Complex Analysis Research

The theory of functions of a complex variable

The theory of functions of a complex variable is beautiful and far-reaching. This theory is one of the great achievements of mathematicians of the late 19th and early 20th centuries.

The theory, to this day, remains very active. It plays an essential role in many branches of mathematics, physics and engineering.

Applications

One notable application is to spectral theory. It is a powerful tool in the study of operators on Hilbert space, both bounded and differential.

We are interested in:

• the application of complex analysis to operators
• the use of operator theory to prove results in complex analysis

Another application is to control theory. For example, take the problem of designing an automatic controller for a class of engineering devices. This can be reduced to the construction of analytic matrix-valued functions on a half-plane meeting some specifications. The latter application makes contact with the theory of functions of several complex variables. This theory has been extensively developed since the 1920s.