Twisted commutative algebras

扭曲交换代数

• 批准号: 1303082
• 负责人: Andrew Snowden
• 金额: $ 14.08万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-09-01 至 2017-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1303082&HistoricalAwards=false
• 关键词: Twisted; commutative; algebras

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.

该项目涉及对某些代数对象的研究，称为扭曲的交换代数（TCA），以及密切相关的概念。 TCA可以定义为换向代数（通常非常大），配备了无限通用线性组的作用。研究这些物体有两个原因。第一个是内部：PI与SAM的联合工作表明TCA的享有丰富的理论。例如，尽管这些代数通常不是有限生成的，但大型动作会减轻这种情况，并在某些方面表现为有限生成的代数。因此，可以尝试从可交换代数来扩展已知结果。第二个原因是外部：TCA可以用于研究具有独立兴趣的对象。通常，这些应用是对物体的无限家族的稳定行为。一些示例应用程序：Segre嵌入式（PI）的共同体，Secant品种方程（Draisma-kuttler），稳定的代表理论（PI和SAM），以及配置空间的共同体学（教堂 - Farb-farb-ellenberg）。在数学中是稳定性的。宽松地说，一个人可能会说，一系列对象最终获得规则行为，它会表现出稳定性。该属性很重要，因为它通常允许对象的整个序列由有限数量的数据描述。近年来，几位研究人员观察到，代数和几何形状在代数中产生的某些不变性序列表现出稳定性。 PI引入了一种工具，称为扭曲的交换代数，以研究其中一些不变性。 PI和SAM的联合工作表明，这些代数本身就是有趣的。该项目的目的是继续发展这些代数的理论和应用。