Index Theory of Perturbed Dirac Operators

点击翻译按钮获取中文摘要

Haskell. Abs Abstract Proposal: DMS-9800782 Principal Investigator: Peter Haskell Professor Haskell plans to study the index theory of perturbed operators of Dirac type on complete and incomplete noncompact manifolds. Index-theoretic invariants of such operators extend the invariants arising from coarse geometry and suggest a formulation of a theory of ends for operator algebras. Some examples of Dolbeault operators with meromorphic perturbations have been used in physical models. Index formulas for more general examples involve the Todd class of the perturbation's possibly singular zero set. The index of an operator that is invariant under and elliptic in directions transverse to a smooth action of a compact Lie group on a closed manifold is usually expressed as a distribution but can be expressed in terms of differences of multiplicities of irreducible representations in the operator's kernel and cokernel. Professor Haskell plans to calculate the latter version using indices of perturbed first-order differential operators on incomplete manifolds. Professor Haskell also plans to develop the index theory associated with self-adjoint perturbed Dirac operators. This will include the role of such operators in the index theory of perturbed Dirac operators on noncompact manifolds with noncompact boundaries. Such work will begin with the investigation of the effect of perturbation growth and end geometry on the spectral theory of self-adjoint perturbed Dirac operators. Because much of physics is about quantities, such as velocity and acceleration, which represent rates of change, mathematical models of physics are often expressed as differential equations, the solutions of which are functions whose rates of change have specified properties. Index theory (for instance, the Atiyah-Singer index theorem) is the study of geometric properties revealed by the comparison of sizes of solution sets of differential equations. Many of the most challenging problems in geometry and topology involve spaces that extend indefinitely. Because many physical models are based on Euclidean space, physicists have studied examples of differential equations on spaces that extend indefinitely. Physical problems involving these differential equations are most tractable in the presence of a constraining force, whose mathematical manifestation is as a potential term in the differential equations (for example, in the Schroedinger equations of quantum physics). Professor Haskell plans to investigate the geometric information carried by differential equations with potential terms in more complicated geometric settings. While understanding the analysis on and geometry of complicated spaces that extend indefinitely is a goal in itself, this work may also have implications for models of geometrically constrained physical systems. Because Professor Haskell's techniques are based on "noncommutative topology" (that is, spaces are studied via algebras of functions on them), this work should also reveal finer structures, such as symmetries, that can be encoded in noncommutative algebras.