Principal Investigator: James E. McClure
The principal investigator plans to work in four quite different areas where homotopy-theoretic methods can be applied. The first area is string topology. This part of the project is joint with Ralph Kaufmann; Kaufmann and McClure will combine their previous work on string topology in order to obtain a lifting of string topology to the chain level for arbitrary genus. The second area is surgery spectra. This is joint with Gerd Laures; Laures and McClure will complete their proof that certain symmetric L-spectra have E-infinity ring structures and that the symmetric signature is an E-infinity ring map. They will also extend their work in order to show that a certain map from algebraic K-theory to symmetric L-theory defined by Weiss and Williams is an E-infinity ring map. The third area is intersection homology. This is joint with Greg Friedman and Scott Wilson; Friedman, McClure and Wilson will work to develop homotopy theoretic structure on intersection chains which should (for example) lead to the development of a rational homotopy theory of stratified spaces. The fourth area is C* algebras. This is joint with Marius Dadarlat; Dadarlat and McClure will work to show that the classical work of Dixmier and Douady on bundles whose fibre is the algebra of compact operators can be extended to bundles whose fibre is a Kirchberg algebra.
The original goal of homotopy theory was to develop methods which could be applied to geometric problems (including problems in dimensions higher than three). Since its origin in the early twentieth century, homotopy theory has developed into an autonomous discipline, but the connection with geometry has always been a central concern. Since about 1960, it has become apparent that homotopy theory has applications to other things besides geometry, for example to analysis, algebra, and more recently to physics. The work of the principal investigator and his coauthors is intended to take the latest developments in homotopy theory (for example, the theory of symmetric spectra) and apply them to other areas, including geometric problems (the classification of higher-dimensional shapes), and problems in mathematical physics (string topology is part of the program being carried on by many people to develop a clearer mathematical understanding of ideas from string theory).
