FEBRUARY 6-7, 2010


9:30-10:00 Coffee and refreshments

10:00-11:00 J.Peter May: What are parametrized spectra good for? (And how an algebraic topologist sees twisted K-theory.)

This will be an attempt to make some old material accessible. In 2006, Johann Sigurdsson and I published a little book (441 pages) entitled ``Parametrized homotopy theory''. The length was caused by annoying but important technicalities that obscure the essential simplicity and rightness of the ideas. I'll explain how even such classical things as Poincare' duality and orientation theory are illuminated by thinking in the setting of spaces and spectra parametrized by a base space B, and I'll place twisted K-theory in a general framework of parametrized homology and cohomology theories. I aim to be intuitive and resolutely nontechnical.

11:15-12:15 Mark Behrens: On the relationship between EO_n and TAF

Abstract: It is well know that for p = 2, the K(1)-localization of KO is EO_1, and for p = 2; 3, the K(2)-localization of TMF is EO_2. When does the K(n)-localization of TAF contain a factor of EO_n? We will provide a complete answer. This is joint work with Mike Hopkins.

2:00-3:00 Dan Freed: An index theorem in differential K-theory

Abstract: In joint work with John Lott we prove a refinement of the Atiyah-Singer index theorem in differential K-theory. It has applications to anomaly cancellations in string theory.

3:00-3:30 Coffee and refreshments

3:30-4:30 Kevin Costello:A geometric construction of the Witten genus

Abstract: I'll explain a construction of the Witten genus of a complex manifold. This construction relies on a rigorous treatment of a certain two-dimensional quantum field theory.

4:45-5:45 Dennis Sullivan: String topology invariants, the form of Thurston geometrization for three-manifolds and some relations with various physical-math conjectures

Abstract: Open string topology puts an algebra-coalgebra structure of degree -1 on chains of paths in a three manifold M with boundary conditions on knots and links inside. From this structure one can derive information about closed string topology of the three manifold M using ideas of Fukaya, Costello and Konsevitch related to categorical string theory. Closed string topology in 3D puts a lie algebra lie coalgebra structure of degree -1 on classes of maps of circles and torii into the three manifold. The lie bracket part of this structure determines the form of the geometrization picture of closed three manifolds and link complements ( work of M.Chas and S.Gadgil) One may from the open string coalgebra part of the discussion derive by homotopical methods an algebra structure on the linearized homology of the resolved version of the coalgebra. This may be related (work of M.Sullivan and J.McGibbon) to the chord algebra which was introduced combinatorially by Lennie Ng to model invariants defined using J-holomorphic curves related to the contact and symplectic structures in the cotangent bundle. These relations support string topology-contact topology conjectures valid in all dimensions(Eliashberg, Latshev and Cielebak) We recently deduced from a talk by Ed Frenkel on physics and the Langlands program these string topology conjectures in turn might be construed as a real version of the composition of two holomorphic conjectures... homological mirror symmetry and a geometric langlands correspondence. This real interpretation depends on understanding better the relation between the string topology (of small loops) and D-modules as suggested by recent work of Costello.

6:30 Dinner Reception in the Upper Mathematics Atrium in East Hall


9:30 Coffee and refreshments

10:00-11:00 Sergei Gukov: Exact Results for Chern-Simons Theory with Complex Gauge Group

Abstract: Chern-Simons gauge theory with complex gauge group has many interesting connections with three-dimensional quantum gravity and geometry of hyperbolic 3-manifolds. Yet, it remains quite a mystery compared to a much better understood theory with compact gauge group. In this talk, I will present several methods that allow to compute all-loop partition functions in perturbative Chern-Simons theory with complex gauge group G_C, sometimes in multiple ways. In the background of a non-abelian irreducible flat connection, perturbative G_C invariants turn out to be interesting topological invariants, which are very different from finite type (Vassiliev) invariants obtained in a theory with compact gauge group G. We shall explore various aspects of these invariants and present an example where we compute them explicitly to very high loop order.

11:15-12:15 Greg Moore: The RR Charge of an Orientifold

Abstract: This work reviews an aspect of my work with J. Distler and D. Freed on orientifolds. We give a mathematical description of the orientifold construction of string theory. After reviewing some aspects of self-dual fields we state a formula for the K-theoretic RR charge of an orientifold (after inverting the prime 2). We compare the Chern character of this charge with previous results in the physics literature.