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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, the spectrum of topological modular forms (also known as "tmf") describes a generalized cohomology theory whose coefficient ring are related to the graded ring of holomorphic modular forms with integral cusp expansions. Indeed, these two rings become isomorphic after inverting 6. tmf is constructed as the global sections of a sheaf of E-infinity ring spectra on the moduli stack of (generalized) elliptic curves. This theory has relations to the theory of modular forms in number theory, the homotopy groups of spheres, and conjectural index theories on loop spaces of manifolds. tmf was first constructed by Mike Hopkins and Haynes Miller; many of the computations can be found in preprints and articles by Paul Goerss, Mike Hopkins, Mark Mahowald, Haynes Miller, Charles Rezk, and Tilman Bauer.