Grant: Canada Research Chairs

Pure mathematics has had a fundamental role in science and many modern technologies. Norbert Wiener created the theories of cybernetics and control, and his work on communication and information is foundational in many signal processing methods. George Zames exploited Hardy spaces to build his H-infinity control theory, the basis of all modern robust control systems. Claude Shannon’s information theory is vital in digital communication systems. Alan Turing is considered the father of theoretical computer science and artificial intelligence, and Steve Smale’s mathematics laid out the mathematical foundation of learning theory. A collection of functions which share a common property, e.g., a restriction on their growth or on the size of their derivatives, is called a function space. For example, the H-infinity space which was used by G. Zames in control theory consists of all bounded analytic functions. There are many celebrated examples of function spaces which bear the name of a prominent mathematicians (Banach, Bergman, Besov, Bloch, de Branres-Rovnyak, Dirichlet, Fock, Hardy, Hilbert, Paley--Wiener, Sobolev). Function spaces are a very active area of mathematics, encompassing the major components of harmonic and functional analysis, operator algebras, approximation theory, several complex variables and real algebraic geometry. They play a dominant and active role in numerous topics in physics, economics, health sciences, data sciences and cybersecurity, and various other branches of engineering such as system theory, signal and image processing, telecommunication, and modern control theory. My long-term plan consists of several diverse projects in the framework of function spaces. These topics, though initially emergent in the classical branch of mathematical analysis, now have connections to numerous other fields. Hence, I envision a multi-disciplinary research group on function spaces, which will bring together scientists and engineers from around the world to work on collaborative projects, to share ideas and resources and to help Université Laval become a world-leading institution in this discipline. The scientific mission of this CRC is to formulate foundational mathematical theories in function spaces that, on one hand, tackle the unsolved challenging mathematical problems and, on the other hand, enrich the fast-developing technologies in engineering domains such as signal processing, mathematical imaging, and image processing.