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

这个建议是基于几何研究的上同调理论。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，表明它们有多少。尽管如此，我们对上同调理论的理解，沿着，以及它们的等变、微分和其他混合变体，仍然有限。这部分是因为除了一些例子，如普通的上同调和K理论，没有很好的几何描述的上同调类的一个给定的理论直接联系到自然发生的对象在几何。即使是在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。理论的几何表示类在许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（上循环水平指标定理）。在这个提议中，PI研究了几种获得K理论几何模型的方法（等变、微分和它们的混合），以及通过使用Bismut Chern特征考虑威尔逊线效应的改进。在研究的一个组成部分中，PI使用Toledo-Tong扭曲分辨率中的Atiyah类的代表研究微分K理论。在另一个，PI的目的是使用他的工作与他的合作者，对等变holonomy的阿贝尔gerbes，研究拓扑不变量的非阿贝尔grebes。这一部分的主要工具是等变拓扑手征同调。这项研究有研究生组成部分。上同调理论，它们的变体和改进是数学分支拓扑学的一部分。上同调不变量可以测量各种各样的现象，从一根绳子绕在一根柱子上的缠绕，到宇宙形状的可能性。上同调的技术和描述在模拟高能物理现象中发挥了核心作用，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。这两个领域之间的比较和相互促进导致了两者的加速丰富，这一趋势继续日益加快。一个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。赠款的几个组成部分吸引了本科生和研究生。

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.