Volume 4 of the series Abel Symposia pp 219277
A Survey of Elliptic Cohomology
 J. LurieAffiliated withDepartment of Mathematics, Massachusetts Institute of Technology Email author
Abstract
This paper is an expository account of the relationship between elliptic cohomology and the emerging subject of derived algebraic geometry. We begin in Sect. 1 with an overview of the classical theory of elliptic cohomology. In Sect. 2 we review the theory of E∞ring spectra and introduce the language of derived algebraic geometry.We apply this theory in Sect. 3, where we introduce the notion of an oriented group scheme and describe connection between oriented group schemes and equivariant cohomology theories. In Sect. 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 Sect. 5 we discuss various applications of these ideas, many of which rely upon a special feature of elliptic cohomology which we call 2equivariance. 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 overmany 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 [1].
 Title
 A Survey of Elliptic Cohomology
 Book Title
 Algebraic Topology
 Book Subtitle
 The Abel Symposium 2007
 Pages
 pp 219277
 Copyright
 2009
 DOI
 10.1007/9783642012006_9
 Print ISBN
 9783642011993
 Online ISBN
 9783642012006
 Series Title
 Abel Symposia
 Series Volume
 4
 Publisher
 Springer Berlin Heidelberg
 Copyright Holder
 SpringerVerlag Berlin Heidelberg
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 Editors

 Nils Baas ^{(ID1)}
 Eric M. Friedlander ^{(ID2)}
 Björn Jahren ^{(ID3)}
 Paul Arne Østvær ^{(ID4)}
 Editor Affiliations

 ID1. Universitet NTNU, Norges Teknisk Naturvitenskap.
 ID2. College of Letters, Arts & Sciences, University of Southern California
 ID3. University of Oslo, Department of Mathematics
 ID4. University of Oslo, Department of Mathematics
 Authors

 J. Lurie ^{(1)}
 Author Affiliations

 1. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA
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