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
- Ian Doust
- Pinhas Grossman
- Galina Levitina
- Milan Pahor
- Denis Potapov
- Fedor Sukochev
- Norman Wildberger
- Dmitriy Zanin
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.
Pinhas Grossman is interested in operator algebras, and in particular von Neumann algebras. He studies algebraic invariants of subfactors.
Galina Levitina is interested in the area of noncommutative (matrix) analysis, particularly in the method of multiple operator integral.
Milan Pahor's research interests include C*-algebras, cross products, group C* algebras, Mathematics education and spatial visualisation.
Denis Potapov works in the area of noncommutative (matrix) analysis. He is particularly interested in the so-called method of multiple operator integrals. This is a relatively modern set of techniques which has recently proved very powerful when applied to a string of hard problems in analysis. He is also very keen on the application of these methods to the area of Mathematical Physics. While multiple operator integrals have been used in Mathematical Physics for the last 50 years or so, the area has recently received new momentum due to fresh ideas brought from vector-valued harmonic analysis and vector-valued Banach theory: Denis has been one of the main contributors to this effort.
Fedor Sukochev has wide interests in non-commutative functional analysis and its applications to non-commutative geometry, particularly those related to quantised calculus and index theorems. He has strong interests in singular (Dixmier) traces and their applications. His current research activity includes non-commutative integration theory, non-commutative probability theory and various aspects of Banach space geometry and its applications.
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.
Dmitriy Zanin studies functions and the functions of functions, with applications ranging from quantum physics to signal processing.