Research
Modern Analysis/Geometry
We have a strong active group in Modern Analysis/Geometry. The topics covered by this include noncommutative geometry (i.e., operator algebras, index theory), geometric theory of boundary value problems, and Riemannian and Finsler geometry. This general area is at the center of the recent flourishing interaction of mathematics and physics. We are fortunate to have strength in this active area since it builds on the connections between the different subjects represented and, hence, encourages and stimulates interaction between all the areas in the Department.
Carl Cowen
Prof. Cowen works in areas related to linear operators on Hilbert spaces of analytic functions such as the Hardy and Bergman spaces. Much of this research deals with the classical Toeplitz, composition, and weighted composition operators and the theory of analytic functions connected to these operators. The goal of this research is to connect the operator theoretic structure of operators with the geometric and analytic structure of the functions defining the operator. Cowen's PhD students have worked in operator theory and related linear algebra and theory of analytic functions.Ronghui Ji
Prof. Ji works in various aspects of noncommutative geometry. His most recent work involves using cyclic cohomology to study problems in pure algebra as well as in C*-algebras, such as the Bass conjecture and the Kadison-Kaplansky conjecture on idempotent in group algebras and the dense subalgebra problem in group C*-algebras. He is also using bivariant charn character to continue his study on index theory for proper actions of discrete groups on manifolds. He has taken on the responsibilities for algebra in the Department and become a resource for the rest of us in this area.
Slawomir Klimek
Prof. Klimek is trained as a mathematical physicist, and his research interests are rooted in problems from that area. His recent work developed a notion of noncommutative harmonic analysis which is likely to have many interesting applications. He has also done substantial work on index theory and dynamics in the context of noncommutative geometry and has recently become interested in the applications of these areas to number theory. He is unique in this area in that he is really trained as a physicist, but has a mathematician's taste in problems and methods.
Zhongmin Shen
Prof. Shen is one of the world-wide leaders in the area of Finsler geometry. His most recent work involves characterizing Finsler metrics of scalar (or constant) flag curvature and studying the global geometric structures of Finsler manifolds with special non-Riemannian curvature properties. He also works in Riemannian geometry as a special case of Finsler geometry. His recent interests involve applications of many aspects of geometry to real world problems. Shen intends to use Finsler geometry in a new area of mathematics called Information Geometry.