Geometric cocycles, differential K-theory, and non-abelian gerbes

这个建议是基于上同调理论的几何研究。上同调理论为拓扑空间提供了有用的不变量，它们的谱分类提供了一个统一的图像，表明它们的数量。然而，我们对上同调理论，以及它们的等变、微分和其他混合变异体的理解仍然有限。这部分是由于这样一个事实，除了一些例子，如普通上同调和k理论，没有很好的几何描述一个给定理论的上同调类与几何中自然发生的物体有直接联系。即使在k理论及其变体的情况下，我们目前的理解也留下了许多未解之谜。几何表示一个理论的类在很多方面都被证明是有用的，包括非同伦不变精化的构造，如微分理论，它们的等变版本，和推进（循环水平指数定理）。在这个提议中，PI研究了几种获得k理论（等变、微分和它们的混合）几何模型的方法，以及通过使用Bismut Chern特征来考虑威尔逊线效应的改进方法。在研究的一个组成部分中，PI使用托莱多-通扭曲分辨率中的Atiyah类代表研究微分k理论。另一方面，PI的目标是利用他和他的合作者在阿贝尔格布的等变完整上的工作，来研究非阿贝尔格布的拓扑不变量。这部分的主要工具是等变拓扑手性同调。这项研究有研究生组成部分。上同调理论及其变体和改进是数学分支拓扑的一部分。上同调不变量可以测量各种各样的现象，从一根绳子缠绕在一根杆子上，到宇宙形状的可能性。上同调技术和描述在模拟高能物理现象方面发挥了核心作用，以至于一些基本概念最初是由物理学家和数学家独立发现的。两个领域之间的比较和交叉施肥导致了两者的加速富集，这一趋势继续日益加强。现代的范畴观点现在成为数学家和物理学家探索上同调思想及其副产品的共同语言。该资助的几个组成部分涉及本科生和研究生。

This proposal is based on a geometric study of cohomology theories. Cohomology theories provide useful invariants for topological spaces, and their classification by the spectra provides a unified picture, indicating how numerous they are. Nevertheless, our understanding of cohomology theories, along with their equivariant, differential, and other mixed variants, remains limited. This is partly due to the fact that aside from a few examples, such as ordinary cohomology and K-theory, there are no good geometric descriptions of the cohomology classes of a given theory with immediate ties to naturally occurring objects in geometry. Even in the case of K-theory and its variants, our current understanding leaves many questions unanswered. Geometrically representing classes of a theory has proven to be useful in many regards, including the construction of non-homotopy invariant refinements, like differential theories, their equivariant versions, and pushforwards (cocycle level index theorems). In this proposal, the PI studies several ways of obtaining geometric models for K-theories (equivariant, differential, and their mixes) as well as refinements that take into account the Wilson line effects by using the Bismut Chern character. In one component of the research, the PI studies differential K-theory using representatives of the Atiyah class in the Toledo-Tong twisted resolution. In another, the PI aims to use his work with his collaborators, on the equivariant holonomy for abelian gerbes, to study topological invariants of non-abelian grebes. The main tool for this part is the equivariant topological chiral homology. This research has graduate student components.Cohomology theories, their variants, and refinements are part of a branch of mathematics called topology. Cohomological invariants can measure a wide variety of phenomena, from wrappings of a piece of rope around a pole, to the possibilities for the shape of the universe. Cohomological techniques and descriptions have taken a central role in modeling high energy physics phenomena to the extent that several fundamental concepts were originally discovered by physicist and mathematicians independently. Comparison and cross-fertilization between the two fields has resulted in an accelerated enrichment of both, a trend that continues to pick up momentum increasingly. A modern categorical point of view now serves as a common language for mathematicians and physicists to explore cohomological ideas and their byproducts. Several components of the grant engage undergraduate and graduate students.