# Algebra

# Algebra

The traditional objects of study in algebra are algebraic structures such as groups, rings, and modules. However, the developments of the last decades have increasingly emphasised the subject's connections with other areas of mathematics and science, such as geometry, topology, classical and quantum field theory, integrable systems, and theoretical computing science.

Here in Glasgow, we study both classical and modern problems and questions in algebra. The research interests of our staff members include geometric group theory and computational semigroup theory, both commutative and noncommutative ring theory, as well as topics in representation theory and homological algebra.

## Staff

#### Dr Spiros Adams-Florou Lecturer

Algebraic K-theory, L-Theory

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

#### Dr Andrew J Baker Reader

Algebraic Topology, especially stable homotopy theory, operations in periodic cohomology theories, structured ring spectra including Galois theory and other applications of algebra, number theory and algebraic geometry.

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

#### Dr Gwyn Bellamy Lecturer

My research interests are in geometric representation theory and its connections to algebraic geometry and algebraic combinatorics. In particular, I am interested in all aspects of symplectic representations, including symplectic reflection algebras, resolutions of symplectic singularties, D-modules and deformation-quantization algebras.

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

**Research student:** Tomasz Przezdziecki

**Postgraduate opportunities:** Deformation-Quantization algebras on K3 surfaces, Homological properties of Deformation-Quantization algebras, Resolutions of symplectic singularities, Sheaves of Cherednik algebras, Rational Cherednik algebras and geometric Langlands

#### Dr Tara Brendle Professor of Mathematics

Geometric group theory; mapping class groups of surface

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

**Research students:** Alan McLeay, Luke Jeffreys

#### Prof Kenneth A Brown Professor of Mathematics

Noncommutative algebra; Hopf algebras; homological algebra

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

**Research students:** Miguel Couto, Paul Gilmartin

#### Dr Mikhail Feigin Senior lecturer

Representations of Cherednik algebras; rings of quasi-invariants

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

**Research student:** Georgios Antoniou

#### Dr Vaibhav Gadre Lecturer

Teichmuller Dynamics, Mapping Class Groups.

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

**Research student:** Luke Jeffreys

#### Dr Christian Korff Research Fellow of the Royal Society/Reader

Representation theoretic and combinatorial aspects in mathematical physics; quantum groups

**Member of other research groups:** Integrable Systems and Mathematical Physics

**Research students:** David Palazzo, Lucia Rotheray

**Postgraduate opportunities:** Integrable quantum field theory and Y-systems, Quantum spin-chains, exactly solvable lattice models and representation theory

#### Dr Ulrich Kraehmer Reader

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

**Research students:** Angela Tabiri, Lucia Rotheray

#### Dr Alan Logan Jack Fellowship

Geometric and combinatorial group theory

#### Dr Ciaran Meachan Lecturer

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

#### Dr Christian Voigt Senior lecturer

Noncommutative geometry; K-theory; Quantum groups

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

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

#### Prof Michael Wemyss Professor of Mathematics

Algebraic geometry and its interactions, principally between noncommutative and homological algebra, resolutions of singularities, and the minimal model program. All related structures, including: deformation theory, derived categories, stability conditions, associated commutative and homological structures and their representation theory, curve invariants, McKay correspondence, Cohen--Macaulay modules, finite dimensional algebras and cluster-tilting theory.

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

#### Dr Stuart White Professor of Mathematics

Rigidity properties for groups; noncommutative geometry

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

**Research student:** Samuel Evington

**Postgraduate opportunities:** Interactions between von Neumann and C*-algebras, Operator Algebras associated to groups

#### Dr Mike Whittaker Lecturer

Operator algebras, self-similar groups, and Zappa-Szep products.

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

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

**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, Analysis

**Research staff:** Joan Bosa

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

## Postgraduates

#### Miguel Couto PhD Student

**Supervisor:** Kenneth A Brown

#### Paul Gilmartin PhD Student

**Research Topic:** The Structure of Infinite Dimensional Hopf Algebras

**Supervisor:** Kenneth A Brown

#### Luke Jeffreys PhD Student

**Supervisors:** Vaibhav Gadre, Tara Brendle

#### Alan McLeay PhD Student

**Research Topic:** Geometric group theory and mapping class groups of surfaces

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

**Supervisor:** Tara Brendle

#### Tomasz Przezdziecki PhD Student

**Supervisor:** Gwyn Bellamy

#### Lucia Rotheray PhD Student

**Supervisors:** Christian Korff, Christian Voigt, Ulrich Kraehmer

#### Angela Tabiri PhD Student

**Research Topic:** Quantum group actions on singular varieties

**Supervisor:** Ulrich Kraehmer

## Postgraduate opportunities

### Quantum spin-chains, exactly solvable lattice models and representation theory (PhD)

**Supervisors:** Christian Korff

**Relevant research groups:** Integrable Systems and Mathematical Physics, Algebra

Quantum spin-chains and 2-dimensional statistical lattice models, such as the Heisenberg spin-chain and the six and eight-vertex models remain an active area of research with many surprising connections to other areas of mathematics.

Some of the algebra underlying these models deals with quantum and Hecke algebras, the Temperley-Lieb algebra, the Virasoro algebra and Kac-Moody algebras. There are many unanswered questions ranging from very applied to more pure topics in representation theory. For example, recently these models have been applied in combinatorial representation theory to compute Gromov-Witten invariants (enumerative geometry) and fusion coefficients in conformal field theory (mathematical physics).

### Integrable quantum field theory and Y-systems (PhD)

**Supervisors:** Christian Korff

**Relevant research groups:** Integrable Systems and Mathematical Physics, Algebra

The mathematically rigorous and exact construction of a quantum field theory remains a tantalising challenge. In 1+1 dimensions exact results can be found by computing the scattering matrices of such theories using a set of functional relations. These theories exhibit beautiful mathematical structures related to Weyl groups and Coxeter geometry.

In the thermodynamic limit (volume and particle number tend to infinity while the density is kept fixed) the set of functional relations satisfied by the scattering matrices leads to so-called Y-systems which appear in cluster algebras introduced by Fomin and Zelevinsky and the proof of dilogarithm identities in number theory.

### 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.

### 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.

### Rational Cherednik algebras and geometric Langlands (PhD)

**Supervisors:** Gwyn Bellamy

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

The geometric Langlands program (google it!) is one of the most profound (conjectural) relations in mathematics, and is often viewed as the very pinnacle of the subject. It turns out thet there is a connection between rational Cherednik algebras and the affine Lie algebra glb n at the critical level (which is the key object of study in geometric Langlands). Very little is known about this relationship and virtually nothing written. The goal of this project would be to remedy this situation.

All projects are subject to the availability of funding.

### Sheaves of Cherednik algebras (PhD)

**Supervisors:** Gwyn Bellamy

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

Soon after Etingof and Giznburg introduced rational Cherednik algebras, Etingof showed that they belong to a much large family of sheaves of Cherednik algebras that can be defined for any smooth variety and finite group acting on this variety. In this generality, very little is know about the algebras. The goal of this project would be to develop tools, as exist in the theory of D-module, to study these algebras. Hopefully one can say something non-trivial about the representation theory of these algebras then.

All projects are subject to the availability of funding.

### Resolutions of symplectic singularities (PhD)

**Supervisors:** Gwyn Bellamy

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

Symplectic singularities appear naturally in representation theory of non-commutative algebras. As such their geometry tells us a lot about the representation theory of these algebras. Through work of Namikawa we now know a great deal about symplectic singularities and their resolutions. However, there are still many open problems. One of the most basic is to count, and explicitly construct, these resolutions. The goal of this project would be to do so for some concrete, but interesting examples.

All projects are subject to the availability of funding.

### Homological properties of Deformation-Quantization algebras (PhD)

**Supervisors:** Gwyn Bellamy

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

In the study of the representation theory of deformation-quantization algebras, several natural categories of sheaves play an important role. Most of these categories should be “smooth” in an appropriate sense. In particular, they should have finite global dimension. The goal of this project would be to understand the homological properties, in particular global dimension, of these categories. The idea here would be to generalize classical (but difficult) results in the theory of D-modules on the global dimension of sheaves of differential algebras.

All projects are subject to the availability of funding.

### Deformation-Quantization algebras on K3 surfaces (PhD)

**Supervisors:** Gwyn Bellamy

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

Smooth K3 surfaces are a natural sources of symplectic manifolds. They have been studied by algebraic geometers for over a century now. One can also study deformation-quantization algebras on them. The goal then would be to 1 relate the properties of these non-commutative algebras, and their representation theory, to the rich underlying geometric properties of the surfaces.

All projects are subject to the availability of funding.