand%20Spin%20Glasses%20a%20Challenge%22
- Bovier's book for sale: http://www.cambridge.org/uk/catalogue/catalogue.asp?isbn=0521849918
- The classic by Mezard, Parisi and Virasoro for sale: http://www.worldscibooks.com/physics/0271.html