Index Theory of Perturbed Dirac Operators

哈斯克尔ABS 摘要 提案：DMS-9800782主要研究者：Peter Haskell 哈斯克尔教授计划研究狄拉克型扰动算子在完备和不完备非紧流形上的指数理论。指数理论的不变量，这样的运营商扩展了不变量所产生的粗糙的几何形状，并建议制定一个理论的结束算子代数。具有亚纯扰动的Dolbeault算子的一些例子已经在物理模型中得到了应用。指数公式更一般的例子涉及托德类扰动的可能奇异零集。 在闭流形上紧李群的光滑作用下不变且在横向方向上椭圆的算子的指数通常表示为分布，但可以表示为算子的核和上核中不可约表示的重数之差。Haskell教授计划使用不完整流形上扰动的一阶微分算子的指数来计算后一个版本。哈斯克尔教授还计划发展与自伴扰动狄拉克算子相关的指数理论。这将包括这些运营商的作用，在指数理论的扰动狄拉克运营商的非紧流形与非紧边界。这样的工作将开始的影响的扰动增长和结束几何的自伴扰动狄拉克算子的谱理论的调查。 由于物理学的大部分内容都是关于速度和加速度等表示变化率的量，因此物理学的数学模型通常表示为微分方程，其解是变化率具有特定性质的函数。指数理论（英语：Index theory）（如Atiyah-Singer指数定理）是研究微分方程解集大小的比较所揭示的几何性质的理论。几何学和拓扑学中许多最具挑战性的问题都涉及到无限延伸的空间。由于许多物理模型都是基于欧几里得空间，物理学家已经研究了无限延伸空间上的微分方程的例子。涉及这些微分方程的物理问题在存在约束力的情况下是最容易处理的，约束力的数学表现形式是微分方程中的一个势项（例如量子物理学的薛定谔方程）。Haskell教授计划在更复杂的几何设置中研究具有势项的微分方程所携带的几何信息。虽然理解无限延伸的复杂空间的分析和几何本身就是一个目标，但这项工作也可能对几何约束的物理系统模型产生影响。由于哈斯克尔教授的技术是基于“非交换拓扑”（即，空间是通过其上的函数的代数来研究的），这项工作也应该揭示更精细的结构，如对称性，可以在非交换代数中编码。

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.