# Complex Analysis and Spectral Theory Conference

### A conference in celebration of Thomas J. Ransford's 60th birthday

## Description

Spectral theory is the branch of mathematics devoted to the study of matrices and their eigenvalues, as well as their infinite-dimensional counterparts, linear operators and their spectra. Spectral theory is ubiquitous in science and engineering, because so many physical phenomena, being essentially linear in nature, can be modelled using linear operators. It draws upon techniques from a variety of other areas of mathematics, and leads to problems in these areas that are of interest in their own right. Complex analysis is the calculus of functions of a complex variable. The roots of the subject go back to the early 19th century, and are associated with the names of Euler, Gauss, Cauchy, Riemann and Weierstrass. Of particular importance are the differentiable functions, usually called analytic or holomorphic functions. They are widely used in mathematics (for example, in Fourier analysis, analytic number theory, and complex dynamics), in physics (potential theory, string theory) and in engineering (fluid dynamics, control theory and theory of communication).

Both topics are related to numerous other domains in mathematics as well as other branches of science and engineering. For example, analytic function spaces arise in various different branches of mathematics and science. The list includes, but is not restricted to, analytical mechanics, physics, astronomy (celestial mechanics), geology (weather modeling), chemistry (reaction rates), biology, population modeling, economics (stock trends, interest rates and the market equilibrium price changes). As a matter of fact, it is hard to find a branch of analysis or applied sciences in which function spaces do not appear. Many mathematicians have studied this domain and contributed to the field and it is rather impossible to provide a list.

As another manifestation, functional analysis is the branch of mathematics concerned with the study of vector spaces and linear mappings acting upon them. The word ‘functional’ refers an operation whose argument is a function, integration for example. Two of the most important names are Hilbert and Banach, and the central notions of the subject are named in their honor: Banach spaces (complete normed vector spaces) and Hilbert spaces (Banach spaces where the norm arises from an inner product). Hilbert spaces, which generalize the notion of Euclidean space to infinite dimensions, are of fundamental importance in many areas, including partial differential equations, quantum mechanics and signal processing. From the earliest days, researchers in functional analysis recognized the importance of studying spaces of functions, as opposed to considering just one function at a time. Together with the development of the Lebesgue integral, this led to new techniques, for example, for analyzing the behavior of analytic functions at the boundary of their domain, and for proving the existence of analytic functions with certain properties, hitherto difficult or impossible to construct. In turn, complex analysis repaid its debt to functional analysis by providing methods for defining functions of operators, for example the square root or the logarithm of an operator or a matrix.

There are many other connections, and in the century that has followed this has become a vast domain of research. In recent years, there have been tremendous amount of work on reproducing kernel Hilbert spaces of analytic functions, on the operators acting on them, as well as on applications in physics and engineering which arise from pure topics like interpolation and sampling.

In this conference, more than thirty analysts, young and prominent, and from Europe and North America, are invited. Many different topics in complex analysis, operator theory, matrix analysis, spectral theory, functional analysis, approximation theory, will be discussed during the invited talks. We expect that this lively meeting will strengthen our understanding of the subjects, how far the applications range, how much is known, and how much is still unknown. The goal of our gathering is to discuss a number of fundamental open problems on Hilbert and Banach spaces of analytic functions and the new ideas that have been developed as well as the recent progress that has been made. We believe this event to be worthwhile, since the ideas involved are of widespread interest in the mathematical analysis community.