Research

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 $E_{n}$. In addition to the relationship indicated in the title, I construct all the $H_{\infty}$ complex orientation on $E_{n}$. This paper has appeared as Duke Math. J. 79 (1995) 423--485.
• Completions of $\mathbf{Z}/ (p)$-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 $E_{n}$ and $E_{n-1}$. This paper has appeared in Geometry and Topology 2 (1998) 145--174
• Power operations in elliptic cohomology and representations of loop groups. 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 K-theory (with N. Strickland). Neil 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 Witten 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) 595--687.
• 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 appeared in 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 lectures.
• Cluster decomposition, T-duality, and gerby CFT's (Appendix) The physics of the main body of the paper suggests a result about twisted equivariant $K$-theory; I explain how this result fits into the framework for twisted equivariant $K$-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 C. Rezk) 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 D. Gepner) 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.

• Multiplicative orientations of KO-theory and of the spectrum of topological modular forms (with M. Hopkins and C. Rezk) We show the spectrum tmf of Topological Modular Forms receives an multiplicative orientation from the bordism spectrum of string manifolds. The paper is essentially complete, although it depends on some results about tmf, which are planned for a book, of which this paper will eventually be a chapter.