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I am a mathematician specializing in algebraic topology.
Here are links to some of my papers.
- Isogenies of formal group laws and power operations
in the cohomology theories . In
addition to the relationship indicated in the title, I construct all
the complex orientation on . This paper has
appeared as Duke Math. J. 79 (1995) 423--485.
- Completions of
-Tate cohomology of
periodic spectra (with J. Morava and H. Sadofsky).
Jack Morava, Hal Sadofsky and I establish a relationship between the
indicated Tate homology spectrum of and .
This paper has appeared in Geometry and
Topology 2 (1998) 145--174
- Power operations in elliptic cohomology and
representations of loop
I study the
extension of isogenies to operations in elliptic cohomology. I also
describe in a precise form the relationship between elliptic cohomology and
reprentations of loop groups. This paper has appeared as
Trans. AMS 352 (2000) 5619--5666
- Weil pairings and Morava
(with N. Strickland).
Strickland and I describe the relationship between the Morava
K-theory of the connected coverings of BU and the theory of
biextensions and Weil pairings. This paper has appeared in
Topology 40 (2001) 127--156.
- Elliptic spectra, the theorem of the cube, and the
genus (with M. Hopkins and N. Strickland).
We show that complex virtual vector bundles with trivializations of
their first and second Chern classes have a canonical orientation in
any elliptic cohomology. In the case of the elliptic cohomology
associated to the Tate elliptic curve, this is the Witten genus. This
paper has appeared as
Invent. math. 146 (2001)
- A renormalized Riemann-Roch formula and the Thom
isomorphism for the free loop space (with J. Morava).
Witten's analysis on the free loop space essentially attaches to a
formal group its quotient by a free cyclic subgroup. This paper has
Topology, geometry, and algebra: interactions
and new directions (Stanford, CA, 1999), 11-36, Contemp. Math., 279,
Amer. Math. Soc., Providence, RI, 2001.
- The Witten genus and equivariant elliptic
cohomology (with M. Basterra).
Maria Basterra and I construct a Thom class in complex
equivariant elliptic cohomology extending the equivariant Witten
genus. This gives a new proof of the rigidity of the Witten genus,
which exhibits a close relationship to my work with Hopkins and
Strickland on non-equivariant orientations of elliptic spectra.
This article has appeared in Math. Z.
- The sigma orienation is an H infinity
map (with M. Hopkins and N. Strickland).
Mike Hopkins, Neil Strickland, and I show that the sigma orienation
constructed in our paper Elliptic spectra... gives an H
infinity map orientation to the Morava E-theory associated to the
deformations of a supersingular elliptic curve. This article has
appeared in Amer. J. Math.
- The sigma orientation
for analytic circle-equivariant elliptic cohomology.
This paper was inspired by the earlier one with Basterra. The
constructions in that paper involved many choices, and it was not
clear that mutliplication by the Thom class was an isomorphism. In
this paper, I construct a canonical Thom isomorphism in
T-equivariant analytic elliptic cohomology, for T-oriented virtual
vector bundles bundles whose Borel-equivariant second Stiefel-Whitney
and second Chern classes vanish. The construction is natural under pull-back of vector bundles and exponential under Whitney sum.
The construction relates the sigma orientation to the representation
theory of loop groups and Looijenga's weighted projective space, and
sheds light even on the non-equivariant case. Rigidity theorems of
Witten-Bott-Taubes including generalizations by Kefeng Liu follow.
This paper has appeared in Geometry and Topology.
- Discrete torsion for the supersingular orbifold
sigma genus (with C. French).
The first purpose of this paper is to examine the relationship between
equivariant elliptic genera and orbifold elliptic genera. We apply
the character theory of Hopkins et. al. to the Borel-equivariant genus
associated to the sigma orientation of Ando-Hopkins-Strickland to
define an orbifold genus for certain total quotient orbifolds and
supersingular elliptic curves. We show that our orbifold genus is
given by the same sort of formula as the orbifold ``two-variable'' genus of
Dijkgraaf et al. In the case of a finite cyclic orbifold group, we
use the characteristic series for the two-variable genus to define an
analytic equivariant genus in Grojnowski's equivariant elliptic
cohomology, and we show that this gives precisely the orbifold
two-variable genus. The second purpose of this paper is to study the
effect of varying the BU<6>-structure in the Borel-equivariant
sigma orientation. We show that varying the BU<6>-structure by a
class in the third cohomology of the orbifold group produces
discrete torsion in the sense of Vafa. This result was
first obtained by Sharpe, for a different orbifold genus and using
different methods. This paper will appear in the proceedings of the
Newton Institute conference on elliptic cohomology.
- Canberra lectures, Summer 2003.
In the summer of 2003 I gave three plenary lectures at the ``International
conference on algebraic geomety and topology'' at the Australian
National University, and I gave a lecture at the ``Kinosaki international
conference on algebraic topology'' in honor of the 60th birthday of
Professor Goro Nishida. I am posting here the slides for some of my
- Cluster decomposition, T-duality, and gerby CFT's
The physics of the main body of the paper suggests a result about
twisted equivariant -theory; I explain how this result fits into
the framework for twisted equivariant -theory of Atiyah and Segal.
- The Jacobi orientation and the two-variable
elliptic genus (with C. French and N. Ganter).
My work with Hopkins, Strickland, and Rezk singles out the
Witten genus-also called the sigma orientation-among elliptic
genera; there is a precise sense in which it is ``initial'' among
elliptic genera. On the other hand, the work on orbifold elliptic
genera, e.g. by Dikjgraaf et al. and Borisov and Libgover, has focused
attention on the two-variable elliptic genus. In this paper, we
explain the relationship between the sigma
orientation and the two-variable elliptic genus. We express
the relationship two ways. The first involves the analysis of
MU<6>-orientations of Ando-Hopkins-Strickland, and gives new insight on
the modularity properties of the two-variable genus. The second uses
the sigma orientation in circle-equivariant elliptic cohomology, and
gives new insight on the ``level N genera'' of Hirzebruch.
- Circle-equivariant classifying spaces and the
rational equivariant sigma genus (with J. Greenlees)
We analyze the circle-equivariant spectrum which is the
equivariant analogue of the cobordism spectrum of stably almost complex
manifolds with vanishing first and second Chern class. We show that
there is a canonical map of ring spectra from this spectrum to the
spectrum constructed by Greenlees representing circle-equivariant
elliptic cohomology, in the case of an analytic ellipic curve. Our
method is a considerable refinement of the methods of
my paper The sigma orientation for analytic... listed above,
and gives a proof of a version of the conjecture of that paper.
- Units of ring spectra and Thom spectra (with
A. Blumberg, D. Gepner, M. Hopkins, and
We develop the theory of orientations for associative ring spectra,
analogous to the theory for commutative ring spectra of May, Quinn,
and Ray. We develop two approaches, one making use of the space-level
anologue of EKMM S-algebras, the other using infinity categories.
- Twists of K-theory and of TMF (with A. Blumberg and
We show how to use the theory of units and Thom spectra developed in
http://arXiv.org/abs/0810.4535 to construct twisted generalized
cohomology. As an application we show how the string orientation
leads to twists of elliptic cohomology by degree-four characteristic classes.
These papers are not yet on the arxiv, but are approaching arxivable form.
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