The structure of invariants in algebra and geometry

本文研究了不变量的概念，包括模和参数空间、阿贝尔和非阿贝尔伽罗瓦上同调、方案的k -上同调、Clifford代数以及1属曲线的周期和指数。特别地，我们将研究某些组态和模空间上的操作结构、伽罗瓦上同调的局部-全局原理、齐次变量的k -上同调与可分代数的k -理论之间的关系、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。一个包含了大量当前数学的标准模型是，人们首先指定一类研究对象，无论它们本质上是“代数的”还是“几何的”，然后试图构造不变量，以便对它们进行区分或分类。这种方法最吸引人的一个方面是不变量和它们测量的对象之间经常出现的相互作用，而且不变量经常有自己的生命，经常成为研究的对象。

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.