Peter Teichner

Math 270, Hot Topics Course
Fall 2006, Th 2:00-3:30 in 2 Evans

The idea of this course is to bring together different groups in the department, consisting of faculty, postdocs, graduate students and advanced undergraduate students, and together learn a current research topic of broad interest.

The first two lectures will be given by the instructor: First an introduction to, and motivation for, the current topic, with lots of background information. In the second meeting, the material will be explained more carefully, and a detailed plan for the semester will be designed. Participants will then pick a topic to present during the semester, roughly in the order of their mathematical age. Each talk should be designed to last about one hour so that there is sufficient time for discussion during and after the talk.

Derived algebraic geometry and topology

Additional Lecture this week:
Tuesday, Sept. 26, 2:10-3:40 in 939 Evans

List of dates, speakers, topics and more references, (many thanks to Chris Douglas for writing this!)
Delta and Simplicial Objects and Chain Complexes, Term paper by Matthias Goerner
Summary of talks. (containing some discussions that happened during the lectures)

This course will be an attempt to go through Jacob Lurie's Survey of Elliptic Cohomology, with an emphasis on the basics of derived geometry used in his approach. We will also study other applications of derived geometry, depending on the interests of the participants.
The basics start with the notion of a quasi-category: a category has objects and morphisms, a 2-category has in addition 2-morphisms, and a quasi-category has in addition k-morphisms for each positive integer k. Examples of quasi-categories come from topological spaces X, namely by using as k-morphisms the maps from the standard k-simplex (or k-ball) to X. To relate such examples to categorical notions, the language of simplicial sets is extremely useful and will be introduced in the first lectures.

To define a derived scheme one needs to replace the usual local objects, namely commutative rings, by quasi-categories with extra structure. There are several versions of such structures, for example differential graded algebras, simplicial rings, E-infinity ring spectra or other structured ring spectra. After looking at some examples how the first two structures are used, we turn to ring spectra, since they are the structure used for Jacob's definition of elliptic cohomology. After introducing some basic notions on structured ring spectra, we can try to understand how they are glued together to give a derived scheme. In the application to elliptic cohomology, we need to study in particular derived elliptic curves.

Common interests of the participants

In the first lecture, I asked for the motivation of the participants to take this course. Here are some replies:

Ishai Dan-Cohen: I'm interested to different extents in various nonabelian things, which might be distantly related to derived algebraic geometry, at least to the extent that it involves homotopical algebra. At the moment I'm starting to study nonabelian Hodge theory. The original papers on the subject don't seem to involve much homotopy theory explicitly, but Martin Olsson's description of the subject does. Less related to my subject, but perhaps a bit more related to the subject of the course, is Grothendieck's description of reconstructing an anabelian space "block by block" from its arithmetic fundamental group. Can this be explained in the context of derived algebraic geometry?

Santiago Canez: I'm interested in this material for two main reasons. First, I'm currently studying Lie groupoids and differentiable stacks and would like to see what the derived world has to offer in terms of understanding these better. Second, it'd be interesting to know if derived algebraic geometry can be used to help explain the various "derived" objects (such as branes) that show up in string theory and homological mirror symmetry.

Raimundo Heluani: There are several variations of the concept of vertex algebras, in the algebraic side, involving representations of infinite dimensional Lie algebras, in the less algebraic side as chiral algebras on curves, and in a more geometric side as Factorization algebras on curves. For several years now, people in either side have been looking for a higher dimensional analog. There is some consense that the "right" definition of a factorization algebra on a surface will be a "derived" version of the one on curves. Aside from that, the hope is to start understanding the homotopy theory of chiral algebras.

David Spivak: Jacob Lurie's theory of infinity categories is both very beautiful and very powerful. I extensively used techniques from his book and ideas from his thesis in my own thesis on derived manifolds. However, as much as I've used his work, I feel as though I've barely scratched the surface, and I look forward to learning a lot more in this course.

Valentin Tonita: I think it's gonna be useful for me to learn about the contribution of this language to intersection theory. Also, I chose the stacks talk because I want to know more about the formal part of the generalization scheme-stack, which a lot of people avoid to give details on.

Christian Blohmann: I have come across n-categories and n-groupoids when I started to work on the following problem: The noncommutative torus algebra is thought to be the replacement for the dual space of the leaf space of the Kronecker foliation of the 2-torus, for which the quotient topology is trivial. As set the quotient space has a natural group structure but the noncommutative torus does not have a Hopf structure as one should expect. This mystery can be solved by looking at the leaf space as smooth stack with a stacky group structure. Are there more mysteries which can be solved with smooth stacks?
Further references include:

Quasi-Categories:
Nichols-Barrer, Combinatorial Quasi-Categories
Jacob Lurie, Higher Topos Theory
Bertrand Toen and Gabriele Vezzosi, Homotopical algebraic geometry I: topos theory, Advances in Mathematics Vol. 193 (2), 2005, p. 257-372

Simplicial homotopy:
Peter May, Simplicial Objects in Algebraic Topology
Goerss-Jardine, Simplicial Homotopy Theory, Birkhauser, Progress in Mathematics

E-infinity ring spectra:
May-Quinn-Ray, E-infinity ring spaces and E-infinity ring spectra , LNM 577
Goerss-Hopkins, Moduli Spaces of Commutative Ring Spectra , Paul Goerss' website

Structured ring spectra:
Elmendorf-Kriz-Mandell-May, Rings, Modules, and Algebras in Stable Homotopy Theory , Mathematical Surveys and Monographs 47, AMS, 1997
Mandell-May-Schwede-Shipley, Model Categories of Diagram Spectra , on Peter May's website

Elliptic cohomology:
Charles Rezk, Notes on topological modular forms , Charles Rezk's website, under 'other writings'
Charles Rezk, Notes on the Hopkins-Miller Theorem , Charles Rezk's website