I studied mathematics as an undergraduate at Princeton University. I began graduate studies at the University of Chicago in 1976, and in 1980 received my Ph.D. under the tutelage of algebraic topologist J. Peter May, writing a thesis on various aspects of iterated loopspace theory.
My early career took me to the
University of Washington (1980-1982), and then to Princeton
University (1982-1986). In 1986, I joined the University of
Virginia as an associate professor, and became a professor in
1991. I am a 2018 Fellow of the American
Visiting positions have included a
position at Northwestern University as the 1982-83 American
Mathematical Society Postdoctoral Fellow, a visiting
professorship at Cambridge University in 1986-87 as a Sloan
Foundation Fellow and again in spring 2006, an appointment to
the Mathematical Science Research Institute in Berkeley during
fall 1989 and again in spring 2014, an appointment to the Centre
National de la Researche Scientifique during a 1994-95 visit to
the University of Paris 13, and a visiting professorship at the
University of Sheffield in spring 2017.
I have greatly enjoyed the international flavor of the mathematical community, and have had the opportunity to give talks in Canada, Mexico, France, Great Britain, Germany, Italy, Poland, Spain, Sweden, Switzerland, Croatia, Japan, Tunisia, and Vietnam.
My research is centered around algebraic topology and homotopy theory. Over the years, my research interests have broadened to include algebraic K-theory and group representation theory.
My work in topology has included work on the development of a character theory for complex oriented cohomology theories, iterated loopspace theory, Goodwillie functor calculus, periodic homotopy, topological realization questions, stable homotopy groups, and model categories.
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 n, p
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.
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.
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.
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
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.
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.
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.
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.
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.