# Selected Projects

1. Cohomology rings E*(BG) when E* is complex oriented.

Generalized group characters and complex oriented cohomology theories, Journal AMS 13 (2000), 553-594. Joint with M. J. Hopkins and D. C. Ravenel.

For each natural number n and each prime p, we construct a character theory for detecting elements in the 'height np local' cohomology of finite groups. This specializes to classical group character theory when n = 1. When n = 2 our work fits in a beautiful way with work by many people on elliptic cohomology. Early versions of this paper were widely circulated in the late 1980's, and HKR characters have been much used and studied.

2. Homotopical algebra: Goodwillie calculus and commutative ring spectra.

Product and other fine structure in polynomial resolutions of mapping spaces, Algebraic and Geometric Topology (2002), 591--647. With S.T. Ahearn.

The McCord model for the tensor product of a space and a commutative ring spectrum, Categorical Decomposition Techniques in Algebraic Topology, Proc. Isle of Skye, Scotland, 2001, Progress in Math (2003), 213--236.

Goodwillie towers and chromatic homotopy: an overview, Algebraic Topology, Proc. Kinosaki, Japan, 2003, Geometry and Topology Monographs 10 (2007), 245--279.

A couple of my major projects since 2000, most obviously the next listed project below, have been supported' by modern aspects of homotopical algebra, most obviously Tom Goodwillie's calculus of homotopy functors.  So some of my work has been on developing homotopical algebra tools.

In the first paper above, we showed how new spectral sequences arising from G. Arone's Goodwillie tower resolutions for classical mapping spaces  (e.g. n-fold loopspaces) are equipped with useful structure generalizing that long known for the classical Eilenberg--Moore spectral sequence for computing the cohomology of  a 1-fold loopspace.

The second paper amounted to a study of the category of commutative S-algebras viewed as a category tensored over spaces (or simplicial sets).  As one application, it was noted that the towers in the first paper arise as S-duals of filtered commutative S-algebras.   Related to this, it is shown that  Topological Andre Quillen homology comes equipped with a canonical filtration.  This filtration plays a major role in current work of Behrens and Rezk, who call this the Kuhn filtration of TAQ(R).

The third paper includes the first exposition of Goodwillie's work written with model category hypotheses, allowing for generalization to many different structured situations.

3. Periodic Homotopy and Goodwillie calculus.

Morava K-theories and infinite loop spaces, Algebraic Topology, Arcata 1986, Springer Lect. Notes Math. 1370 (1989), 243-257.

Tate cohomology and periodic localization of polynomial functors, Invent. Math. 157 (2004), 345-371.

Localization of Andre-Quillen-Goodwillie towers, and the periodic homology of infinite loopspaces, Advances Math. 201 (2006), 318-378.

A guide to telescopic functors, Johns Hopkins Conference on Algebraic Topology, Proc. Baltimore, 2007, Homology, Homotopy, and Applications 10 (2008), 572--602.

The first paper, an early application of the Nilpotence and Periodicity Theorems of Devanitz-Hopkins-Smith, exposed a curious relation between the stable and unstable worlds of homotopy, as viewed through the eyes of height n homology theories. My constructions, and Bousfield's more refined versions, seem to now be called Bousfield-Kuhn telescopic functors. Pete made much use of these ideas, and the telescopic functors underlie Rezk's logarithmic' cohomology operations, and, in the case n = 2, were used in an essential way in the first announced proof by Hopkins and collaborators of the rigidification of the string bordism elliptic genus.  Currently, they play a starring role in exciting recent papers generalizing Sullivan's work on rational homotopy to higher chromatic heights.

The second and third papers represent a major project of mine of the past decade. Using the telescopic functors, I have shown that Goodwillie calculus interacts in striking ways with periodic stable homotopy. In the third paper, I construct a highly structured splitting, after periodic localization, of the Goodwillie tower associated to the suspension spectrum of the O-th space of a spectrum, and use this to say much about the Morava K-theory of infinite loop spaces. In the second paper, I show that all polynomial endofunctors of spectra split after periodic localization. Enroute, I reprove and strengthen results of Greenlees-Sadofsky-Hovey, and Mahowald-Shick on Tate cohomology.

The fourth paper is a modern guide to the Bousfield-Kuhn functors, including characterizations via their properties.

4. Whitehead's Symmetric Products of Spheres Conjecture and the Goodwillie tower of the circle.

A Kahn-Priddy sequence and a conjecture of G. W. Whitehead, Math. Proc. Camb. Phil. Soc. 92(1982), 467-483.

The transfer and Whitehead's conjecture, Math. Proc. Camb. Phil. Soc. 98}(1985), 459--480. Joint with S.B.Priddy.

The Whitehead Conjecture, the Tower of S^1 Conjecture, and Hecke algebras of type A, Journal of Topology 8 (2015), 118-146.

In the first two papers, related to work of Stewart Priddy and Steve Mitchell, I proved a long-standing conjecture about the stable homotopy groups of spheres and related spaces. The novel aspects of the work was the use of algebra related to Hecke algebras and Steinberg modules to guide geometric constructions involving transfers and Hopf invariants. Predating the current great interest in all things Koszul, at the heart of this work are connections between 'braided' algebras and Koszul algebras. This is made more clear in the last, much more recent, paper, where I  simutaneously give a state-of-the-art proof of the Whitehead conjecture and verify a long suspected property of the Goodwillie tower resolution of the circle, as analyzed by Arone and his collaborators.

5. Generic Representation Theory and Functor Cohomology.

Generic representations of the finite general linear groups and the Steenrod algebra: I, II, III, Amer. J. Math. 116(1994), 327-360, K-theory J. 8(1994), 395-428, K-theory J. 9(1995), 273-303.

Rational cohomology and cohomological stability in generic representation theory, Amer. J. Math. 120(1998), 1317-1341.

A stratification of generic representation theory and generalized Schur algebras, K-Theory Journal 26 (2002), 15-49.

Generic representation theory of finite fields in nondescribing characteristic, Advances Math. 272 (2015), 598-610.

The three-paper series develops the modular representation theory of the general linear groups over finite fields from a certain categorical 'generic' point of view. My original interest in this was to develop the algebra of cohomology operations (the Steenrod algebra) in a more unified and conceptual manner. Within a few years, connections to questions in algebraic K-theory became clear. The next paper overlaps with work by K-theorists Friedlander and Suslin (part of Suslin's Cole Prize citation) and can also be viewed as being in parallel with older work by my representation theorist colleagues, Parshall and Scott. The `stratification' paper applies some of these ideas to strengthen the classical link between the general linear groups and the symmetric groups. There is renewed interest in functor categories, and my recent paper in this area is a reflection of this.

6. Topological Realization Questions.

On topologically realizing modules over the Steenrod algebra, Annals of Math. 141(1995), 321-347.

Topological nonrealization results via the Goodwillie tower approach to iterated loopspace homology, Algebraic and Geometric Topology 8 (2008), 2109--2129.

The first paper proposed, and partially verified, various conjectures concerning the 'size' of spaces. These take the form: the mod p cohomology of a topological space must be either 'very small' or 'very large,' where cohomology is organized by using both the nilpotent and Krull filtrations of the category of unstable modules over the Steenrod algebra. In 2010, Lionel Schwartz and Gerald Gauden elegantly proved my conjectures. The second paper gives an alternative proof of some of Schwartz' earlier work on this problem at the prime 2, using a spectral sequence associated to a Goodwillie tower, rather than Schwartz' iterated use of the Eilenberg Moore spectral sequence.

7. Classifying spaces and the Cohomology of Finite Groups.

Chevalley group theory and the transfer in the homology of symmetric groups, Topology 24(1985), 247--264.

Stable decompositions of classifying spaces of finite abelian p-groups, Math. Proc. Camb. Phil. Soc. 103(1988), 427-449. Joint with J. C. Harris.

Primitives and central detection numbers in group cohomology, Advances in Math. 216 (2007), 387--442.

The nilpotent filtration and the action of automorphisms on the cohomology of finite p-groups, Math. Proc. Camb. Phil. Soc. 144 (2008), 575--602.

Nilpotence in group cohomology, Proc. Edin. Math. Soc. 56 (2013), 151--175.

The first paper, which supported my work on the Whitehead conjecture, includes a theorem for computing finite group cohomology referred to as the Cardenas-Kuhn Theorem in the book by Milgram and Adem. The second paper served as a model for many papers by Martino, Priddy, Benson, Feshbach, and others studying the classifying spaces of finite groups. This paper is really a paper on representation theory and Mackey functors; thanks to G. Carlsson's work on the Segal Conjecture, it has definitive topological interpretation. One novelty was the use of modular representation theory of finite semigroups.

The three recent papers reflect my ongoing interest in group cohomology. The first of these connects a previously mysterious invariant of the cohomology of finite groups defined by topologists Henn, Lannes, and Schwartz to duality results of group theorists Benson and Carlson, and to Benson's Regularity Conjecture. This was subsequently proved by Symonds, and the third paper makes use of this.