Cycles, Differential Characters and Global Problems in Geometry

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Abstract for DMS - 0102525 (Blaine Lawson)This project is concerned with global problems in geometry and inparticular with the study of cycles residues and differential characters.It focuses on the relationship between certain important families of cyclesin a space and the geometry of the space itself. Of particular interestare algebraic cycles and the cycles associated to singularities of mappingsor the higher order contact of geometric structures. These objects -- ofimportance in themselves -- have been shown to have ties to other areas ofmathematics. A major aim here is the discovery and development of suchties. The proposal has several interrelated parts. The first concerns groupsof algebraic cycles and cocycles on a projective variety. A theory ofhomology-type based on cycles has been developed by the proposer andothers. It will be used to study concrete questions about algebraicspaces. 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 thequaternionic analogues will be studied. A second part of the proposal concerns differential characters, objects which mediate between cycles and smooth data, and lead to importantgeometric invariants. Recent discoveries have been made concerning them --for example, the existence of a fundamental duality theorem. Furtherdevelopment of the theory is proposed. Geometric results will be sought bybringing 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 globalgeometry of singularities, and its relation to characteristic classes anddifferential 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 gaugefield theory and gravity in Physics as well as many areas of geometry andalgebra. A concept of central importance in geometry is that of a ``cycle''.In algebraic geometry a cycle corresponds to the simultaneous solution of asystem of polynomial equations. In differential geometry cycles arise inmany 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 isconcerned with the study of certain important classes of cycles which arisein geometry. Part of the study aims at relating them to fundamentallarge-scale geometry of the surrounding space. In the algebraic case thishas led to the establishment of surprizing and important relationshipsbetween spaces of algebraic cycles and fundamental constructions inalgebraic topology that have led to new insights in both fields. This workwill be continued with the intent of obtaining further concreteapplications. A second part of the proposal concerns differential characters,objects which mediate between cycles and smooth data. They lead toimportant geometric invariants and have appeared in discussions of the``Mirror Symmetry Conjecture'' from modern physics. The proposer has madesome recent discoveries about characters, including a basic DualityTheorem. Further development of the theory and its applications isproposed. Another area of investigation is concerned with relationsbetween cycles and geometry which arise from connections. Connections arefundamental in mathematics, where they constitute differentiation laws, andin physics, where they represent the fundamental forces of nature at theclassical level. The investigator has developed a theory of singular connections whichencompasses much previously unrelated phenomena and has applications tomany areas of geometry. The proposal will continue this work withemphasis on applications. Yet another area of the proposal is concernedwith very special cycles in geometry which relate to gauge field theory andgravity 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 anundergraduate educational effort aimed at fostering mathematicalindependence and developing interactive environments.