Cycles, Residues & Global Problems in Geometry

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AbstractProposal: DMS 9802054Principal Investigators: H. B. Lawson, Jr. and M.-L. MichelsohnThis multi-part project is concerned with the study of cycles andresidues in geometry. One part concerns the groups of algebraiccycles and cocycles on a projective variety X and aims to relate thesegroups to the global structure of X. A theory of homology type foralgebraic varieties based on homotopy groups of cycle spaces has beendeveloped, and will be used to study concrete questions aboutalgebraic spaces. A second part of the proposal concerns cycles inprojective space, which have surprizing connections to fundamentalconstructions in algebraic topology. Some of the resulting questionsconcern spaces of real and quaternionic cycles related tocharacteristic classes and representation theory. Others concerncycles under the action of a finite group. Here the spaces have ledto new equivariant cohomology theories whose development andapplication will be explored. A third area of investigation concernssingular connections and characteristic currents, a generalization ofclassical Chern-Weil theory which relate singularities of mappings tocharacteristic forms in a canonical analytic way; applications anddevelopments of the theory include a new approach to Morse Theory. Afourth area concerns calibrated cycles in geometry: special Lagrangiancycles in Calabi-Yau manifolds, cycles related to existence ofp-Kaehler spaces, and cycles appearing in M-brane theory. Thisproject 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 andhas several interrelated parts. The first aims at furthering ourunderstanding of the spaces which arise as solutions of systems ofalgebraic equations (so called ``algebraic cycles''). These spaceshave a long history and play a central role in many areas ofmathematics, applied mathematics and physics. Breakthroughs over thepast ten years have given fresh insights into the subject and a richlystructured theory has emerged. The proposed research will forge newlinks between algebra and geometry/topology, and lead towards settlingsome important conjectures in the field. Another area ofinvestigation is concerned with relations between cycles and geometrywhich arise from connections. Connections are fundamental inmathematics, where they constitute differentiation laws, and inphysics, where they represent the fundamental forces of nature at theclassical level. The investigators have developed a theory ofsingular connections which encompasses many previously unrelatedphenomena and has applications to several areas of geometry. Thisproject will continue this work with emphasis on applications. Instudying the least area problem one of the investigators developed atheory of calibrated cycles which currently plays an important role inphysical theories. This new relationship has raised some importantquestions and conjectures that will be studied.