2. Determinants, differential cocycles and statement of results 5

Abstract not found

The work to be presented in this paper has been inspired by several of Professor Graeme Segal’s papers. Our search for a geometrically defined elliptic cohomology theory with associated elliptic objects obviously stems from his Bourbaki seminar [Se88]. Our readiness to form group completions of symmetric monoidal categories

Abstract not found

The problem of describing the homotopy groups of spheres has been fundamental to algebraic topology for around 80 years. There were periods when specific computations were important and periods when the emphasis favored theory. Many

Abstract. In [AHS01] the authors constructed a natural map, called the sigma orientation, from the Thom spectrum MU〈6 〉 to any elliptic spectrum in the sense of [Hop95]. MU〈6 〉 is an H ∞ ring spectrum, and in this paper we show that if (E, C, t) is the elliptic spectrum associated to the universal deformation of a supersingular elliptic curve over a perfect field of characteristic p> 0, then the sigma orientation is a map of H ∞ ring spectra.

where E is a suitable complete periodic complex oriented theory and G is a finite group: we describe its variety in terms of the formal group associated to E, and the category of abelian p-subgroups of G. Our results considerably extend those of Hopkins-Kuhn-Ravenel [16], and this enables us to obtain information about the associated homology of BG. For example if E is the complete 2-periodic version of the Johnson-Wilson theory E(n) the irreducible components of the variety of the quotient E (BG)=I k by the invariant prime ideal I k = (p; v 1; : : : ; v k\Gamma1) correspond to conjugacy classes of abelian p-subgroups of rank n \Gamma k. Furthermore, if we invert v k the decomposition of the variety into irreducible pieces corresponding to minimal primes becomes a decomposition

Abstract. We 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 recent work on non-equivariant orientations of elliptic spectra. 1.

1.1. Notation and conventions 3 1.2. Even periodic ring spectra 3 2. Schemes 3

This paper is an expository account of the relationship between elliptic cohomology and the emerging subject of derived algebraic geometry. We begin in §1 with an overview of the classical theory of elliptic cohomology. In §2 we review the theory of E∞-ring spectra and introduce the language of derived algebraic geometry. We apply this theory in §3, where we introduce the notion of an oriented group scheme and describe connection between oriented group schemes and equivariant cohomology theories. In §4 we sketch a proof of our main result, which relates the classical theory of elliptic cohomology to the classification of oriented elliptic curves. In §5 we discuss various applications of these ideas, many of which rely upon a special feature of elliptic cohomology which we call 2-equivariance. The theory that we are going to describe lies at the intersection of homotopy theory and algebraic geometry. We have tried to make our exposition accessible to those who are not specialists in algebraic topology; however, we do assume the reader is familiar with the language of algebraic geometry, particularly with the theory of elliptic curves. In order to keep our account readable, we will gloss over many details, particularly where the use of higher category theory is required. A more comprehensive account of the material described here, with complete definitions and proofs, will be given in [21]. In carrying out the work described in this paper, I have benefitted from the ideas of many people. I