Analysis is an extremely broad mathematical discipline. In Glasgow, research in analysis encompasses partial differential equations, harmonic analysis, complex analysis and operator algebras. The group currently consists of three members of academic staff. More information about our research interests can be found through the links below and information about postdocs, research students, grants and collaborators through the links on the right.
Noncommutative geometry; K-theory; Quantum groups
The application of new mathematical ideas and new computational paradigms to material science, with an emphasis on self-assembling nano-materials; analysis and numerical analysis of partial differential equations arising in Continuum Physics; Material Science and Geometry.
Member of other research groups: Continuum Mechanics - Modelling and Analysis of Material Systems
C* and von Neumann algebras; rigidity and similarity properties of operator algebras
Operator algebras, topological dynamical systems, and noncommutative geometry.
C*-algebras, their classification and amenability properties; special examples of C*-algebras; K-theory and non commutative topology, noncommutative dynamical systems, geometric group theory with applications to C*-algebras.
Supervisor: Joachim Zacharias
Research Topic: Quantum groups and the Baum-Connes Conjecture
Supervisor: Christian Voigt
Supervisor: Mike Whittaker
Operator Algebras associated to groups (PhD)
Operator algebras (both C*-algebras and von Neumann algebras) arise naturally from groups and provide a framework for the study of unitary representations. A driving question is how the structure of the operator algebra reflects that of the original group. This direction of research arguably dates back to the foundational papers of Murray and von Neumann and has repeatedly generated profound new insights over the years. Two possible avenues for PhD research are described below.
Rigidity asks to what extent the group is `remembered' by the operator algebra. There has been dramatic progress in recent years following the discovery of the first examples of von Neumann rigid groups by Ioana, Popa and Vaes in 2012. In the setting of C*-algebras, questions of rigidity are very tantalising; both in the setting of amenable groups, and also finding von Neumann rigid groups which are C*-simple.
A striking new connection to boundary actions developed by Kalentar and Kennedy provides a new framework to examine the situation when a reduced group C*-algebra is simple: in particular how much extra information does one gain from knowing that there is an underlying group? To what extent can one view these reduced group C*-algebras as mirroring behaviour of the associated von Neumann factor?
One particularly nice feature of working with operator algebras associated to groups is that there are concrete examples you can compute with in order to develop intutiution and understanding.
Interactions between von Neumann and C*-algebras (PhD)
Operator algebras are closed self-adjoint subalgebras of the bounded operators on a Hilbert space. They come in two distinct types: von Neumann algebras and C*-algebras depending on which topology one uses to take the closure in. Every abelian C*-algebra is the algebra of continuous functions vanishing at infinity on a locally compact space, while abelian von Neumann algebras are the algebras of essentially bounded functions on some measure space. These flavours of topology and measure theory persist in the non-commutative setting: notions of homotopy and pervade the study of C*-algebras, while arguments involving von Neumann algebras often have a measure theoretic style.
There has been dramatic recent progress in the structure theory of simple amenable C*-algebras driven by striking parallels with the deep structural results for injective von Neumann factors due to Connes, Haagerup and Popa in the 70's. A key theme has been the development of ``finitely coloured'' C*-algebraic versions of von Neumann results allowing for the presence of topological phenomena. This idea sparks a number of exciting new directions for research: what other von Neumann notions can be succesfully coloured? How do coloured results fit into the setting of the Toms-Winter regularity conjecture? Can this idea be succesfully used outside the amenable setting to develop new connections there?