# Analysis

# Analysis

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.

## Staff

#### Dr Christian Voigt Senior lecturer

Noncommutative geometry; K-theory; Quantum groups

**Member of other research groups:** Geometry and Topology, Algebra

**Research students:** Jamie Antoun, Andrew Monk, Samuel Evington, Lucia Rotheray

#### Dr Stephen J Watson Lecturer

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

#### Dr Stuart White Professor of Mathematics

C* and von Neumann algebras; rigidity and similarity properties of operator algebras

**Member of other research groups:** Geometry and Topology, Algebra

**Research student:** Samuel Evington

**Postgraduate opportunities:** Interactions between von Neumann and C*-algebras

#### Dr Mike Whittaker Lecturer

Operator algebras, topological dynamical systems, and noncommutative geometry.

**Member of other research groups:** Geometry and Topology, Algebra

**Research students:** Dimitrios Gerontogiannis , Mustafa Ozkaraca, Jamie Antoun

**Postgraduate opportunities:** Aperiodic substitution tilings and their C*-algebras., Operator algebras associated to self-similar actions.

#### Dr Joachim Zacharias Reader

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.

**Member of other research groups:** Integrable Systems and Mathematical Physics, Geometry and Topology, Algebra

**Research staff:** Joan Bosa

**Research students:** Luke Hamblin, Dimitrios Gerontogiannis

## Postgraduates

#### Samuel Evington PhD Student

**Research Topic:** W*-bundles

**Supervisors:** Stuart White, Christian Voigt

#### Dimitrios Gerontogiannis PhD Student

**Supervisors:** Mike Whittaker, Joachim Zacharias

#### Luke Hamblin PhD Student

**Supervisor:** Joachim Zacharias

#### Andrew Monk PhD Student

**Research Topic:** Quantum groups and the Baum-Connes Conjecture

**Supervisor:** Christian Voigt

#### Mustafa Ozkaraca PhD Student

**Supervisor:** Mike Whittaker

## Postgraduate opportunities

### Operator algebras associated to self-similar actions. (PhD)

**Supervisors:** Mike Whittaker

**Relevant research groups:** Geometry and Topology, Analysis, Algebra

This project will focus on self-similar groups and their operator algebras. The primary aim will be to examine a new class of groups that act self-similarly on the path space of a graph and to study the noncommutative geometry of a natural class of operator algebras associated to these self-similar groups.

### Interactions between von Neumann and C*-algebras (PhD)

**Supervisors:** Stuart White

**Relevant research groups:** Analysis

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?

### Aperiodic substitution tilings and their C*-algebras. (PhD)

**Supervisors:** Mike Whittaker

**Relevant research groups:** Geometry and Topology, Analysis