Math 541 : Introduction to Mathematical Physics - Fall 2008
>>>>>>>>>>>>Problem session : Monday at 5:00 in Math 402 <<<<<<<<<<<<<<
Meeting time and place: The course meets Tu/Th 11:00 in
Math East 246
Instructor:
Tom Kennedy
Course notes
The dates are the dates they were posted
so you can tell if they have been updated since you last downloaded
them.
Notes from Werner's Park City lectures on percolation
nn
Applets, pictures
Ising model
applet
Course description
Physics provides a wealth of interesting mathematical models,
conjectures about these models, and even ideas for proving the
conjectures. The goal of this course is to introduce some of the
important models, conjectures and theorems
that have come from classical statistical mechanics.
Particular models will include the
Ising model (the prototypcial
model for magnetic phase transitions),
percolation (originally introduced to model connectivity properties
of random media),
and various random walks including self-avoiding walks
(which are a model for polymers in solution).
Physically, all of these models exhibit phase transitions in
which an infinitesimal change in a parameter (e.g., temperature)
produces a global qualitative change in the system.
An important idea from physics is ``universality'' which says that
many features of these phase transitions do not depend on the
microscopic details of the model but only on some qualitative
features of the model. Physicists's understanding of this universality
is based on a set of ideas known as the renormalization group (RG).
We will study this set of ideas in several concrete settings.
If time permits we will also study how one does Monte Carlo
simulations of these models and a rigorous expansion technique
(polymer expansions).
Math 541 may be taken several times for credit since the
topics of this course typically change from year to year.
In particular, the topics of Math 541 this coming Fall will have little
overlap with the topics covered the last two times the course was offered
(Fall '07 and Fall '05). So students who have taken the course
before are encouraged to take it again!
Tentative Syllabus
1. Ising model : def and existence of a phase transition
2. Percolation : def and existence of a phase transition
3. Renormalization group ("real space transformations") for the Ising model
4. Random walk models
a. Ordinary random walk, Brownian motion and the invariance principle
b. A renormalization group approach to the central limit theorem
c. Self avoiding random walks
d. A renormalization group approach to self-avoiding walks
5. Gaussian processes and fields
6. Lattice models with continuous variables : XY and Heisenberg models
a. role of symmetry in phase transitions - Mermin Wagner theorem
b. effect of symmetry on critical phenomena
7. Euclidean field theory as classical statistical mechanics
a. Free field theories as Gaussian processes, interacting field theories as ?
b. Renormalization group for field theories
8. Monte Carlo simulation of these models
a. Ising model : slow methods (Glauber) and
fast methods (Swendsen-Wang)
b. Self-avoiding walk
9. Polymer or cluster expansions
Prerequisites:
This course is primarily intended for graduate students in the math, applied
math or physics programs. An undergrad course in probability would
be helpful, but is by no means essential. Strong undergrads who are
interested in the course should talk to the instructor.