Mathematical Sciences: Homotopy Theory of Spectra

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9505130 Smith Spectra have been important in homotopy theory for many years. Recently the investigator has constructed a category of spectra with a symmetric monoidal smash product. The construction is elementary and will provide an introduction to spectra which will not scare away the novice. The strict associativity of the smash product means that many other definitions regarding spectra can be greatly simplified. For example, one can easily discuss ring spectra, commutative ring spectra, module spectra. The investigator is using this new definition of ring spectra in the study of topological Hochschild homology and topological cyclic homology. Spectra have been developed over the last thirty years as a bridge between the geometric problems that the algebraic topologist studies and the algebraic techniques which he uses. The earliest examples were given by Rene Thom in his work on the cobordism classification of manifolds, where he showed that manifolds, the geometric objects common to mathematics and physics, are classified up to cobordism by certain spectra. Over the years, as one needed spectra having more and more properties, many different constructions have been given. These constructions have became increasingly complicated, until only a few experts could understand them. However, recently the investigator has found a much easier way to construct spectra that nonetheless have all the properties one needs for the applications. This makes the field far more accessible, especially to graduate students, and should lead to much fuller appreciation of this powerful tool. ***