Math 8390
Topics in Mathematical Physics
Time: 12:20pm-01:10 p.m. MWF
Room: VinH 206
Instructor: Sasha Voronov
Topics: operads, conformal field theory, graph homology, string
topology
In this one-semester course, I plan to give an introduction to operad
theory, aiming at recent developments in mathematics related to
quantum field theory (QFT), enhanced by applying operadic
methods. Examples of QFT's include conformal field theory (CFT),
topological CFT, string theory, and quantum gravity. The theory of
operads is an area of algebraic topology, which proved to be useful in
the study of loop spaces in the seventies and is now undergoing a
period of renaissance, mainly because of recent applications to
homotopy structures in algebra and topology, deformation quantization,
and theoretical physics.
You can see some lectures of a similar course I gave at M.I.T. posted
at
http://www.math.umn.edu/~voronov/18.276/index.html
The course at the U is revised to include graph homology and string
topology.
Prerequisites: Some familiarity with complex algebraic curves (or
Riemann surfaces) and basic homology theory will be helpful. No
knowledge of physics or operads is required - all necessary notions
will be introduced along the way.
Outline of the course: Introduction to operads; operads related to
moduli spaces of algebraic curves; cyclic and modular operads and
PROP's; the little disks operad; the Fulton-MacPherson
compactification; algebras related to moduli spaces; homotopy
algebras, including A_infty-, L_infty-, and G_infty-algebras;
deformation quantization; introduction to CFT's, modular functors, and
other 2d QFT's, such as topological QFT's and CFT's and quantum
gravity; graph homology; string topology.