Generalised Smooth Cohomology Theories

上同调理论是研究空间的重要工具。然而，它们是相当粗糙的，并且通常希望有细化，可以捕捉小尺度的几何形状。这些改进被称为光滑上同理论，也是我们提出的研究课题。我们的目标是构建上同调理论的光滑改进，并追求其新的应用。我们的四个主要重点是考虑到空间所具有的对称性（特别是等变/ c理论）的上同调理论的光滑改进的构建。我们还希望构造二元理论的光滑精化，以及空间光滑x理论的新几何模型。我们感兴趣的光滑上同理论的主要数学应用是它们作为几何指数理论中出现的次要不变量的自然家园的推测作用。我们想要使这个概念精确，特别是证明Atiyah-Singer指标定理的光滑细化。我们也想把我们的工作应用到物理学上。量子场论中的某些场被解释为光滑上同理论中的类。我们希望利用这种洞察力来研究弦理论中的深层对称性，即所谓的t二象性。

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.