Cohomology and Actions of Groups

要研究的一个非常经典的东西，可以追溯到古希腊，就是物体的对称性。抽象地说，这些被称为群；它们可能非常复杂，难以理解。有许多方法可以研究这些抽象的对称群，并试图给它们施加一些秩序；其中之一是给群关联一个新的代数不变量，上同调就是一个很好的例子。然而，这些新的不变量很难理解，甚至很难计算，部分原因是它们是无限的。通过使用一种称为规律性的量，就有可能表明，这些无限的物体是由它们自身特定的有限部分控制的，这样它们就变得容易处理得多了。该项目旨在使这些技术可用于比以前更大的群类，并通过这样做，获得关于这些群的结构及其上同调的更详细的信息。

A very classical thing to study, going back to the Ancient Greeks, is the symmetries of an object. In the abstract these are known as groups; they can be very complicated and difficult to understand. There are many ways of investigating these abstract groups of symmetries and trying to impose some order on them; one of these is to associate to the group a new algebraic invariant, of which cohomology is a prime example. However, these new invariants can be very difficult to understand, or even to calculate, partly because they are infinite. By using a quantity called the regularity it is possible to show that these infinite objects are controlled by a specific finite part of themselves and they then become much easier to deal with. This project aims to make these techniques available for much larger classes of groups than before and, in so doing, obtain much more detailed information about the structure of these groups and their cohomology.