Twisted commutative algebras

这个项目是关于被称为扭曲交换代数(TCA)的某些代数对象的研究，以及与之密切相关的概念。TCA可以被定义为一个交换代数(通常非常大)，配备有无限一般线性群的作用。研究这些物体有两个原因。首先是内部：PI与SAM的联合工作表明TCA拥有丰富的理论。例如，虽然这些代数通常不是有限生成的，但大群作用减轻了这一点，在某些方面它们表现为有限生成的代数；因此，人们可以尝试推广交换代数的已知结果。第二个原因是外部的：TCA可以用来研究独立感兴趣的对象。通常，这些应用是针对无限族对象的稳定行为的。一些例子应用：Segre嵌入的合子(PI)、割变方程(Drisma-Kuttler)、稳定表示理论(PI和Sam)和配置空间的上同调(Church-Farb-Ellenberg)。粗略地说，有人可能会说，如果一个对象序列最终获得了一种规则的行为，它就会表现出稳定性。该属性很重要，因为它通常允许使用有限数量的数据来描述整个对象序列。近年来，一些研究人员观察到，在代数和几何中出现的某些不变量序列具有稳定性。PI引入了一种工具，称为扭曲交换代数，来研究其中的一些不变量。PI和SAM的联合工作表明，这些代数本身就很有趣。本项目的目的是继续发展这些代数的理论和应用。

This project is concerned with the study of certain algebraic objects called twisted commutative algebras (tca's), and closely related notions. A tca can be defined as a commutative algebra (typically very large), equipped with an action of the infinite general linear group. There are two reasons for studying these objects. The first is internal: joint work of the PI with Sam indicates that tca's enjoy a rich theory. For example, although these algebras are typically not finitely generated, the large group action mitigates this and in some ways they behave as finitely generated algebras; one can therefore attempt to extend known results from commutative algebra. The second reason is external: tca's can be used to study objects of independent interest. Typically, these applications are to the stable behavior of infinite families of objects. Some example applications: syzygies of Segre embeddings (PI), equations of secant varieties (Draisma--Kuttler), stable representation theory (PI and Sam), and cohomology of configuration spaces (Church--Farb--Ellenberg).An important theme in mathematics is that of stability. Loosely speaking, one might say that a sequence of objects exhibits stability if it eventually acquires a regular behavior. This property is important because it often allows the entire sequence of objects to be described by a finite amount of data. In recent years, it has been observed by several researchers that certain sequences of invariants arising in algebra and geometry exhibit stability. The PI introduced a tool, called twisted commutative algebras, to study some of these invariants. Joint work of the PI and Sam has shown that these algebras are interesting in their own right. The purpose of this project is to continue to develop the theory and applications of these algebras.