Mathematical Sciences: Index Invariants for Operator Range and Commuting Tuples of Operators on Hilbert Space

凯里教授将继续他对索引的研究 运算符范围的不变量，这是对 Fredholm算子的经典指标理论他还将 从事多变量光谱理论的研究， 与相干层相关联的Koszul复杂的一个 交换n元组运算符。这一捆， 通过代数拓扑过滤的补的 n元组的基本谱，产生一个不变量， 凯里教授将研究的属性。 这是数学研究中的算子理论 Hilbert空间运算符可以被认为是丰富的数， 遵守与普通数相同的算术定律， 值得注意的例外：运算符的乘法取决于 取因子的顺序，而不是每个非零 运算符有一个逆。对于有限维算子，即 数字的方阵，这些现象解释了 线性代数学科的丰富性。在无限的- 希尔伯特空间的维数设置，还有 非零指数的现象，运营商遵守的法律， 算术的数字，除了误差项是小的， 一个适当的意义，并包含重要的信息， 无论是什么情况，都会引起运营商的关注。指数理论 这种感觉是凯里教授项目的基本主题。

Professor Carey will continue his investigation of index invariants for operator ranges, a far-reaching extension of the classical index theory for Fredholm operators. He will also pursue a project in multivariable spectral theory having to do with the coherent sheaf associated with the Koszul complex of a commuting n-tuple of operators. This sheaf, after it has been filtered through the algebraic topology of the complement of the essential spectrum of the n-tuple, yields an invariant whose properties Professor Carey will study. This is mathematical research in the theory of operators on Hilbert space. Operators may be thought of as enriched numbers, obeying the same laws of arithmetic as ordinary numbers with two notable exceptions: multiplication of operators depends upon the order in which the factors are taken, and not every non-zero operator has an inverse. For finite-dimensional operators, i.e. square matrices of numbers, these phenomena account for the richness of the subject of linear algebra. In the infinite- dimensional setting of Hilbert space, there is furthermore the phenomenon of non-zero index, of operators obeying the laws of arithmetic for numbers except for error terms that are small in an appropriate sense and that contain important information about whatever situation gave rise to the operators. Index theory in this sense is the underlying theme of Professor Carey's project.