Elliptic Cohomology: Geometry, Applications, and Higher Chromatic Analogues

doi:10.1017/CBO9780511721489

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.

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Contents

Frontmatter
pp i-iv
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Contents
pp v-vi
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Preface
pp vii-xii
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Charles Thomas, 1938–2005
pp xiii-xvi
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1 - Discrete torsion for the supersingular orbifold sigma genus
pp 1-25
- By Matthew Ando, Christopher P. French
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2 - Quaternionic elliptic objects and K3-cohomology
pp 26-43
- By Jorge A. Devoto
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3 - The M-theory 3-form and E8 gauge theory
pp 44-88
- By Emanuel Diaconescu, Daniel S. Freed, Gregory Moore
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4 - Algebraic groups and equivariant cohomology theories
pp 89-113
- By John P. C. Greenlees
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5 - Delocalised equivariant elliptic cohomology (with an introduction by Matthew Ando and Haynes Miller)
pp 114-121
- By Ian Grojnowski
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6 - On finite resolutions of K(n)-local spheres
pp 122-169
- By Hans-Werner Henn
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7 - Chromatic phenomena in the algebra of BP*BP-comodules
pp 170-203
- By Mark Hovey
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8 - Numerical polynomials and endomorphisms of formal group laws
pp 204-213
- By Keith Johnson
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9 - Thom prospectra for loopgroup representations
pp 214-238
- By Nitu Kitchloo, Jack Morava
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10 - Rational vertex operator algebras
pp 239-254
- By Geoffrey Mason, UNIVERSITY OF CALIFORNIA, SANTA CRUZ
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11 - A possible hierarchy of Morava K-theories
pp 255-264
- By Norihiko Minami
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12 - The motivic Thom isomorphism
pp 265-285
- By Jack Morava
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13 - Toward higher chromatic analogs of elliptic cohomology
pp 286-305
- By Douglas C. Ravenel
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14 - What is an elliptic object?
pp 306-317
- By Graeme Segal, ALL SOULS COLLEGE, OXFORD
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15 - Spin cobordism, contact structure and the cohomology of p-groups
pp 318-324
- By C. B. Thomas
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16 - Brave New Algebraic Geometry and global derived moduli spaces of ring spectra
pp 325-359
- By Bertrand Toën, Gabriele Vezzosi
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17 - The elliptic genus of a singular variety
pp 360-364
- By Burt Totaro
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Elliptic Cohomology

Geometry, Applications, and Higher Chromatic Analogues

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Contents

Frontmatter
pp i-iv

Contents
pp v-vi

Preface
pp vii-xii

Charles Thomas, 1938–2005
pp xiii-xvi

1 - Discrete torsion for the supersingular orbifold sigma genus
pp 1-25

2 - Quaternionic elliptic objects and K3-cohomology
pp 26-43

3 - The M-theory 3-form and E8 gauge theory
pp 44-88

4 - Algebraic groups and equivariant cohomology theories
pp 89-113

5 - Delocalised equivariant elliptic cohomology (with an introduction by Matthew Ando and Haynes Miller)
pp 114-121

6 - On finite resolutions of K(n)-local spheres
pp 122-169

7 - Chromatic phenomena in the algebra of BP*BP-comodules
pp 170-203

8 - Numerical polynomials and endomorphisms of formal group laws
pp 204-213

9 - Thom prospectra for loopgroup representations
pp 214-238

10 - Rational vertex operator algebras
pp 239-254

11 - A possible hierarchy of Morava K-theories
pp 255-264

12 - The motivic Thom isomorphism
pp 265-285

13 - Toward higher chromatic analogs of elliptic cohomology
pp 286-305

14 - What is an elliptic object?
pp 306-317

15 - Spin cobordism, contact structure and the cohomology of p-groups
pp 318-324

16 - Brave New Algebraic Geometry and global derived moduli spaces of ring spectra
pp 325-359

17 - The elliptic genus of a singular variety
pp 360-364

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Elliptic Cohomology

Geometry, Applications, and Higher Chromatic Analogues

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