Index Theory of Perturbed Dirac Operators

Haskell。项目负责人：Peter Haskell教授计划研究完全非紧流形和不完全非紧流形上Dirac型微扰算子的指标理论。这类算子的指标论不变量扩展了由粗糙几何产生的不变量，并提出了算子代数的端点理论的表述。一些具有亚纯微扰的Dolbeault算子的例子已经在物理模型中得到应用。更一般例子的指标公式涉及到微扰可能的奇异零集的Todd类。在闭流形上紧李群的光滑作用的横方向上不变性和椭圆性算子的指标通常表示为一个分布，但可以用算子核和核中不可约表示的多重性的差异来表示。Haskell教授计划用不完全流形上的扰动一阶微分算子的指标来计算后一个版本。Haskell教授还计划发展与自伴随摄动狄拉克算子相关的指标理论。这将包括这些算子在具有非紧边界的非紧流形上的摄动狄拉克算子的指标理论中的作用。这项工作将从研究摄动增长和端几何对自伴随摄动狄拉克算子谱理论的影响开始。因为很多物理学都是关于量的，比如速度和加速度，它们代表变化率，所以物理学的数学模型通常用微分方程来表示，微分方程的解是变化率具有特定性质的函数。指数理论（例如，Atiyah-Singer指数定理）是通过比较微分方程解集的大小揭示几何性质的研究。几何和拓扑中许多最具挑战性的问题都涉及无限延伸的空间。由于许多物理模型都是基于欧几里得空间，物理学家研究了无限扩展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一个约束力的情况下是最容易处理的，它的数学表现是微分方程中的一个潜在项（例如，在量子物理学的薛定谔方程中）。Haskell教授计划在更复杂的几何环境中研究具有位势项的微分方程所携带的几何信息。虽然理解对无限延伸的复杂空间的分析和几何本身是一个目标，但这项工作也可能对几何约束物理系统的模型产生影响。因为Haskell教授的技术是基于“非交换拓扑”（也就是说，空间是通过其上的函数代数来研究的），这项工作也应该揭示更精细的结构，比如对称，可以用非交换代数来编码。

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.