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

Carey教授将继续他对算子值域的指数不变量的研究，这是对Fredholm算子的经典指数理论的深远扩展。他还将从事多变量谱理论的一个项目，该项目与与可交换n元算符的Koszul复形相关的相干束有关。在经过n元组本质谱的补集的代数拓扑过滤后，这一束产生了一个不变量，凯里教授将研究它的性质。这是对Hilbert空间上算子理论的数学研究。运算符可以被认为是丰富的数，遵守与普通数相同的算术法则，但有两个明显的例外：运算符的乘法取决于因子的取值顺序，并且并不是每个非零运算符都有逆运算。对于有限维运算符，即数的平方矩阵，这些现象解释了线性代数主题的丰富性。在Hilbert空间的无穷维集中，还存在非零指数的现象，即算子遵守算术法则，但误差项在适当的意义上是小的，并且包含关于产生算子的任何情况的重要信息。从这个意义上讲，指数理论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.