All right. I've slowly come to the conclusion that the paper(adsbygoogle = window.adsbygoogle || []).push({});

A Survey of Elliptic Cohomology

J. Lurie

www.math.harvard.edu/~lurie

mentioned on various physics blogs already, along with anything else Lurie might have written, is going to be well worthwhile spending time on. Lurie is undoubtedly one of those very rare young wizards that find their natural home surfing the waves on the deserted beach of future mathematics. Funnily enough, he was mentioned to me only recently by a very respectable older mathematician whom Lurie had managed to impress. I'd never heard of him.

It would be nice to leave him to it, but it looks like he's quite keen on things like higher dimensional toposes. This is frustrating, because he's really a homotopy theorist and treats categories accordingly.

To put the reader in the picture: Urs Schreiber at http://golem.ph.utexas.edu/string/index.shtml has been talking quite a bit lately about Elliptic Cohomology. This is because the non-perturbative Stringy people, such as the authors of

Type II string theory and modularity

Igor Kriz, Hisham Sati

http://arxiv.org/abs/hep-th/0501060

have become interested inrefining the K-theory partition function in type II string theories using elliptic cohomology(to quote the abstract). It turns out that to be 'anomaly-free' one needs a pretty fancy type of cohomology called TMF (topological modular forms). This is reviewed in

Algebraic topology and modular forms

Michael J. Hopkins

http://arxiv.org/abs/math/0212397

but the more mathematically inclined should look in Lurie's paper to see how it arises from the search for some kind of universal elliptic theory - and universal is meant in the categorical sense. This is all in part 1 of the paper. By part 2 we're thrown into Derived Algebraic Geometry, which is what Lurie'sthesisis on. He wants a G-equivariant type of general cohomology theory. Recall that ordinary G-equivariant cohomology came crashing together with quantum field theories when Witten looked at 2D Yang-Mills (in particular) path integrals in such terms.

Back later

Kea

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# Elliptic Cohomology making a splash

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