M392C: Topics in Geometry and Quantum Physics
Announcements
Basic Information
Professor: Dan Freed, RLM 9.162
Class Meetings: TTh 11:00-12:30, RLM
10.176
Office Hours: To be announced
This is the first semester of a year-long topics course. I will treat topics
from geometry and topology which are relevant for quantum field theory and
string theory. The emphasis is on the mathematics, though I will also
explain some basics about the physics as well. The course is designed to be
of interest to all geometry students.
Roughly, the first semester will cover geometry which enters into
semi-classical descriptions of quantum systems. These include fiber bundles
and connections, symplectic and Poisson geometry, Riemannian geometry, spin
geometry, etc. Lectures will introduce the basic ideas and prove some
theorems about them in geometry as well as discuss their use in the physics.
The second semester will cover topics more directly relevant to quantum
theory, so perhaps more functional analysis and algebraic topology.
The formal work for the course is a term paper each semester.
Summer Reading
Some students have asked what they can do to prepare for the course. I
strongly recommend strong grounding in basics smooth manifolds, particularly
calculus on manifolds. This includes some topics beyond the prelim class,
such as Lie derivatives (including forms), the Frobenius theorem, basics
about Lie groups, etc. Here are some suggested texts:
Volume 1 of Michael Spivak's "Comprehensive Introduction to Differential
Geometry"
Chapters 1-4 of Frank Warner's "Foundations of Differentiable
Manifolds and Lie Groups"
Jack Lee's
book on smooth manifolds
Notes on smooth manifolds by Nigel Hitchin:
1
2
3
4
Appendix
If you'd like to go further, you can learn some algebraic topology. The
classic "Differential Forms in Algebraic Topology" by Bott and Tu is highly
recommended. You should also know some algebraic topology from a more
traditional point of view, as
in Hatcher's
book.
Finally, if you'd like to read up on the physics background I recommend
learning about special relativity and electromagnetism. The Feynman Lectures
on Physics make great reading, including the third volume on quantum
mechanics.
Please do not feel you need to read all of this to attend and follow the
lectures! These are suggestions. Also, it will be more fun if you form
study groups with other students; email
me if you need help finding other students registered for the
course.
Have a great summer!