CYCLES DIFFERENTIAL CHARACTERS AND GLOBAL PROBLEMS IN GEOMETRY

Abstract for DMS - 0102525 (Blaine Lawson) This project is concerned with global problems in geometry and in particular with the study of cycles residues and differential characters. It focuses on the relationship between certain important families of cycles in a space and the geometry of the space itself. Of particular interest are algebraic cycles and the cycles associated to singularities of mappings or the higher order contact of geometric structures. These objects -- of importance in themselves -- have been shown to have ties to other areas of mathematics. A major aim here is the discovery and development of such ties. The proposal has several interrelated parts. The first concerns groups of algebraic cycles and cocycles on a projective variety. A theory of homology-type based on cycles has been developed by the proposer and others. It will be used to study concrete questions about algebraic spaces. In a variant of the theory involving real algebraic cycles, surprizing connections to equivariant homotopy theory have been found. The implications for real algebraic geometry will be explored, and the quaternionic analogues will be studied. A second part of the proposal concerns differential characters, objects which mediate between cycles and smooth data, and lead to important geometric invariants. Recent discoveries have been made concerning them -- for example, the existence of a fundamental duality theorem. Further development of the theory is proposed. Geometric results will be sought by bringing the calculus of variations to bear in this domain. A third area of the proposal concerns the study of singularities and characteristic forms. The subject includes a generalization of Chern-Weil theory which gives canonical homologies between singularities of bundle maps and characteristic forms. Many applications concerning the global geometry of singularities, and its relation to characteristic classes and differential characters, will be investigated. A forth area is concerned with special cycles in geometry: Special Lagrangian cycles in Calabi-Yau manifolds, and associative and Cayley cycles in G(2) and Spin(7) spaces. These latter subjects relate to gauge field theory and gravity in Physics as well as many areas of geometry and algebra. A concept of central importance in geometry is that of a ``cycle''. In algebraic geometry a cycle corresponds to the simultaneous solution of a system of polynomial equations. In differential geometry cycles arise in many ways: as the large scale solutions of certain differential equations, and as the level sets and singularity sets of differentiable mappings. Curves and surfaces in space are simple examples. This proposal is concerned with the study of certain important classes of cycles which arise in geometry. Part of the study aims at relating them to fundamental large-scale geometry of the surrounding space. In the algebraic case this has led to the establishment of surprizing and important relationships between spaces of algebraic cycles and fundamental constructions in algebraic topology that have led to new insights in both fields. This work will be continued with the intent of obtaining further concrete applications. A second part of the proposal concerns differential characters, objects which mediate between cycles and smooth data. They lead to important geometric invariants and have appeared in discussions of the ``Mirror Symmetry Conjecture'' from modern physics. The proposer has made some recent discoveries about characters, including a basic Duality Theorem. Further development of the theory and its applications is proposed. 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 investigator has developed a theory of singular connections which encompasses much previously unrelated phenomena and has applications to many areas of geometry. The proposal will continue this work with emphasis on applications. Yet another area of the proposal is concerned with very special cycles in geometry which relate to gauge field theory and gravity in Physics as well as many areas of geometry and algebra. This project will also be concerned with graduate student development. Students will be part of the research team. There will also be an undergraduate educational effort aimed at fostering mathematical independence and developing interactive environments.