University of Chicago
Department of Mathematics

There are two different ways --- due, evidently independently, to C. Simpson and C. Rezk --- to use G. Segal's delooping machine to produce, for any n, a theory of (infinity,n)-categories such that the i-morphisms are isomorphisms for i>n. As I shall explain, the category of these ``(infinity,n)-categories'' carries a number of model structures of interest, and one can use the homotopy theory of these (infinity,n)-categories to formulate the theory of symmetric monoidal (infinity,n)-categories and that of (infinity,n)-stacks. Using these ideas, I shall discuss some surprising applications of homotopy theory to algebraic geometry. In particular, I sketch the proof of a powerful new local-to-global principle for complexes of modules, and for an algebraic variety X, I describe a kind of motivic homotopy type X_M, the representations of the loop space of which correspond naturally to cohomology theories on X.

See http://www.math.uchicago.edu/~wahl/midwest.html for more details

See http://www.math.uchicago.edu/~wahl/midwest.html for more details

The stable homotopy groups of spheres posses a filtration called the chromatic filtration. This filtration is given by the height filtration on the moduli space of formal groups. I will describe how to decompose the second layer of this filtration using the category of supersingular elliptic curves, and the building for GL_2(Q_l). For p=3 and l=2 this recovers a decomposition studied by Goerss, Henn, Mahowald, and Rezk.

I will outline recent work of Michael Ching and myself concerning two closely related problems -- the higher chain rule in Goodwillie's homotopy calculus of functors and a classification of approximating towers in terms of derivatives. Together, these ideas lead to an interesting question about operads and EKMM spectra whose answer would have deep implications about the structure of calculus of functors.

In topological quantum field theory one is interested in studying functors from a topological category of n-dimensional cobordisms into the category of vector spaces. In two dimensions such functors are very well understood. In fact, specifying a (symmetric monoidal) functor from the 2-dimensional cobordism category 2-Cob into Vect is equivalent to specifying a commutative Frobenius algebra. This makes the study of 2-dimensional TQFT's particularly simple. Recent developments in string theory have prompted many to consider topological quantum field theories using a more interesting version of the 2-dimensional cobordism category, namely one that allows for cobordisms between 1-manifolds with boundary. In this talk I will define a category of planar cobordisms between `open strings' and show that functors from this category into Vect are equivalent to (not necessarily commutative) Frobenius algebras. This result arises naturally by considering adjunctions in 2-categories. If time permits, I will also sketch how this process can be generalized to higher-dimensional surfaces using higher-dimensional category theory. N.B. This talk is intended to be accessible; knowledge of category theory will not be assumed.

There is an interesting relation between Lie 2-algebras, the Kac-Moody central extensions of loop groups, and the group String(n). A Lie 2-algebra is a categorified version of a Lie algebra where the Jacobi identity holds up to a natural isomorphism called the "Jacobiator". Similarly, a Lie 2-group is a categorified version of a Lie group. If G is a simply-connected compact simple Lie group, there is a 1-parameter family of Lie 2-algebras g_k each having g as its Lie algebra of objects, but with a Jacobiator built from the canonical 3-form on G. There appears to be no Lie 2-group having g_k as its Lie 2-algebra, except when k = 0. However, for integral k there is an infinite-dimensional Lie 2-group whose Lie 2-algebra is equivalent to g_k. The objects of this 2-group are based paths in G, while the automorphisms of any object form the level-k Kac Moody central extension of the loop group of G. The nerve of this 2-group gives a topological group that is an extension of G by the Eilenberg-MacLane space K(Z,2). When k = +/- 1, this topological group can also be obtained by killing the third homotopy group of G. Thus, when G = Spin(n), it is none other than String(n).