Summary: Math 845, Spring 2009, Topics in Mathematical Physics
Instructor: Abdelmalek Abdesselam, email@example.com. Phone: 434-924-
4926. Office: KER 227.
Location and times: MWF from 10 am to 10:50 am, in KER 326.
Office hours: MW from 1pm to 2pm and by appointment.
Prerequisites: Math 731 or instructor permission. Recommended but not re-
quired: Math 732, Math 745.
Topics: The main object of study in this course are functional integrals, i.e., mea-
sures on spaces of functions or distributions (Sobolev spaces, Schwartz spaces)
which arise in modern statistical mechanics and quantum field theory. Such
measures also arise as invariant measures for some partial differential equations,
i.e., infinite-dimensional dynamical systems. The goal of this course is to provide
you with the essential tools needed in order to tackle research problems in the
area. These are Feynman diagrams, the cluster and Mayer expansions, pertur-
bative renormalization and the rigorous theory of the renormalization group. A
typical example of measure one would like to study is that of a random function
q : Rd
R with a probability density distribution of the form exp(-V (q))Dq.
Here Dq should be thought of as the Lebesgue measure on the function space to
which q belongs. V (q) is the integral dx over Rd