Cycles, characters and global geometry

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AbstractAward: DMS-0404766Principal Investigator: H. Blaine Lawson, Jr.This project is concerned with the study of cycles, residues,boundaries and differential characters. The proposal has severalinterrelated parts. The first concerns the groups of algebraiccycles and cocycles on a projective variety $X$. The aim is torelate these groups to the global structure of $X$. Theinvestigator has, with others, established a theory of homologytype for algebraic varieties based on the homotopy groups ofcycles spaces. This theory will be used to study concretequestions about algebraic spaces. Implications for realalgebraic geometry will be explored. Striking onnections touniversal constructions in topology which emerged in priorresearch will also be investigated. A second part of the proposalconcerns cycles which bound complex subvarieties in a projectivemanifold. Several new conjectures relate these cycles toapproximation theory, pluripotential theory, and the projectivespectrum of Banach graded algebras. A third area of the proposalconcerns the study of singularities and characteristicforms. This subject includes a generalization of Chern-Weiltheory which gives canonical homologies between singularities ofbundle maps and characteristic forms. It includes a usefulanalytic tool -- geometric atomicity -- which will be studied,and it yields a new approach to Morse Theory. Applicationsrelating singularities to global geometry remain to beinvestigated. The forth part of the proposal concerns sparks andspark complexes. This recently developed framework for the studyof differential characters has yielded interestinggeneralizations which extend Deligne cohomology and arithmeticChow groups. They are essentially secondary invariants whichmediate between cycles and smooth data. Further development ofthe theory and its application to the study of cycles isproposed. A fifth area is concerned with special cycles ingeometry, in particular Special Lagrangian cycles in Calabi-Yaumanifolds, and associative and Cayley cycles in $G_2$ andSpin$_7$ spaces. These latter subjects relate to mirror symmetryconjectures and to M-theory in Physics as well as many areas ofgeometry and algebra. This project will also be concerned withstudent development, including an undergraduate educationaleffort aimed at fostering mathematical independence anddeveloping interactive enviornments.A concept of central importance in geometry is that of a``cycle''. In algebraic geometry a cycle corresponds to thesimultaneous solution of a system of polynomial equations. Indifferential geometry they arise in many ways: as the large scalesolutions of certain differential equations, and as the levelsets and singularity sets of differentiable mappings. Curves andsurfaces in space are simple examples. Cycles with a particulargeometry also play a fundamental role in modern physical theoriesThis proposal is concerned with the study of cycles across thisbroad spectrum. In the algebraic setting cycles have been relatedto fundamental large-scale geometry of their surroundingspace. This discovery has revealed surprizing and importantrelationships between spaces of algebraic cycles and fundamentalconstructions in algebraic topology and has led to new insightsin both fields. This work will be continued.Another area of investigation concerns cycles which form theboundary of subsets with special geometric structure. Theyrepresent non-linear versions of classical boundary valueproblems in analysis. Such questions arise in many contexts.Recently the proposer has formulated conjectures relating certainimportant classes of such cycles to questions in approximationtheory and Banach algebras. Successful resolution should producesignificant new insights in several fields of mathematics.A third area of study concerns a mathematical apparatus developedby the proposer to detect subtle relationships between cycles andthe global structure of the space they live in. This apparatusencompasses some of the most effective tools historicallydeveloped for this purpose, and it is much more general. Furtherdevelopment of this theory and its applications will be persued.A fourth domain of investigation is concerned with special cyclesin 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 inPhysicsThis project will also be concerned with graduate studentdevelopment. Students will be part of the research team. Therewill also be an undergraduate educational effort aimed atfostering mathematical independence and developing interactiveenvironments.