Office: MSB M312 Phone: (860)
486-3850 e-mail: kaufmann@math.uconn.edu
Office hours: TuTh
Topics in Geometry and Topology
I
TuTh
TuTh
Syllabus
Short description: The
course will cover several areas of topology and geometry which are also
related
to physics. The topics will include: graphs, operads, co-bordisms and
genera,
Hopf algebras and possibly gauge theory. We will start with co-bordisms
and
genera and then pass on to discuss their relationship to so-called
topological
field theories and Frobenius algebras. We will then concentrate on
operads and
study examples of these based on graphs. These topics are of
independent
mathematical interest, but they are also key in a modern mathematical
formulation
of quantum field theory and provide a basis for research in this area.
After
this we will turn to Hopf algebras. These algebras turn up when
regarding
certain topological spaces (hence the name) and have been instrumental
in the
study of renormalization. Lastly, time permitting we will turn to gauge
theory.
Audience: Graduates with basic knowledge about topology,
geometry
and algebra. Basically you should know what a topological space is, and
have
some familiarity with rings and algebras. It is a plus if you know what
a
manifold is.
News:
The first Homework is up
Course progression:
I. Cobordism rings and
genera
The
cobordism ring
Characteristic
classes
Genera:
The definition and relations to power series
Elliptic
genera
Fiber
bundles (a short review)
The
Atiyah-Singer index theorem
Equivariant
signature of loop space and the Witten Genus.
II. Cobordism category
and TFT
Functors
and equivalences of categories, braided and tensor categories
TFT, co-bordims and Frobenius algebras
2-categories, boundary TFT and
characters
III. Hopf algebras
Bialgebras, Anti-podes
Milnor-Moore theorem
IV.
Operads
Functors and Operads
Little discs, framed little discs and
their algebras
The Hopf algebra of an operad and
renormalization
Formality, Deligne's conjecture and
deformation quantization.
Homework
References
II. Cobordism category
and TFT :