Topological modular forms

In mathematics, the spectrum of topological modular forms (tmf) describes a generalized cohomology theory whose coefficient ring is 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.

Construction

The original construction of tmf uses the obstruction theory of Hopkins, Miller, and Paul Goerss, and is based on ideas of Dwyer, Kan, and Stover. In this approach, one defines a presheaf O^top ("top" stands for topological) of multiplicative cohomology theories on the etale site of the moduli stack of elliptic curves and shows that this can be lifted in an essentially unique way to a sheaf of E-infinity ring spectra. This sheaf has the following property: to any etale elliptic curve over a ring R, it assigns an E-infinity ring spectrum (a classical elliptic cohomology theory) whose associated formal group is the formal group of that elliptic curve.

TMF

TMF may refer to:

TMF (UK & Ireland)

TMF (The Music Factory) was a music video and entertainment channel in the United Kingdom and Ireland. The channel was owned by Viacom International Media Networks and was originally a Dutch channel. Formed after the two other TMF stations, which are based in mainland Europe, the channel was created to counter against EMAP's The Hits channel (now 4Music) on the new free-to-air digital terrestrial television service Freeview in 2002.

Upon launch, TMF's description was "TMF is designed to replicate the rhythms of the whole family, playing the best pop videos with the biggest pop stars to become the sound track to the British family life." However, it had since broadened its content to show programmes from its sister channels MTV, VH1, Nick Jr. and Comedy Central, so no longer relied on just music videos.

TMF broadcast on Freeview, Sky and Virgin Media and as well as in some Irish cable packages. The channel was the most viewed music video related station in the UK, according to BARB ratings.