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, becoming Professor in 1991.

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, an appointment to the Mathematical Science Research Institute in Berkeley during the fall of 1989, and an appointment to the Centre National de la Researche Scientifique during a 1994-95 visit to various Paris universities.

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, Switzerland, Japan, and Vietnam.

A strong personal connection to the world of mathematics has roots in my family history: both my father, Harold Kuhn, and uncle, Leon Henkin, are well-known mathematicians, and I have many early childhood memories of Princeton's 'old' Fine Hall. I have been interested in algebraic topology since my undergraduate days, and have enjoyed witnessing, and being part of, the many dramatic new developments in this subject since 1980.

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.

My algebraic work has been on the topics of modern Steenrod algebra technology over all finite fields, group cohomology, generic representation theory of the finite general linear groups, rational cohomology, and homological stability questions.

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

   In this paper, and in related work with 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. In a supporting paper, I proved a theorem for computing finite group cohomology referred to as the Cadenas-Kuhn Theorem in the book by Milgram and Adem. Due to its connections with Arone and Mahowald's work on resolutions of spheres arising from Goodwillie calculus, there is renewed interest in my Whitehead Conjecture constructions.

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

   This 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.

   
   
3. 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. Versions of this paper have been widely circulated since the late 1980's, and it has been much cited and used. It was given a 'Featured Review' by Math Reviews.

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

   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 has made remarkable use of these ideas, and the telescopic functors in the case n = 2 are used in an essential way in the announced proof by Hopkins and collaborators of the rigidification of the string bordism elliptic genus.

   The second and third papers represent a major project of mine of the last few years. Using the telescopic functors in a critical way, 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.

   
   
5. 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.

   This project represents a major part of my research in the 1990s. 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 listed 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 last paper applies some of these ideas to strengthen the classical link between the general linear groups and the symmetric groups.

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

   This 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. Beginning with my reductions, the most basic of my conjectures has been confirmed by Lionel Schwartz, and he and his students in Paris have made much progress on the others.