Mathematical Sciences: Noncommutative Differential Geometry and Operator Algebras

纪将研究非交换微分几何上 与离散群相关联的C ~*-动力算子代数 系统和余切空间的严格变形量子化 一个光滑的流形。 本研究的核心是构建 光滑且稠密的子代数。 这些项目的完成将使季羡林能够阐明某些深刻的 算子代数的性质，如部分解 广义Kadison猜想，进一步理解 非交换Toeplitz指数理论，以及某些 非对易流形 这个项目的一般数学领域有其基础 希尔伯特空间算子的代数理论。 运营商 可以被认为是复数的有限或无限矩阵 号码 特殊类型的运算符通常放在 代数，自然称为算子代数。 这些抽象 物体的应用范围之广令人惊讶。 比如说， 它们在纽结理论中起着关键作用，而纽结理论目前 被用来研究DNA的结构， 在非对易几何中的重要性， 在物理学中变得越来越重要。

Ji will investigate noncommutative differential geometry on operator algebras associated with discrete groups, C*-dynamical systems, and strict deformation quantization of a cotangent space of a smooth manifold. Central to this study is the construction of smooth and dense subalgebras of the corresponding C*-algebras. Completion of the projects will enable Ji to elucidate certain deep properties of operator algebras, such as a partial solution to the generalized Kadison conjecture, a further understanding of the noncommutative Toeplitz index theory, and a new birth of certain noncommutative manifolds. 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.