Mathematical Sciences: Analytical and Combinatorial Aspects of Subfactors

Bisch将利用子因子和交换平方的组合方面构造和分析超有限II_1因子的不可约子因子。他还将研究子因子之间的联系、量子物理和低维拓扑。最后，他将研究自由群因子的子因子。这里的关键技术是Voiculescu的自由概率论。这个项目的一般数学领域在希尔伯特空间算子的代数理论中有其基础。运算符可以被认为是有限或无限的复数矩阵。特殊类型的运算符通常放在一个代数中，自然地称为运算符代数。这些抽象对象有各种各样的应用。例如，它们在结理论中发挥着关键作用，而结理论目前正被用于研究DNA的结构，它们在非交换几何中具有重要的基础意义，而非交换几何在物理学中正变得越来越重要。

Bisch will construct and analysis irreducible subfactors of the hyperfinite II_1 factor using the combinatorial aspects of subfactors and commuting squares. He will also study connections between subfactors, quantum physics, and low dimensional topology. Finally, he will study subfactors of free group factors. The key technique here will be Voiculescu's free probability theory. The general area of mathematics of this project has its basis in the theory of algebras of Hilbert space operators. Operators can be thought of as finite or infinite matrices of complex numbers. Special types of operators are often put together in an algebra, naturally called an operator algebra. These abstract objects have a surprising variety of applications. For example, they play a key role in knot theory, which in turn is currently being used to study the structure of DNA, and they are of fundamental importance in noncommutative geometry, which is becoming increasingly important in physics.