Topics in Topology and Geometry

9803254 Burghelea D. Burghelea 计划基于线性代数“冯·诺依曼”方法和椭圆算子的正则化行列式，从事拓扑、几何分析和动力学方面的工作。他计划：1）解决有关 L2 不变量、扭转不变量、扭转与动力学之间的关系的许多开放性问题，2）开发新的数学工具，例如与 Morse-Bott 函数相关的几何复形 Hodge 理论、具有对称性和参数的 Witten-Hellfer-Sjostrand 理论，以及 3）针对几何分析和谱几何中的许多开放性问题测试这些新工具。这项研究将更多地阐明 L2 不变量的本质和功效，并将大大提高几何分析中一些成功技术的通用性，并有望解决拓扑和谱几何中的一些开放问题。上述工作探讨了几何对象（黎曼流形）的形状（体现在其拓扑和几何形状中）与声音（体现在与之相关的各种拉普拉斯算子的频谱中）之间的关系 - 从振动的固有频率角度思考。它还研究了形状和声音对动力学基本定性元素（例如闭合轨迹、吸引子、排斥子和鞍点）施加的约束。使用的方法涉及冯·诺依曼引入的异常量，例如非整数的维度和无限大物体的体积。该研究将调查有关形状和声音以及几何物体动力学的附加信息的类型，这些信息可以通过使用这些不寻常的量来获得。 ***

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. ***