The Strategic University Program in Pure Mathematics (Suprema) at the Department of Mathematics, University of Oslo, organizes a Masterclass with Jacob Lurie in August-September 2006, on Elliptic Cohomology and Derived Algebraic Geometry. Lurie holds the 2004 American Institute of Mathematics Five-Year Fellowship, and works at Harvard University.
The masterclass will consist of a series of introductory lectures over three weeks, followed by a one-week concluding workshop. The masterclass lectures will be aimed at faculty, postdocs and advanced doctoral students with a general knowledge of algebraic topology and/or algebraic geometry. The concluding workshop will build on this material, and will include further invited speakers.
In algebraic topology, the representing objects for multiplicative cohomology theories like singular cohomology, topological K-theory, elliptic cohomology and cobordism theory are known as ring spectra. Sufficiently nice ring spectra are the "brave new rings" of modern stable homotopy theory. These have classical underlying rings, given by the (either zeroth or graded) coefficient ring of the theory, and ring spectra can be thought of as topological enrichments of their underlying rings. Derived algebraic geometry is the study of algebraic spaces locally ringed by "brave new rings", rather than by ordinary rings. This perspective leads to the construction and study of a derived moduli stack of elliptic curves. Its global sections is the topological modular forms ring spectrum tmf of Hopkins et al, whose coefficient ring is close to, but subtly and interestingly different from, the ring of integral modular forms. For example, the Borcherds congruences satisfied by the theta functions of even unimodular lattices express that these integral modular forms lift to topological modular forms. For another example, the Witten genus of string theory, associating a modular form to each string manifold, can be constructed as a ring spectrum map from string bordism to tmf.
The masterclass lectures, held by Jacob Lurie, are planned to consist of a 3-hour session on each of the following days:
Plan: Start with the problem of trying to construct equivariant cohomology theories. Use this to motivate the introduction of derived algebraic geometry. Then some time discussing "rings" (E-infty / strictly commutative / differential graded algebras), how to glue them to make schemes, some examples. Close by describing some applications of these ideas to classical algebraic geometry (virtual fundamental classes/deformation theory).
Plan: Brief review of derived algebraic geometry. Return to the problem of trying to construct equivariant cohomology theories. Definition of an oriented algebraic group, and explanation of how to pass from these to cohomology theories and (in good cases) back. Detailed analysis of the case of the multiplicative group: K-theory / Snaith's theorem / equivariant K-theory / complex conjugation involution. Detailed analysis of the case of the additive group: periodic rational cohomology / action of G_m, recovery of nonperiodic cohomology. Brief mention of elliptic cohomology.
Plan: Brief review of the second lecture. Remarks about classification of algebraic groups / recollections about elliptic curves. Definition of the moduli stack of elliptic curves / Deligne-Mumford stacks in general. Then, derived Deligne-Mumford stacks. Describe the original Hopkins-Miller construction, statement of its universal property. Application of second lecture to construct equivariant elliptic cohomology, and some words about what it looks like. Compactifying the moduli stack. Tate K-theory and the "point at infinity". Change of gears: more geometric discussion about Dirac operator on loop space, Witten genus, et cetera. Existence of "topologial Witten genus" as an advertisement for the geometric relevance of tmf. A few words about the conjectural relationship between tmf and conformal field theory.
In addition, there will be preparatory lectures on some of the background material that may be good to know:
Expected participants include: Nils A. Baas (NTNU), Clark Barwick, Håkon Schad Bergsaker, Anna Blechingberg, Christin Borge, Morten Brun (UiB), Ulrik Buchholtz (Copenhagen), Fabrizio Donati (Trieste), Bjorn Ian Dundas (UiB), Thomas Gregersen, Idar Hansen (NTNU), Runar Ile (UiB), Dan Isaksen (Wayne State), Bjørn Jahren, Finn Faye Knudsen (NTNU), Nadia S. Larsen, Sergey Neshveyev, Paul Arne Østvær, Kristian Ranestad, John Rognes, Christian Schlichtkrull (UiB), Olav Skutlaberg, Arne B. Sletsjøe and Brian Smithling (Chicago).
The concluding workshop will take place