Functional analysis and harmonic analysis both arose out of the study of the differential equations of mathematical physics. Wave and diffusion phenomena are highly amenable to the techniques of these areas, and so functional and harmonic analysis continue to find new applications in fields such as quantum mechanics and electrical engineering. While harmonic analysis focusses on the behaviour of a particular function, functional analysis considers the properties of large collections of functions.
The interests of the group range from classical problems, analysis on Lie groups, single operator theory in Banach spaces through semigroup theory to von Neumann and C*-algebras.
Michael Cowling's diverse interests include heat and wave and other semigroups of operators, oscillatory and singular integrals, multipliers, analysis on Lie groups, uniformly bounded representations, lattices and rigidity. He is also studying harmonic analysis on locally compact groups and associated operator algebras.
Ian Doust works in single operator theory, specifically the spectral theory of linear operators acting in Banach spaces. His work on the interaction between the geometry of Banach spaces and functional calculi has lead naturally to integral representations of operators.
Valentyn Golodets retired from the Institute of Low Temperature Physics, in Kharkov. He is a Visiting Fellow in the School who is working with Tony Dooley on ergodic theory.
Pinhas Grossman is interested in operator algebras, and in particular von Neumann algebras. He studies algebraic invariants of subfactors.
Hendrik Grundling deals with C*-algebra problems arising from Mathematical Physics (especially Quantum Field Theory). With his co-authors, he recently developed and analysed two C*-algebras with which to model bosonic quantum systems. He also works on the problem of extending group algebras and crossed products to topological groups which are not locally compact.
Brian Jefferies has interests in the theory of vector measures with applications to functional integration and Feynman path integrals. He is currently working on functional calculi for non-commuting systems of operators. His monograph Spectral Properties of Noncommuting Operators, Lecture Notes in Mathematics Vol. 1843 was published by Springer in 2004.
Milan Pahor's research interests include C* Algebras, Cross Products, Group C* Algebras, Mathematics Education and Spatial Visualisation.
Denis Potapov has research interests in the fields of noncommutative analysis, Schur Multipliers and Double Operator Integrals.
Fedor Sukochev has wide interests in non-commutative functional analysis and its applications to non-commutative geometry, in non-commutative integration theory and in Banach space geometry and its applications.
Denis Potapov and Fedor Sukochev are interested in vector-valued harmonic analysis and non-commutative analysis, especially the area concerning operator inequalities and operator-smooth functions, where the key role is played by the operator integral representation. Denis and Fedor also have interests in the concepts of the spectral shift function and spectral flow, and their roles in non-commutative geometry and mathematical physics.
Norman Wildberger has worked on the orbit method and moment map on compact and nilpotent Lie groups. He has recently been developing the theory of hypergroups which impacts significantly in this area, and also has applications to diophantine equations.