Instructor: Peter Teichner

Lectures: We 2:00-3:30, 891 Evans

Course Control Number: 55027

Recommended Reading: Papers by Segal: QFT.pdf, Symplectic.pdf, Stanford.pdf, Locality.pdf.  Papers by Witten: Elliptic Genera.pdf, TQFT.pdf

Papers by Stolz-Teichner: Elliptic-Objects.pdf, Survey.pdf

Schedule and References:

Sept. 3 Chris Schommer-Pries, Chris1.pdf

Optional Pre-reading:

(1) Review the usual non-topological group cohomology and especially the interpretation in degrees 0,1,2. Also helpful would be to review sheaf cohomology and the relationship between sheaf cohomology and Cech cohomology. Good references are probably:

Charles A. Weibel. An introduction to homological algebra.

Kenneth S. Brown. Cohomology of Groups.

(2) If you want to read about Segal's version of group cohomology, it appears here: Graeme Segal. Cohomology of topological groups. In Symposia Mathematica, Vol. IV (INDAM, Rome, 1968/69), pages 377–387. Academic Press, London, 1970.

Sept. 17 Chris Schommer-Pries, Chris2.pdf

Optional Pre-reading:

(0) Bicategories. It is just a fact of life that a modern mathematician has to learn to work with bicategories. I leave it up to you to find a reference you consider good, although a succinct one is the following: Tom Leinster, Basic-Bicategories.pdf

(1) It would be useful to read about 2-groups and the Baez-Crans-Schreiber-Stevenson model of the String group. 

John C. Baez and Aaron D. Lauda. Higher-dimensional algebra. V. 2-groups. 

Theory Appl. Categ., 12:423–491 (electronic), 2004. arXiv:math/0307.5200.

John C. Baez, Danny Stevenson, Alissa S. Crans, and Urs Schreiber. From 

loop groups to 2-groups. Homology, Homotopy Appl., 9(2):101–135, 2007. 

arXiv:math/0504.5123

Also, on general principle, Peter's students should read about the Stolz-Teichner model of the String group in section 5 (especially 5.4) of the Elliptic Objects paper.

(2) I don't particularly like most of the presentations of bibundle bicategory that I've seen. They are sometimes from the wrong point of view and sometimes erroneous. Section 6 of the following has all the definitions.

N. P. Landsman. Bicategories.pdf of operator algebras and Poisson manifolds. In 

Mathematical physics in mathematics and physics (Siena, 2000), volume 30 of 

Fields Inst. Commun., pages 271–286. Amer. Math. Soc., Providence, RI, 2001

I also like the following recent paper. It does a good job surveying/comparing various localizations of the category of Lie groupoids and motivating why one should use the bibundle bicategory instead.

Eugene Lerman. Orbifolds as stacks? 2008. arXiv:0806.4160v1.

(3) The classical Dold-Kan correspondence. Review how this works and what the functors are as presented in section 8.4 of:

Charles A. Weibel. An introduction to homological algebra, volume 38 of Cambridge 

Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1994.

(4) Simplicial Spaces and classifying spaces. If you've never seen it, it is worth looking at Segal classic paper:

Graeme Segal. Classifying-spaces .pdf and spectral sequences. Inst. Hautes  Etudes Sci. 

Publ. Math., (34):105–112, 1968.

And if you still haven't had enough take a look at:

John C. Baez and Danny Stevenson. The classifying space of a topological 2-group. 

2008. arXiv:math/0801.3843

Sept. 24 Chris Schommer-Pries, Chris3.pdf

Oct. 1 Christian Zickert, Christian.pdf

Oct. 8 Kevin Walker, Kevin1.pdf

Oct. 15 Kevin Walker, Part 2

Oct. 29 Chris Schommer-Pries, Chris4.pdf

Nov. 5 Dmitri Pavlov, A product theorem for Lp-spaces, dmitri1.pdf

Nov. 12 Dmitri Pavlov, A product theorem for Lp-spaces, Part 2

Nov. 19 Dmitri Pavlov, A product theorem for Lp-spaces, Part 3