Proposal: DMS 9802054 Principal Investigators: H. B. Lawson, Jr. and M.-L. Michelsohn
This multi-part project is concerned with the study of cycles and residues in geometry. One part concerns the groups of algebraic cycles and cocycles on a projective variety X and aims to relate these groups to the global structure of X. A theory of homology type for algebraic varieties based on homotopy groups of cycle spaces has been developed, and will be used to study concrete questions about algebraic spaces. A second part of the proposal concerns cycles in projective space, which have surprizing connections to fundamental constructions in algebraic topology. Some of the resulting questions concern spaces of real and quaternionic cycles related to characteristic classes and representation theory. Others concern cycles under the action of a finite group. Here the spaces have led to new equivariant cohomology theories whose development and application will be explored. A third area of investigation concerns singular connections and characteristic currents, a generalization of classical Chern-Weil theory which relate singularities of mappings to characteristic forms in a canonical analytic way; applications and developments of the theory include a new approach to Morse Theory. A fourth area concerns calibrated cycles in geometry: special Lagrangian cycles in Calabi-Yau manifolds, cycles related to existence of p-Kaehler spaces, and cycles appearing in M-brane theory. This project is also concerned with graduate student development, especially interaction at the research level among graduate students.
This project concerns questions of global structure in geometry and has several interrelated parts. The first aims at furthering our understanding of the spaces which arise as solutions of systems of algebraic equations (so called ``algebraic cycles''). These spaces have a long history and play a central role in many areas of mathematics, applied mathematics and physics. Breakthroughs over the past ten years have given fresh insights into the subject and a richly structured theory has emerged. The proposed research will forge new links between algebra and geometry/topology, and lead towards settling some important conjectures in the field. Another area of investigation is concerned with relations between cycles and geometry which arise from connections. Connections are fundamental in mathematics, where they constitute differentiation laws, and in physics, where they represent the fundamental forces of nature at the classical level. The investigators have developed a theory of singular connections which encompasses many previously unrelated phenomena and has applications to several areas of geometry. This project will continue this work with emphasis on applications. In studying the least area problem one of the investigators developed a theory of calibrated cycles which currently plays an important role in physical theories. This new relationship has raised some important questions and conjectures that will be studied.
