Topics in Topology and Geometry

拓扑与几何专题

• 批准号: 9803254
• 负责人: Dan Burghelea
• 金额: --
• 依托单位: Ohio State University
• 依托单位国家: 美国
• 项目类别: Continuing grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-07-15 至 2002-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9803254&HistoricalAwards=false
• 关键词: Topics; Topology; Geometry

9803254 Burghelea D. Burghelea plans work in topology, geometric analysis and dynamics, based on methods of linear algebra ``a la von Neumann'' and on regularized determinants of elliptic operators. He plans: 1) to attack a number of open problems on L2-invariants, torsion invariants, relation between torsion and dynamics, 2) to develop new mathematical tools, such as the Hodge theory of geometric complexes associated to Morse-Bott functions, Witten-Hellfer-Sjostrand theory with symmetry and with parameters, and 3) to test these new tools against a number of open problems in geometric analysis and spectral geometry. The research will shed more light on the nature and the power of the L2 invariants and will considerably increase the generality of some successful techniques in geometric analysis and hopefully solve some open problems in topology and spectral geometry. The above work explores the relationship between the shape of a geometric object (a Riemannian manifold), as embodied in its topology and geometry, and sound, as embodied in the spectra of various Laplace operators associated to it -- think in terms of natural frequencies of vibration. It also investigates the constraints imposed by the shape and sound on basic qualitative elements of dynamics such as closed trajectories, attractors, repulsors, and saddle points. The methods used involve unusual quantities introduced by von Neumann, like dimensions that are not integers and volumes of infinitely large objects. The research will investigate the type of additional information about the shape and sound, and about the dynamics on geometric objects, that can be obtained by using these unusual quantities. ***

9803254 Burghelea D. Burghelea计划在拓扑，几何分析和动力学方面工作，基于线性代数的方法``a la von noumann''和椭圆算子的正则决定因素。 He plans: 1) to attack a number of open problems on L2-invariants, torsion invariants, relation between torsion and dynamics, 2) to develop new mathematical tools, such as the Hodge theory of geometric complexes associated to Morse-Bott functions, Witten-Hellfer-Sjostrand theory with symmetry and with parameters, and 3) to test these new tools against a number of open problems in geometric analysis and spectral geometry.该研究将更多地了解L2不变性的性质和力量，并将大大提高一些成功技术的一般性，并希望解决拓扑和光谱几何学方面的一些开放问题。上述工作探讨了几何对象的形状（riemannian歧管）的形状，如其拓扑和几何形状所体现的，声音在与其相关的各种拉普拉斯操作员的光谱中所体现 - 用天然振动频率进行思考。它还研究了由封闭轨迹，吸引子，驱动器和马鞍点等动力学基本定性元素所施加的限制。所使用的方法涉及冯·诺伊曼（von Neumann）引入的异常数量，例如不是整数和无限大对象的体积的维度。该研究将研究有关形状和声音的其他信息，以及有关几何对象的动力学的其他信息，这些信息可以通过使用这些异常数量来获得。 ***