## 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 the staff members listed below. More information about our research interests can be found by clicking on staff links, and information about research opportunities and seminars through the links on the right.

# Information about

## Staff

#### Dr Kevin Aguyar Brix : Carlesberg and DFF Fellow

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

**Research student:** Owen Tanner

**Postgraduate opportunities:** Interactions between groups, topological dynamics and operator algebras

#### Prof Christian Voigt : Professor

Noncommutative geometry; K-theory; Quantum groups

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

**Research student:** Owen Tanner

#### 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

#### Prof Mike Whittaker : Professor of Mathematics

Operator algebras, topological dynamical systems, and noncommutative geometry.

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

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

#### Dr Runlian Xia : Lecturer

#### Prof Joachim Zacharias : Professor

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

## Postgraduate opportunities

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

**Supervisors:** Mike Whittaker

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

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.

Self-similar groups are an important and active new area of group theory. The most famous example is the Grigorchuk group, which was the first known example of a group with intermediate growth. This makes investigating C*-algebras associated to them particularly interesting. In particular, these groups are often defined by their action on a graph, and the associated C*-algebra encodes both the group and path space of the graph in a single algebraic object, as well as the interaction between them.

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

**Supervisors:** Mike Whittaker

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

A tiling is a collection of subsets of the plane, called tiles, for which any intersection of the interiors of two distinct tiles is empty and whose union is all of the plane. A tiling said to be aperiodic if it lacks translational periodicity. The most common method of producing aperiodic tilings is to use a substitution rule; a method for breaking each tile into smaller pieces, each of which is a scaled down copy of one of the original tiles, and then expanding so that each tile is congruent to one of the original tiles.

This project will focus on a natural class of operator algebras associated with an aperiodic substitution tiling. These algebras were first considered by Kellendonk and reflect the symmetries of a tiling in an algebraic object that allows up to consider invariants in a noncommutative framework. A key area of study are spectral triples associated with aperiodic tilings, which allow us to think of tilings as noncommutative geometric objects.

### Interactions between groups, topological dynamics and operator algebras (PhD)

**Supervisors:** Xin Li

**Relevant research groups:** Analysis

The goal of this project is to develop a better understanding of the 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, continuous orbit equivalence is also closely related to the concepts of quasi-isometry in geometric group theory and Cartan subalgebras in C*-algebras. Hence we expect that progress made in the context of this project will have an important impact on establishing a fruitful interplay between operator 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.