The d-bar-Neumann operator and related topics

The topic of this project lies at the interface of several subfields of mathematics. Complex analys is (differential and integral calculus in several variables over the complex numbers) is complemented by methods in infinite-dimensional vector spaces (Hilbert spaces). A partial differential equation is of special importance: the Cauchy-Riemann differential equation, which is responsible for the serious differences to real analysis. The d-bar Neumann operator provides important insights into the solu- tion of the homogeneous and inhomogeneous Cauchy-Riemann differential equation and gives deep information on how the properties of holomorphic functions describe the geometry of the domains on which the functions are defined. It is also interesting to see connections to problems in mathematical physics, such as Schrodinger operators with magnetic field and the uncertainty relation in quantum physics. The creation and annihilation operators of quantum mechanics gain an elegant mathematic a l description in special Hilbert spaces of holomorphic functions. These Hilbert spaces have special re- producing kernels whose analysis is particularly challenging. An important generalization of Cauchys integral formula for several complex variables is Lerays integral formula, which is analyzed by elegant methods of the last fifty years. Finally, modern differential geometry is also used to describe the duality in Hilbert spaces of holomorphic functions in many details.