Directions in Index Theory and Riemannian Geometry

指标理论与黎曼几何的方向这个建议涉及各种空间的几何问题，即光滑黎曼流形、度量空间、奇异空间和非对易空间.光滑黎曼流形的问题涉及曲率条件，例如当一个有边界的紧致流形允许一个度量在其边界上具有非负的数量曲率和非负的平均曲率，关于度量空间的问题涉及Ricci张量在度量空间中的可能扩张。 奇异空间的问题涉及椭圆分析，非对易几何的问题涉及一般叶理指数定理的局部证明，随着数学和物理学的发展，空间的概念也在发展。 物理学中的一些明显的例子是从平坦空间到弯曲空间的演化，以及从经典相空间到量子力学希尔伯特空间的演化。 也有数学内在的原因导致人们扩展空间的概念。当然，人们不想随意定义，而是想考虑那些自然地脱离真实的问题的概念。一个更近的空间概念福尔斯属于“非对易几何”的范畴，在这个范畴中，传统的空间首先被认为是根据其上的函数来定义的，然后将函数环从交换的情形推广到非交换的"函数“环。 非对易几何植根于几何分析的一个分支，称为指数理论。有时，关于普通“对易”空间的问题，如叶理，可以转化为关于非对易空间的问题，并从这个角度进行攻击。在这个建议中考虑的许多问题都有其物理根源。 多年来，数学和物理之间的相互作用卓有成效，而我们所提出的工作也有望对这种相互作用有所贡献。特别是，关于正标量曲率的问题在广义相对论中出现，非对易几何涉及现代理论物理的许多领域。

ABSTRACTDIRECTIONS IN INDEX THEORY AND RIEMANNIAN GEOMETRYThis proposal concerns geometric questions about varioustypes of spaces, namely smooth Riemannian manifolds, measuredmetric spaces, singular spaces and noncommutative spaces.The questions about smooth Riemannian manifolds concerncurvature conditions, such as when a compact manifold-with-boundarycan admit a metric having nonnegative scalar curvature on the interiorand nonnegative mean curvature on the boundary, and when a closedmanifold can admit a metric with positive p-curvature tensor.The questions about measured metric spaces concern possibleextensions of the Ricci tensor to such spaces. Thequestions about singular spaces concern elliptic analysis.The questions about noncommutative geometry concern local proofsof the general foliation index theorem.As mathematics and physics have evolved, the concept of space has alsoevolved. Some evident examples from physics are the evolution fromflat space to curved space, and the evolution from the classical phasespace to the quantum mechanical Hilbert space. There have also beenreasons intrinsic to mathematics that lead one to extendthe notion of space. Of course, one does not want to make arbitrarydefinitions, but one rather wants to consider concepts that arisenaturally from real problems.A more recent notion of space falls under the heading of``noncommutative geometry'', in which a traditional space isfirst considered to be defined in terms of the functions on it, and thenthe ring of functions is generalized from the commutative case to allowfor a noncommutative ring of ``functions''. Noncommutative geometryis rooted in the branch of geometric analysis called index theory.Sometimes problems about ordinary ``commutative'' spaces, such asfoliations, can be translated intoproblems about noncommutative spaces and attacked from that angle.Many of the questions considered in thisproposal have their roots in physics. Throughout the years there hasbeen much fruitful interaction between mathematics and physics, andthe proposed work will hopefully contribute to this interaction.In particular, questions about positive scalar curvature arise ingeneral relativity theory, and noncommutative geometry touches on manyareas of modern theoretical physics.