Edward Witten once said that Elliptic Cohomology was a piece of 21st Century Mathematics that happened to fall into the 20th Century. He also likened our understanding of it to what we know of the topography of an archipelago; the peaks are beautiful and clearly connected to each other, but the exact connections are buried, as yet invisible. This very active subject has connections to algebraic topology, theoretical physics, number theory and algebraic geometry, and all these connections are represented in the sixteen papers in this volume. A variety of distinct perspectives are offered, with topics including equivariant complex elliptic cohomology, the physics of M-theory, the modular characteristics of vertex operator algebras, and higher chromatic analogues of elliptic cohomology. This is the first collection of papers on elliptic cohomology in almost twenty years and gives a broad picture of the state of the art in this important field of mathematics.

Elliptic cohomology is a very active field of mathematics, with connections to algebraic topology, theoretical physics, number theory and algebraic geometry. All these connections are represented in the sixteen papers in this volume, which offer a variety of distinct perspectives. Topics covered include equivariant complex elliptic cohomology, the physics of M-theory, the modular characteristics of vertex operator algebras, and higher chromatic analogues of elliptic cohomology. This is the first collection of papers on elliptic cohomology in almost twenty years and offers a broad picture of this important field of mathematics.

Preface; 1. Discrete torsion for the supersingular orbifold sigma genus Matthew Ando and Christopher P. French; 2. Quaternionic elliptic objects and K3-cohomology Jorge A. Devoto; 3. Algebraic groups and equivariant cohomology theories John P. C. Greenlees; 4. Delocalised equivariant elliptic cohomology Ian Grojnowski; 5. On finite resolutions of K(n)-local spheres Hans-Werner Henn; 6. Chromatic phenomena in the algebra of BP*BP-comodules Mark Hovey; 7. Numerical polynomials and endomorphisms of formal group laws Keith Johnson; 8. Thom prospectra for loopgroup representations Nitu Kitchloo and Jack Morava; 9. Rational vertex operator algebras Geoffrey Mason; 10. A possible hierarchy of Morava K-theories Norihiko Minami; 11. The M-theory 3-form and E8 gauge theory Emanuel Diaconescu, Daniel S. Freed and Gregory Moore; 12. The motivic Thom isomorphism Jack Morava; 13. Toward higher chromatic analogs of elliptic cohomology Douglas C. Ravenel; 14. What is an elliptic object? Graeme Segal; 15. Spin cobordism, contact structure and the cohomology of p-groups C. B. Thomas; 16. Brave New Algebraic Geometry and global derived moduli spaces of ring spectra Bertrand Toen and Gabriele Vezzosi; 17. The elliptic genus of a singular variety Burt Totaro.