Topics in Geometry and Topology I 337

Ralph M. Kaufmann

Office: MSB M312    Phone: (860) 486-3850     e-mail:
Office hours:  TuTh
2:30-3:30 pm and by appointment

Topics in Geometry and Topology I

TuTh TuTh  12:30 - 1:45 PM  MSB 307 (! new room)

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

                        Basic defnitions and constructions
                        Functors and Operads
                        Little discs, framed little discs and their algebras

                        Cyclic operads and correlation functions
                        The Hopf algebra of an operad and renormalization
                        Formality, Deligne's conjecture and deformation quantization.








Oct 26






I. Cobordism rings and genera :
1.   Hirzebruch, Friedrich. “Topological methods in algebraic geometry”  Classics in Mathematics. Springer-Verlag, Berlin, 1995
2.  Hirzebruch, Friedrich; Berger, Thomas; Jung, Rainer “Manifolds and modular forms”. Aspects of Mathematics, E20. Friedr. Vieweg & Sohn, Braunschweig, 1992.

II. Cobordism category and TFT :

1. Gelfand, Sergei I.; Manin, Yuri I. Methods of homological algebra. Second edition. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2003. xx+372 pp.
2. Kassel, Christian Quantum groups. Graduate Texts in Mathematics, 155. Springer-Verlag, New York, 1995. xii+531 pp
3. Atiyah, Michael Topological quantum field theories. Inst. Hautes Études Sci. Publ. Math. No. 68 (1988), 175--186 (1989).  (pdf)
III Hopf algebras
1. Kassel, Christian Quantum groups. Graduate Texts in Mathematics, 155. Springer-Verlag, New York, 1995. xii+531 pp
2. Montgomery, Susan. Hopf algebras and their actions on rings. CBMS Regional Conference Series in Mathematics, 82. American Mathematical Society, Providence, RI, 1993. xiv+238 pp.
IV. Operads
1.Markl, Martin; Shnider, Steve; Stasheff, Jim Operads in algebra, topology and physics. Mathematical Surveys and Monographs, 96. American Mathematical Society, Providence, RI, 2002. x+349 pp.
2. Kaurmann, R, Operads, Stings and Deligne's conjecture.