Commutative Algebraic Methods in Operator Theory

该项目旨在发展一个Fredholm理论，该理论包括单个算子和作用于公共Hilbert空间的交换算子的元组。为了将交换代数中的思想和方法引入算子理论，我们将重点介绍Koszul复方法。该策略是在算子理论中引入Samuel多重性等稳定不变量来计算多变量Fredholm指数。这与已有的诸如Bergman空间和Dirichlet空间的不变子空间的余维数、Hardy空间上的Toeplitz算子、对称Fock空间上的Arveson曲率不变量、高维域上Hardy空间上的函数论、Apostol的三角表示等问题有很多联系。线性算子的Fredholm指标是数学中研究得最深入的数值不变量之一。著名的Atiyah-Singer指数定理是数学中最深奥的成果之一，其内容是关于如何计算一类特殊算子的Fredholm指数。本项目将重点发展传统单变量Fredholm理论的多变量版本。这使得人们可以将交换代数（一个看似遥远的数学领域）的思想引入多变量算子理论，从而为单变量算子理论中的传统问题提供新的思路。

The project aims at developing a Fredholm theory encompassing both a single operator and a tuple of commuting operators acting on a common Hilbert space. The Koszul complex approach will be emphasized in order to introduce ideas and methods in commutative algebra to operator theory. The strategy is to introduce stabilized invariants such as the Samuel multiplicities into operator theory to calculate the multivariable Fredholm index. This has many connections with existing topics such as the codimension of invariant subspaces of the Bergman space and the Dirichlet space, Toeplitz operators on the Hardy space, Arveson's curvature invariant on the symmetric Fock space, function theory on Hardy spaces over higher dimensional domains, Apostol's triangular representation.The Fredholm index of a linear operator is one of the most intensely studied numerical invariants in mathematics. The content of the celebrated Atiyah-Singer Index Theorem, one of the deepest results in mathematics, is on how to calculate the Fredholm index of a special class of operators. This project will focus on developing a multivariable version of the traditional one variable Fredholm theory. This allows one to introduce ideas from commutative algebra, a seemingly distant area in mathematics, into multivariable operator theory, which in turn shed new light on traditional problems in one variable operator theory.