Special Meeting: Fields Program in Geometric Applications of Homotopy Theory - International US Participation

Stanford University, Stanford CA

Investigators

Abstract

The applications of homotopy theory have always defined the most compelling aspects of the subject. The purpose of the Fields program is to study and develop new applications of homotopy theory in algebraic geometry, mathematical physics and related disciplines. The homotopy theories of simplicial sheaves and presheaves are the foundation for motivic homotopy theory, and as such have contributed to the major calculational successes of the last decade in algebraic K-theory. In recent years, presheaves of spectra on the stack of formal group laws have emerged in the definitions of elliptic cohomology theories and topological modular forms. Simplicial sets and related combinatorial constructions are also the foundation for higher category theory. All of these subjects have both current and intended applications in mathematical physics. Similar ideas are present in new applications of homotopy theory in algebraic combinatorics and computer science, particularly in analysis of hyperplane arrangements or graph colouring, and in models for concurrent behaviour of parallel processing systems, computational geometry, and complexity. The program will take place at the Fields Institute, which will mount an intensive program on Geometric Applications of Homotopy Theory during the period January-June 2007. NSF funding will allow a significant number of young U.S. researchers (postdocs, junior faculty, and graduate students) to participate in this program, and receive training in this area. Homotopy theory is a core topic within the mathematical subject of topology, but as described above, it impacts on a broad range of fields, including algebraic geometry, logic and computer science, and mathematical physics (eg in topological quantum field theory). Bringing together researchers in homotopy theory with those from the areas of application will stimulate exciting progress in both fields.