Conference for Bob Stong
Schedule

Saturday, November 10

9:15-9:45
Coffee and conference registration

9:45-10:45
Jack Morava (Johns Hopkins University)
Title: The Madsen-Tillmann Spectrum for Spin Four-Manifolds
Abstract: The category of one-manifolds, with 2-D cobordisms as morphisms, seems to be of great importance in string theory, and (through its connection with the Riemann moduli space) in algebraic geometry.

Recently Madsen, Tillmann, et al., identified the spectra associated to such categories in great generality. Spin-manifolds in dim 3 and 4 seem especially interesting; I'll discuss some algebraic properties of their associated spectra, in particular a representation (away from 2) of the 4-D spin cobordism spectrum as homotopy-theoretic endomorphisms of the 3-D spin cobordism spectrum.

11:00-12:00
Dev Sinha (University of Oregon)
Title: A Complete Understanding of Rational Hopf Invariants
Abstract: In joint work with Ben Walter, we give a new, definitive treatment of the rational "homotopy period problem." That is, we explicitly define rational functionals on homotopy groups given appropriate cochain data. This is of course a classical question, tracing its roots to Hopf as well as an integral formula for the classical Hopf invariant due to J. H. C. Whitehead. The question was addressed, slightly incorrectly, in Sullivan's seminal paper on minimal models in rational homotopy theory. Hain gave one solution in his thesis, where he relied on the Milnor-Moore theorem which identifies rational homotopy within loopspace homology, and the problem has been revisited by Novikov. Our approach is correct, explicit, simple, and potentially generalizable to characteristic p since we do not rely on the Milnor-Moore theorem. We start with a model for the cooperad governing Lie coalgebras which arises naturally from the study of the cohomology of configuration spaces. A cobar construction over this operad gives a "Lie coalgebra model" for a rational space, in the sense of Quillen. But we give an elementary way to associate homotopy periods to cycles in this cobar construction. The underlying geometry, which would be natural to any cobordism theorest, allows us for example to show how Boardman-Steer's Hopf invariants fit into the picture.

Lunch

2:00-3:00
Peter Landweber (Rutgers University)
Title: Cobordism and Equivariant Bordism at Virginia
Abstract: Pierre Conner and Ed Floyd launched their study of cobordism and its application to group actions in the mid-1960's. After recalling this golden era, I will continue by discussing work done by Bob Stong on equivariant bordism. It will be impossible to resist adding some reflections on the joint projects I've had the good fortune to work on with Bob.

3:15-4:15
Serge Ochanine (University of Kentucky)
Title: A Mirror Symmetry Formula for the Elliptic Genus of Complete Intersections
Abstract: I will discuss an elementary proof of a physics inspired summation formula for the elliptic genus of complete intersections. Joint work with Vassily Gorbounov.

4:30-5:30
Doug Ravenel (University of Rochester)
Title: Some Early and Late Mathematical Work of Bob Stong
Abstract: I will talk about some dazzling work that Bob did in the '60s and the '80/90s, namely
* (early work) his papers on the connective covers of BO and BU, the Stong-Hattori theorem and cobordism of maps plus his book on cobordism theory;
* (late work) his papers with Landweber and myself on the elliptic genus and elliptic cohomology.
[SLIDES FOR THE TALK]

6:00-9:00
Banquet

Sunday, November 11

9:30-10:30
Mark Behrens (MIT)
Title: v₂-Periodicity at the Prime 2
Abstract: I will describe the construction of a minimal v₂-periodic self map on the finite complex M(p,v₁⁴). This map gives rise to periodic families of 2-primary elements in the stable homotopy groups of spheres with period 192. This is joint work with Michael Hill, Michael Hopkins, and Mark Mahowald.

10:45-11:45
Ralph Cohen (Stanford University)
Title: Geometric Cobordism Theory
Abstract: The mathematical notion of a quantum field theory can be viewed as a functor from a cobordism category to a linear category. Recent work of Galatius, Madsen, Tillmann, and Weiss identified the homotopy type of topological cobordism categories. By a "topological cobordism category," I mean a category whose objects are closed manifolds, and whose morphisms are cobordisms, perhaps endowed with certain structure on their tangent bundles. In a variety of important modern field theories. However, the cobordism categories have geometric structures. That is, the manifolds are endowed with structure defined in terms of Riemannian metrics. In this lecture, I will discuss the homotopy type of three such geometric cobordism categories: (1) The symplectic cobordism category; (2) the category of holomorphic curves in CPⁿ; and (3) the category of flat connections on Riemann surfaces. This represents joint work with S. Galatius and N. Kitchloo, as well as work of my student, D. Ayala.