Mathematical Sciences: Noncommutative Differential Geometry and Operator Algebras

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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.