Index Theory of Perturbed Dirac Operators

哈斯克尔。 Abs 摘要提案：DMS-9800782 首席研究员：Peter Haskell Haskell 教授计划研究完全和不完全非紧流形上狄拉克型摄动算子的指数理论。此类算子的索引理论不变量扩展了由粗略几何产生的不变量，并提出了算子代数的终结理论的公式。具有亚纯扰动的 Dolbeault 算子的一些例子已在物理模型中使用。更一般示例的索引公式涉及扰动的可能奇异零集的托德类。 在闭流形上紧致李群的平滑作用的横向方向上不变且椭圆的算子的索引通常表示为分布，但也可以用算子核和协核中不可约表示的重数差来表示。哈斯克尔教授计划使用不完全流形上的扰动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。这将包括此类算子在具有非紧边界的非紧流形上的扰动狄拉克算子的指数理论中的作用。此类工作将从研究扰动增长和末端几何对自伴扰动狄拉克算子谱理论的影响开始。 由于物理学的大部分内容都是关于数量的，例如代表变化率的速度和加速度，因此物理学的数学模型通常表示为微分方程，其解是变化率具有指定属性的函数。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来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最容易处理，其数学表现形式是微分方程中的潜在项（例如，在量子物理的薛定谔方程中）。哈斯克尔教授计划研究更复杂的几何设置中具有潜在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形状本身就是一个目标，但这项工作也可能对几何约束物理系统的模型产生影响。由于哈斯克尔教授的技术基于“非交换拓扑”（即通过空间上的函数代数来研究空间），因此这项工作还应该揭示可以用非交换代数编码的更精细的结构，例如对称性。

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.