Generalised Smooth Cohomology Theories

广义光滑上同调理论

• 批准号: 147846739
• 负责人: Dr. Alexander Rudolf Kahle
• 金额: --
• 依托单位: Max-Planck-Institut für Mathematik
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2009
• 资助国家: 德国
• 起止时间: 2008-12-31 至 2011-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/147846739?language=en
• 关键词: Generalised; Smooth; Cohomology; Theories

Cohomology theories are important tools in studying spaces. They are however rather coarse, and it is often desirable to have refinements that can capture the small scale geometry. These refinements take the name of smooth cohomology theories, and are the subject of our proposed research. Our goal is both to construct new smooth refinements of cohomology theories, and to pursue novel applications thereof One ofour main thrusts is the construction of smooth refinements of cohomology theories that take into account the symmetries a space possesses (in particular equivariant /C-theory). We also wish to construct smooth refinements of bivariant theories, and new geometric models for the smooth X-theory of a space. The principal mathematical application for smooth cohomology theories that we are interested in is their conjectured role as the natural home for the secondary invariants that appear in geometric index theory. We would like to make this notion precise, and in particular prove a smooth refinement of the Atiyah-Singer index theorem. We would also like to apply our work to physics. Certain fields in Quantum Field Theory have been interpreted as classes in smooth cohomology theories. We would like to use this insight to study a deep symmetry in string theory, the so called T-duality.

上同调理论是研究空间的重要工具。然而，它们相当粗糙，并且通常期望具有可以捕获小尺度几何形状的细化。这些改进的光滑上同调理论的名称，是我们提出的研究的主题。我们的目标是构建新的光滑上同调理论的改进，并追求新的应用，我们的主要目标之一是考虑到空间的对称性（特别是等变/C-理论）的上同调理论的光滑改进的建设。我们还希望构造双变理论的光滑加细，以及空间的光滑X-理论的新几何模型。我们感兴趣的光滑上同调理论的主要数学应用是它们作为几何指数理论中出现的二级不变量的自然家园的角色。我们想使这一概念精确，特别是证明一个光滑的改进的Atiyah-Singer指标定理。我们也想把我们的工作应用到物理学上。量子场论中的某些领域被解释为光滑上同调理论中的类。我们想用这种洞察力来研究弦理论中的一种深层对称性，即所谓的T对偶性。