## 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 Bonicke : Lecturer

C*-algebras, groupoids, K-theory

#### Dr Samuel Evington : Research Assistant

#### Prof Xin Li : Chair of Mathematical Analysis

**Postgraduate opportunities:** Topological full groups and continuous orbit equivalence

#### Dr Christian Voigt : Senior lecturer

Noncommutative geometry; K-theory; Quantum groups

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

**Research student:** Jamie Antoun

#### Dr James Walton : Research Assistant

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

#### 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 Mike Whittaker : Lecturer

Operator algebras, topological dynamical systems, and noncommutative geometry.

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

**Research students:** Dimitrios Gerontogiannis , Kate Gibbins, Mustafa Ozkaraca, Jamie Antoun, Cheng Chen

#### Dr Runlian Xia : Lecturer

**Research student:** Kate Gibbins

#### 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 student:** Dimitrios Gerontogiannis

## Postgraduates

## Postgraduate opportunities

### Topological full groups and continuous orbit equivalence (PhD)

**Supervisors:** Xin Li

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

This proposed PhD project is part of a research programme whose aim is to develop connections between C*-algebras, topological dynamics and geometric group theory which emerged recently.

More specifically, the main goal of this project is to study topological full groups, which are in many cases complete invariants for topological dynamical systems up to continuous orbit equivalence. Topological full groups have been the basis for spectacular developments recently since they led to first examples of groups with certain approximation properties, solving long-standing open questions in group theory. The goal of this project would be to systematically study algebraic and analytic properties of topological full groups. This is related to algebraic and analytic properties of topological groupoids, the latter being a unifying theme in topological dynamics and operator algebras. A better understanding of the general construction of topological full group -- which has the potential of solving deep open questions in group theory and dynamics -- goes hand in hand with the study of concrete examples, which arise from a rich variety of sources, for instance from symbolic dynamics, group theory, semigroup theory or number theory.

Another goal of this project is to develop a better understanding of the closely related concept of continuous orbit equivalence for topological dynamical systems. This new notion has not been studied in detail before, and there are many interesting and important questions which are not well-understood, for instance rigidity phenomena. Apart from being interesting on its own right from the point of view of dynamics, the concept of continuous orbit equivalence is also closely related to Cartan subalgebras in C*-algebras and the notion of quasi-isometry in geometric group theory. Hence we expect that progress made in the context of this project will have an important impact on establishing a fruitful interplay between C*-algebras, topological dynamics and the geometry of groups.

The theme of this research project has the potential of shedding some light on long-standing open problems. At the same time, it leads to many interesting and feasible research problems.