Mathematical Sciences: Index Theory on Noncompact Manifolds

数学科学：非紧流形指标论

• 批准号: 9500724
• 负责人: Peter Haskell
• 金额: $ 5.33万
• 依托单位: Virginia Polytechnic Institute and State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1995
• 资助国家: 美国
• 起止时间: 1995-06-15 至 1998-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9500724&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Index; Theory; Noncompact

9500724 Haskell This project involves the study of the index theory of Dirac-Schrodinger operators on noncompact manifolds. In this area of index theory both incomplete and complete noncompact manifolds can be considered from a single perspective. The perspective combines techniques of regular singular theory and operator algebras. It encompasses self-adjoint Dirac-Schrodinger operators as well as those with nonzero Fredholm indices. It addresses problems in the index theory of invariant operators elliptic in directions transverse to actions of compact Lie groups as well as problems involving singular Riemannian foliations and related questions on manifolds with conical singularities. These problems all involve Hilbert space operators and their relation to the geometric properties of manifolds, the higher dimensional analogues of surfaces. Partial differential operators play a large role in this work. These operators occur naturally in the study of applied mathematics and mathematical physics. The overall goal is to compute invariants associated with these operators which simultaneously yield geometric information and information about the nature of solutions to important partial differential equations. ***

小行星9500724 本项目主要研究非紧流形上的Dirac-Schrodinger算子的指数理论。在这一领域的指数理论都不完全和完全非紧流形可以考虑从一个单一的角度。该观点结合了经常奇异理论和算子代数的技术。它包括自伴Dirac-薛定谔算子以及具有非零Fredholm指数的算子。它解决了问题的指标理论不变的运营商椭圆形的方向横向行动的紧凑李群，以及问题涉及奇异黎曼叶和相关问题的流形与锥形奇点。 这些问题都涉及希尔伯特空间算子及其与流形几何性质的关系，流形是曲面的高维类似物。偏微分算子在这项工作中发挥了很大的作用。这些算子在应用数学和数学物理的研究中自然出现。总体目标是计算与这些运营商，同时产生几何信息和重要的偏微分方程的解决方案的性质的信息相关的不变量。 ***