Summary of our research interests
The Analysis research group at the Delft Institute of Applied Mathematics focuses on functional analysis and operator theory and its applications to the study of (partial) differential equations, both deterministic and non-deterministic, arising from models in e.g. quantum physics, fluid dynamics, and mathematical biology. It connects with a diverse range of disciplines in Analysis. Research topics include: operator theory, including semi-groups of operators, functional calculus, and non-commutative analysis; stochastic analysis; spectral theory and variational methods; harmonic analysis; representation theory and quantum groups; differential geometry and topology. Besides offering the core subjects in Analysis, our section contributes to the Applied Mathematics curriculum through a rich variety of electives connecting with our research, both on the bachelor and master level. It also offers a course on the History and Philosophy of Mathematics.
Stochastic partial differential equations with random perturbations can be studied in a functional analytic framework as stochastic evolution equations governed by a semigroup generator in an infinite-dimensionalspace. For Hilbert spaces, the theory of stochastic evolution equations has witnessed a rapid developement during the past two decades. One of its ingredients is the extension to the Hilbert space context of the classical Ito stochastic integral. In collaboration with Mark Veraar (Delft, now Karlsruhe) and L. Weis (Karlsruhe), the classical stochastic Ito integral has been extended to UMD Banach spaces. This permits a full treatment of semilinear stochastic evolution equations in the L^p scale. Current work focusses on understanding qualitative properties of solutions of stochastic evolution equations and understanding the connections between stochastic analysis on the one hand and harmonic analysis on the other. stochastic integral to stochastic integrands and on exploring connections between stochastic integration in Banach spaces and the geometry of the underlying space. This project is currently supported by and NWO VIDI grant (Mark Veraar). Previous grants include an NWO VIDI and NWO VICI grant (Jan van Neerven) and an NWO VENI grant (Mark Veraar). (NWO).
Operator theory concerns the theory of bounded operators on a Hilbert space. The study of algebras of such operators leads to the
subarea of operator algebras (C*-algebras and von Neumann algebras). Operator algebras have been invented because of their eminence in
quantum mechanics and they have many strong connections to a wide variety of subjects within mathematics and mathematical physics.
The aim of the current research project is to import tools coming from harmonic analysis and use these to solve long-standing open
problems in operator theory. Such tools are for instance the development of non-commutative Riesz transforms, harmonic analysis
beyond abelian groups and new applications of Calderon-Zygmund theory and singular integrals. The harmonic analysis involves basically
all aspects: concrete harmonic analysis, abstract harmonic analysis and representation theory. Typical problems in operator theory
that can (potentially) be solved involve sharp commutator estimates as well as approximation and rigidity properties of operator algebras.
This project is supported by an NWO Vidi grant.