Topics in Topology and Geometry

9803254 Burghelea D. Burghelea计划工作在拓扑学，几何分析和动力学，基于线性代数"a la von Neumann“方法和椭圆算子的正则化行列式。他计划：1）解决L2-不变量、挠率不变量、挠率与动力学之间的关系等方面的一些开放问题; 2）开发新的数学工具，如与Morse-Bott函数相关的几何复形的Hodge理论、具有对称性和参数的Witten-Hellfer-Sjostrand理论; 3）针对几何分析和谱几何中的一些开放问题测试这些新工具。该研究将进一步揭示L2不变量的性质和能力，并将大大增加几何分析中一些成功技术的通用性，并有望解决拓扑学和谱几何中的一些开放问题。上面的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结构中）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-从振动的自然频率方面考虑。它还研究了形状和声音对动力学基本定性元素的约束，如闭合轨迹，吸引子，排斥子和鞍点。所使用的方法涉及冯·诺依曼引入的不寻常的量，例如不是整数的尺寸和无限大物体的体积。该研究将调查有关形状和声音的附加信息的类型，以及几何物体的动力学，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. ***