Homotopy Theory and Its Applications

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9802516Ravenel Professor Ravenel's current research is focused on the telescopeconjecture of stable homotopy theory, one of several conjectures heformulated 20 years ago, and the only one that is not well understood atthis time. Other conjectures he made then have since evolved into thenilpotence theorem of Devinatz-Hopkins-Smith, the periodicity theorem ofHopkins-Smith, and the the thick subcategory theorem. All of these arepart of the chromatic approach to homotopy theory, which continues toattract the interest of numerous algebraic topologists. Morespecifically, Ravenel has constructed and will study the ThomifiedEilenberg-Moore spectral sequence, which includes the usualEilenberg-Moore spectral sequence, the Adams spectral sequence, and theAdams-Novikov spectral sequence as special cases. He also has a programfor the computation of the Morava K-theory of interated loop spaces ofspheres. Stable homotopy theory is a branch of algebraic topology, which hasbeen central to pure mathematics since its founding by Poincare a centuryago. It has been a continuing source of new ideas in algebra and geometry,as seen in the work of Lefschetz on complex algebraic varieties in the1930's, the efforts leading to the proof of the Weil conjectures ofarithmetic algebraic geometry in the 1960's and 1970's, and most recently,in the successful application of motivic cohomology to the Milnorconjecture in algebraic K-theory by Voevodsky in 1997. It has also foundnumerous applications in differential geometry, in graph theory, and intheoretical physics. The University of Rochester is one of the leadingcenters of algebraic topology in the world. ***