Timeline for What are some interesting problems in the intersection of Algebraic Number Theory and Algebraic Topology?
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 1, 2012 at 2:27 | comment | added | Tyler Lawson | Paul Siegel is correct; the connection originally came through the Witten genus, and the fact that it was was supposed to associate modular forms to certain manifolds. This led into the study of elliptic genera, and their associated lifts to elliptic cohomology theories. | |
| Jul 29, 2012 at 16:51 | history | made wiki | Post Made Community Wiki by François G. Dorais | ||
| Jul 28, 2012 at 1:48 | comment | added | Paul Siegel | ... While nobody can make rigorous sense of this interpretation, it is at least known that the Witten genus plays the same role in TMF that the $\hat{A}$ genus plays in K-theory. | |
| Jul 28, 2012 at 1:44 | comment | added | Paul Siegel | @Yosemite Sam: I think the connection has to do with the so-called "Witten genus". Physicists - motivated I think by the path integral formulation of quantum field theory - became interested in the geometry and topology of loop spaces of manifolds. The Witten genus is an invariant which is supposed to be the loop space counterpart of the $\hat{A}$ genus in ordinary topology in the sense that it would probably be the index of the Dirac operator on a loop space if such a thing were known to exist... | |
| Jul 27, 2012 at 23:57 | comment | added | Yosemite Sam | could you perhaps expand a bit more on (or suggest a reference for) the connections to physics? | |
| Jul 27, 2012 at 7:22 | comment | added | Mark Grant | This is a great answer! The only thing missing is a precise definition of the field of study "chromatic homotopy theory". Could you say, for example, exactly which objects are studied there? Or does it refer more to the technique of working a prime at a time? | |
| Jul 27, 2012 at 6:22 | history | answered | Tyler Lawson | CC BY-SA 3.0 |