The structure of invariants in algebra and geometry

这项研究集中在许多不同伪装下的不变量的概念：模和参数空间，交换和非交换Galois上同调，方案的K上同调，Clifford代数，以及亏格1曲线的周期和指数。特别是，它将研究某些配置和模空间上的算子结构，通过域修补的Galois上同调的局部-整体原理，齐次簇的K-上同调与可分代数的K-理论之间的关系，以及方案的有限态射的Clifford代数不变量，以获得关于周期指数问题的新结果。一个囊括了大量当前数学的标准模型是，一个人从指定一类研究对象开始，无论它们本质上是“代数”还是“几何”，然后试图构造不变量，以便区分它们或对它们进行分类。这种方法最吸引人的方面之一是不变量和它们所测量的对象之间经常出现的相互作用，以及不变量通常实现自己的生命，经常成为研究对象的事实。

This research focuses on the concept of invariants in a number of different guises: moduli and parameter spaces, abelian and non-abelian Galois cohomology, K-cohomology of schemes, Clifford algebras, and the period and index of a genus 1 curve. In particular, it will investigate operad structures on certain configuration and moduli spaces, local-global principles for Galois cohomology via field patching, relations between the K-cohomology of homogeneous varieties and the K-theory of separable algebras, and Clifford algebra invariants of finite morphisms of schemes to obtain new results concerning the period-index problem. A standard model encapsulating a good deal of current mathematics is that one begins by specifying a class of objects of study, be they `algebraic' or `geometric' in nature, and then one attempts to construct invariants in order to distinguish or classify them. One of the most fascinating aspects of this approach is the interplay that often arises between the invariants and the objects which they measure, and the fact that the invariants often achieve a life of their own, often becoming objects of study in themselves.