# 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

#### 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, Operator Algebras associated to groups

#### 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:** 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.

### Operator Algebras associated to groups (PhD)

**Supervisors:** Stuart White

**Relevant research groups:** Algebra, Analysis

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)

**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:** 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.