Topological Modular Forms

Christopher L. Douglas, John Francis, Andre G. Henriques, Michael A. Hill (Herausgeber)

Buch | Hardcover

315 Seiten

2014
American Mathematical Society (Verlag)
978-1-4704-1884-7 (ISBN)

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The theory of topological modular forms is an intricate blend of classical algebraic modular forms and stable homotopy groups of spheres. The construction of this theory combines an algebro-geometric perspective on elliptic curves over finite fields with techniques from algebraic topology, particularly stable homotopy theory. It has applications to and connections with manifold topology, number theory, and string theory.

This book provides a careful, accessible introduction to topological modular forms. After a brief history and an extended overview of the subject, the book proper commences with an exposition of classical aspects of elliptic cohomology, including background material on elliptic curves and modular forms, a description of the moduli stack of elliptic curves, an explanation of the exact functor theorem for constructing cohomology theories, and an exploration of sheaves in stable homotopy theory. There follows a treatment of more specialized topics, including localization of spectra, the deformation theory of formal groups, and Goerss--Hopkins obstruction theory for multiplicative structures on spectra. The book then proceeds to more advanced material, including discussions of the string orientation, the sheaf of spectra on the moduli stack of elliptic curves, the homotopy of topological modular forms, and an extensive account of the construction of the spectrum of topological modular forms. The book concludes with the three original, pioneering and enormously influential manuscripts on the subject, by Hopkins, Miller, and Mahowald.

Christopher L. Douglas, Oxford University, United Kingdom. John Francis, Northwestern University, Evanston, IL, USA. Andre G. Henriques, Utrecht University, Netherlands. Michael A. Hill, University of Virginia, Charlottesville, VA, USA.

Elliptic genera and elliptic cohomology by C. Redden
Ellliptic curves and modular forms by C. Mautner
The moduli stack of elliptic curves by A. Henriques
The Landweber exact functor theorem by H. Hohnhold
Sheaves in homotopy theory by C. L. Douglas
Bousfield localization and the Hasse square by T. Bauer
The local structure of the moduli stack of formal groups by J. Lurie
Goerss-Hopkins obstruction theory by V. Angeltveit
From spectra to stacks by M. Hopkins
The string orientation by M. Hopkins
The sheaf of E ring spectra by M. Hopkins
The construction of tmf by M. Behrens
The homotopy groups of tmf and of its localizations by A. Henriques
Ellitpic curves and stable homotopy I by M. J. Hopkins and H. R. Miller
From elliptic curves to homotopy theory by M. Hopkins and M. Mahowald
1 E ring spectra by M. J. Hopkins
Glossary by C. L. Douglas, J. Francis, A. G. Henriques, and M. A. Hill

Reihe/Serie	Mathematical Surveys and Monographs
Verlagsort	Providence
Sprache	englisch
Maße	178 x 254 mm
Gewicht	778 g
Themenwelt	Mathematik / Informatik ► Mathematik ► Algebra
Themenwelt	Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
ISBN-10	1-4704-1884-3 / 1470418843
ISBN-13	978-1-4704-1884-7 / 9781470418847
Zustand	Neuware