Principal bundles in algebraic geometry

当面对一个复杂的问题时，一种被证明有用的技术是利用它的对称性。在许多自然发生的问题中，对称的集合通常是相当大的，本质上是无限的。为了理解这种情况下的对称集合，看看对称集合是否有额外的结构是很有用的。其中一种结构是集合由一些多项式方程定义。这在数学中被称为代数群。这个建议是关于研究代数群的几何和它们所作用的空间。几何图形通常非常复杂，我们需要适当的机制来描述这些空间。同调代数就是这样一种机器。本提案旨在利用这种同调技术来进一步研究代数群及其相关空间。这项工作最直接的影响将体现在数学方面。数学物理在最近几十年里与代数几何融合在一起。该提案的一部分是关于有理代数变体的。这些对象在计算机时代得到了应用，例如，在计算机辅助设计中，它们可以用来快速计算十字路口。

When confronted with a complex problem one technique that has proven useful is to exploit its symmetries. In many naturally occurring problems the collection of symmetries is often quite large, infinite in nature. To understand the collection of symmetries in this situation, it is useful to see if the collection symmetries, has extra structure. One such structure is that the collection be defined by some polynomial equations. This is known as an algebraic group in mathematics. This proposal is about study the geometry of algebraic groups and the spaces that they act upon. The geometry can often be very complicated and we need the appropriate machinery to be able to say anything about these spaces. One such machine is homological algebra. This proposal seeks to use such homological techniques to further the study of algebraic groups and their associated spaces. The most immediate consequences of this work will be felt in mathematics. Secondary effects will be felt in mathematical physics which has seen a merger with algebraic geometry in recent decades. Part of the proposal is about rational algebraic varieties. These objects have seen applications in the computer age, for example to computer aided design where they can be used to compute intersections quickly.