By Christopher L. Douglas, John Francis, André G. Henriques, Michael A. Hill, Andre G. Henriques

The speculation of topological modular types is an difficult mixture of classical algebraic modular kinds and good homotopy teams of spheres. the development of this concept combines an algebro-geometric viewpoint on elliptic curves over finite fields with ideas from algebraic topology, quite good homotopy conception. It has functions to and connections with manifold topology, quantity concept, and string thought. This e-book offers a cautious, obtainable creation to topological modular kinds. After a short heritage and a longer evaluation of the topic, the booklet right commences with an exposition of classical elements of elliptic cohomology, together with historical past fabric on elliptic curves and modular varieties, an outline of the moduli stack of elliptic curves, an evidence of the precise functor theorem for developing cohomology theories, and an exploration of sheaves in sturdy homotopy conception. There follows a remedy of extra really expert themes, together with localization of spectra, the deformation thought of formal teams, and Goerss-Hopkins obstruction concept for multiplicative buildings on spectra. The booklet then proceeds to extra complicated fabric, together with discussions of the string orientation, the sheaf of spectra at the moduli stack of elliptic curves, the homotopy of topological modular varieties, and an in depth account of the development of the spectrum of topological modular kinds. The ebook concludes with the 3 unique, pioneering and vastly influential manuscripts at the topic, by way of Hopkins, Miller, and Mahowald.

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