Twisted commutative algebras

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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.