1999 AMS-IMS-SIAM Summer Research Conference on

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Brenda Johnson

Union College

Constructing and Characterizing Degree n Functors

Let F be a functor from a basepointed category with finite coproducts to a category of chain complexes over an abelian category. Such a functor is homologically degree n if its (n+1)-st cross effect (in the sense of Eilenberg and Mac Lane) has trivial homology. We describe a method for constructing, by means of cotriples associated to the cross effects of F, a universal tower,

...   --->   P_{n+1}  F   --->   P_n F  --->  ... ---> P_1 F  --->  P_0 F = F(*),

in which each functor P_n F is homologically degree n. This construction arose from the study of Goodwillie's Taylor tower in the case of functors of modules over a ring. Using this model, we will characterize homologically degree n functors in terms of modules over a certain DGA, and discuss some related constructions due to Eilenberg-Mac Lane, and Dold-Puppe. This is joint work with Randy McCarthy.

Max Karoubi

Université Paris 7 (France)

Quantum methods in Algebraic Topology

We sketch a new cohomology theory of "quantum" differential forms which generalizes Sullivan's work for rings of arbitrary characteristic. This new approach may be extended to other contexts, like the cohomology with values in a sheaf. As applications we propose

1. A much simpler description of cup-i products (and of Steenrod operations)

2. Thanks to the recent work of Kriz, Mandell and May, a new approach to homotopy type through the concept of "neo-algebra" (a generalisation of a commutative DGA in positive characteristic)

3. An explicit computation of the cohomology of iterated loop spaces in terms of this neo-algebra approach.

Igor Kriz

University of Michigan

A brief introduction to the work of J. Peter May

Gaunce Lewis

Syracuse University

Mackey functor commutative algebra

(joint work with Florian Luca)

Let G be a compact Lie group, R be a commutative G-Mackey functor ring, and R(G) be the value of R at G. There is a topology on the set Spec(R) of Mackey functor prime ideals of R which is an obvious generalization of the Zariski topology on the spectrum of an ordinary commutative ring. This space Spec(R) carries a significant of information about R. In particular, the spectrum Spec(R(G)) of the ordinary commutative ring R(G) is a retract of Spec(R). Moreover, there is a function from the set Spec(R) to the set Conj(G) of conjugacy classes of subgroups of G, which can be used to determine the strongest possible induction theory satisfied by R.

This talk will be devoted to a discussion of the properties of Spec(R). Examples of Spec(R) for various groups G and rings R will also be discussed.

Ib Madsen

Århus Universitet (Denmark)

On the homotopical structure of diffeomorphisms of surfaces

Mark Mahowald

Northwestern University

EO_2 resolutions and isogenies of elliptic curves

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