Wang will continue his research on K-homology of smooth algebras, investigating its striking application in operator theory, its deep connection with geometry and dynamical systems, and with quantization theories. The machineries developed in noncommutative differential geometry in the last decade, and the techniques and concepts in quantized functional analysis are the key tolls in this investigation. 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.
