# 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 Alex Bartel Senior lecturer

**Algebraic number theory:**- Galois module structures, e.g. the structure of the ring of integers of a number field as a Galois module, or of its unit group, or of the Mordell-Weil group of an abelian variety over a number field;
- Arithmetic statistics, especially the Cohenâ€•Lenstra heuristics on class groups of number fields and their generalisations;
- Arithmetic of elliptic curves over number fields.

**Representation theory of finite groups:**- Integral representations of finite groups;
- Connections between the Burnside ring and the representation ring of a finite group;
- Applications of the above to number theory and geometry.

**Geometry and topology:**- Actions of finite groups of low-dimensional manifolds.

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

#### Dr Gwyn Bellamy Senior 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 students:** Niall Hird, Tomasz Przezdziecki, Kellan Steele, Kellan Steele

**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

#### Dr Kenneth A Brown Professor of Mathematics

Noncommutative algebra; Hopf algebras; homological algebra

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

**Research student:** Miguel Couto

#### Dr Ben Davison Lecturer

Algebraic geometry, geometric representation theory, cluster algebras, Higgs bundles, Nakajima quiver varieties

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

**Research student:** Ogier Van Garderen

#### Dr Sam Dean Lecturer

#### 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 students:** Maali Alkadhem, Georgios Antoniou

#### Dr Jamie Gabe Honorary Research Fellow

**Member of other research groups:** Mathematical Biology, Geometry and Topology

#### Dr Vaibhav Gadre Lecturer

Teichmuller Dynamics, Mapping Class Groups.

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

**Research student:** Luke Jeffreys

#### Dr Sira Gratz Lecturer

representation theory of algebras, cluster algebras and cluster categories, triangulated categories

**Postgraduate opportunities:** Cluster Algebras and their implementation on the computer

#### Dr Dimitra Kosta LKAS Fellowship

Markov bases of toric ideals; graphs; matroids; factoriality of threefolds.

**Member of other research groups:** Statistical Methodology, Environmental Statistics, Biostatistics and Statistical Genetics, Geometry and Topology

#### Dr Alan Logan Jack Fellowship

Geometric and combinatorial group theory

#### Dr Ciaran Meachan Lecturer

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

#### Dr Greg Stevenson 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:** Jamie Antoun, Andrew Monk, Samuel Evington

#### 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

**Research students:** Sarah Kelleher (Mackie), Ogier Van Garderen

#### 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

#### 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, Jamie Antoun

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

#### Dr Billy Woods Lecturer

#### 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

#### Niall Hird PhD Student

**Supervisor:** Gwyn Bellamy

#### Luke Jeffreys PhD Student

**Supervisors:** Vaibhav Gadre, Tara Brendle

#### Sarah Kelleher (Mackie) PhD Student

**Supervisor:** Michael Wemyss

#### 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

#### Kellan Steele PhD Student

**Supervisor:** Gwyn Bellamy

#### Kellan Steele PhD Student

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

**Supervisor:** Gwyn Bellamy

#### Angela Tabiri PhD Student

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

**Supervisor:** Ulrich Kraehmer

#### Ogier Van Garderen PhD Student

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

**Supervisors:** Michael Wemyss, Ben Davison

## 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.

### 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.

### Algebraic Combinatorics and Symbolic Computation (PhD)

**Supervisors:** Christian Korff

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

This project will look at topics in algebraic combinatorics, for example the ring of symmetric functions, the Robinson-Schensted-Knuth correspondence, crystal graphs, and their implementation in symbolic computational software. An example can be found here:

http://demonstrations.wolfram.com/KirillovReshetikhinCrystals/

Emphasis of the project will be on the efficient design and writing of algorithms with the aim to provide data for open research problems in algebraic combinatorics.

The ideal candidate should have a dual interest in mathematics and computer science and, in particular, already possess some programming experience. As part of the project it is envisaged to spend some time with a private software company to explore a possible commercialisation of the resulting computer packages.

### Cluster Algebras and their implementation on the computer (PhD)

**Supervisors:** Sira Gratz, Christian Korff

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

This project will look at cluster algebras and their implementation in symbolic computational software. An example can be found here:

http://demonstrations.wolfram.com/ClusterAlgebras/

Emphasis of the project will be on the efficient design and writing of algorithms with the aim to provide data for open research problems in cluster algebras as well as their visualisation, e.g. in the context of triangulations of surfaces.

The ideal candidate should have a dual interest in algebra and computer science and, in particular, already possess some programming experience. As part of the project it is envisaged to spend some time with a private software company to explore a possible commercialisation of the resulting computer packages.