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September 6, 2009

A Seminar on a Survey of Elliptic Cohomology

Posted by Urs Schreiber

My position in Hamburg is about to terminate, currently I am based at the MPI Bonn and about to start a postdoc position in Ieke Moerdijk’s group in Utrecht.

In Bonn we are preparing for a program by Stephan Stolz and Peter Teichner on, you know, QFT, cohomology and such things. There are lots of students of them around and they are starting to run a

Seminar on Elliptic Cohomology

This is supposed to essentially go through, guess what, Jacob Lurie’s notes. On that occasion I started a corresponding nnLab entry

nnLab:A Survey of Elliptic Cohomology

So far this contains just a table of contents, a summary of one of the main ideas and a few links, as far as existent. I am planning to expand this entry as we go along in the Seminar, and hopefully others will join in, too.

Posted at September 6, 2009 2:23 PM UTC

TrackBack URL for this Entry:   http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/2052

14 Comments & 4 Trackbacks

cohomology theories

Here are the notes for today’s session:

A Survey of Elliptic Cohomology - cohomology theories.

This is a bit rough at the moment, handle with care.

Next time some of the things roughly indicated there will be discussed in more detail.

Go latest changes to see which parts of this is turning into separate entries.

(Help is appreciated. I don’t really have the time to wiki-fy all this nicely. But should be a start.)

Posted by: Urs Schreiber on September 7, 2009 6:30 PM | Permalink | Reply to this

Re: A Seminar on a Survey of Elliptic Cohomology

It might be worth noting (as Jacob did) that even theories are nice when evaluated on CP^\infty, because the Atiyah-Hirzebruch spectral sequence is then supported in even total degree, making all differentials trivial.

Posted by: Scott Carnahan on September 7, 2009 8:42 PM | Permalink | Reply to this

Re: A Seminar on a Survey of Elliptic Cohomology

Hi Scott!

Thanks. Please add that to the wiki!

It’s just as easy as dropping a comment here. Your comment here will eventually be washed away by the waves of time. On the wiki it will remain.

I am quite aware that there are loads of things that still ought to be mentioned at that wiki entry. Loads of things. The more people join in in adding stuff, the greater the chance is that we get something approximately comprehensive.

Posted by: Urs Schreiber on September 7, 2009 9:15 PM | Permalink | Reply to this

formal group laws in cohomology

While I was away at the Interactive Science Symposium unfortunately missing the second session of our seminar, Ryan Grady kindly wikified some notes:

Part II: A Survey of Elliptic Cohomology – formal groups and cohomology

Great, thanks!!

Everybody please feel free and encouraged to add/improve whatever still needs being added and improved.

(Hopefully somebody breaks down and creates the waiting Chern class and elliptic curve before we have these grayish unsatisfied links all over the place…)

Posted by: Urs Schreiber on September 10, 2009 6:58 PM | Permalink | Reply to this

elliptic curves

The remaining bit of the second session is now filled in in the second part of

A Survey of Elliptic Cohomology - formal groups and cohomology

starting with the bit on elliptic curves and formal group laws.

Notice the disclaimer at the beginning of the entry: this is still looking for somebody to take care of polishing it more properly.

Posted by: Urs Schreiber on September 14, 2009 6:51 PM | Permalink | Reply to this

E-infinity rings and derived schemes

The beginning of the material of the third and probably fourth session is appearing here:

A Survey of Elliptic Cohomology - E E_\infty-rings and derived schemes.

Posted by: Urs Schreiber on September 14, 2009 6:55 PM | Permalink | Reply to this

Re: E-infinity rings and derived schemes

Transcript of the second part of the sessions on E E_\infty-rings and derived schemes is now here:

A Survey of Elliptic Cohomology - E E_\infty-rings and derived schemes: part 2

This talks about (,1)(\infty,1)-categories, stable (,1)(\infty,1)-categories, symmetric monoidal (,1)(\infty,1)-categories, commutative monoids in symmetric monoidal (,1)(\infty,1)-categories and finally gives the coherent-homotopy-style definition of E E_\infty-ring.

This time most of the keywords that appeared already had their nnLab entries, so not that much new raw material. Though there might be some stuff that might usefully be added even to the existing entries.

There’ll be a break for a week now, as next weak we have the Stolz-Teichner school on the proof of how the partition function of a Euclidean supersymmetric 2d QFT is a topological modular form.

Posted by: Urs Schreiber on September 16, 2009 5:36 PM | Permalink | Reply to this

Re: E-infinity rings and derived schemes

The point of the summer school being run by Stolz and Teichner is to prove the following: The partition function of a supersymmetric Euclidean field theory is a (weakly) holomorphic integral modular function. This means that it is SL(2,Z) invariant and has at worst a pole at infinity. Further, the q-expansion has integer coefficients.

It should be noted that this is the degree 0 case of a theorem which says that the partition function of a SUSY EFT of degree -n is a weakly integral modular form of weight n/2.

Posted by: Ryan Grady on September 21, 2009 6:21 PM | Permalink | Reply to this
Read the post A Seminar on Gromov-Witten Invariants
Weblog: The n-Category Café
Excerpt: A mathematician's seminar on Gromov-Witten theory.
Tracked: September 18, 2009 4:56 PM
Read the post A Seminar on a Geometric Model for TMF
Weblog: The n-Category Café
Excerpt: A seminar on geometric models for tmf cohomology theory by Stephan Stolz and Peter Teichner.
Tracked: September 21, 2009 8:02 PM

Brave New Notes

The Stolz-Teichner-workshop is over and so we are back from geometric models to brave new formalism.

Today’s installment is here:

A Survey of Elliptic Cohomology – E E_\infty-rings and derived schemes – Part 3: brave new schemes.

Most of the actual material, though, was or is by now wikified and hence elsewhere. Go to nnLab: generalized scheme and see there the examples-section and follow links to your heart’s content.

Posted by: Urs Schreiber on September 28, 2009 7:17 PM | Permalink | Reply to this

elliptic curves

After all the abstract machinery, today a hands-on down-to-earth introductory talk on elliptic curves and their moduli spaces:

A Seminar on a Survey of Elliptic Cohomology – elliptic curves

I am dreaming that some kind soul will at some point enjoy going through this and increasing his or her nnLab page number by creating separate entries for the keywords appearing and copy-and-pasting the material there.

That’s anyway what I think the purpose of these typed up notes should be: raw material for further entries.

Posted by: Urs Schreiber on September 30, 2009 7:20 PM | Permalink | Reply to this

equivariant cohomology

today the seminar started going through the chapter on equivariant coohomology.

quick and rough notes are here:

A Survey of Elliptic Cohomology – equivariant cohomology

Posted by: Urs Schreiber on October 7, 2009 6:11 PM | Permalink | Reply to this

Re: equivariant cohomology

I see that Ryan Grady kindly filled in the material that I lost due to my browser dying at one point in the seminar.

Thanks!

Posted by: Urs Schreiber on October 8, 2009 1:46 PM | Permalink | Reply to this
Read the post Path-Structured Smooth (∞,1)-Toposes
Weblog: The n-Category Café
Excerpt: A talk on a context for higher synthetic differential geometry.
Tracked: October 13, 2009 1:59 AM
Read the post Notions of Space
Weblog: The n-Category Café
Excerpt: A survey of Jacob Lurie's "Structured Spaces".
Tracked: November 4, 2009 3:00 PM

Re: A Seminar on a Survey of Elliptic Cohomology

This seminar is now complete. All lecture notes have been posted and they could use some polishing if anyone is interested.

Posted by: Ryan Grady on December 15, 2009 5:10 PM | Permalink | Reply to this

Re: A Seminar on a Survey of Elliptic Cohomology

Thanks, Ryan, that’s awesome!!

I see that where I left off with the derived moduli stack of elliptic curves

you now added

towards a proof

and

compactifying the derived moduli stack

and

descent ss and coefficients.

Eventually somebody will go through this material and polish it further, I am sure. (For me, this will most likely happen when I get into a situation where I need this material for something, say for some presentation. )

Posted by: Urs Schreiber on December 15, 2009 5:48 PM | Permalink | Reply to this

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