School of Mathematics and Statistics

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.

Dr Andrew J Baker Reader

Hopf algebras and formal groups; homological algebra; Galois theory of commutative rings

Member of other research groups: Geometry and Topology

Dr Gwyn Bellamy Lecturer

representation theory; symplectic reflection algebras; D-modules

Member of other research groups: Geometry and Topology

Dr Tara Brendle Lecturer

Geometric group theory; mapping class groups of surface

Member of other research groups: Geometry and Topology
Research students: Neil Fullarton, Charalampos Stylianakis, Alan Logan

Prof Kenneth A Brown Professor of Mathematics

Noncommutative algebra; Hopf algebras; homological algebra

Member of other research groups: Geometry and Topology
Research students: Astrid Jahn, Paul Gilmartin

Dr Mikhail Feigin Lecturer

Representations of Chrednik algebras; rings of quasi-invariants

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

Dr James Griffin Research Associate

Operads;  Loop spaces; (Co)homology theories of algebras

Member of other research groups: Geometry and Topology
Supervisor: Ulrich Kraehmer

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

Dr Ulrich Kraehmer Lecturer

Noncommutative geometry; Hopf algebras and quantum groups; homological algebra

Research staff: James Griffin
Research students: Jake Goodman, Ana Rovi, Paul Slevin

Prof Stephen J Pride Professor of Mathematics

Combinatorial group theory; semigroup theory; automata theory

Research student: Alan Logan

Dr Anne Thomas Lecturer

Geometric group theory; rigidity; buildings; Coxeter groups

Member of other research groups: Geometry and Topology

Dr Christian Voigt Lecturer

Noncommutative geometry; K-theory; Quantum groups

Member of other research groups: Geometry and Topology, Analysis

Dr Stuart White Senior Lecturer

Rigidity properties for groups; noncommutative geometry

Member of other research groups: Geometry and Topology, Analysis
Research students: Tomasz Pierzchala, Jorge Castillejos Lopez , Liam Dickson

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 student: Gabriel Tornetta

Neil Fullarton PhD Student

Research Topic: Mapping class groups and related groups
Member of other research groups: Geometry and Topology
Supervisor: Tara Brendle

Paul Gilmartin PhD Student

Research Topic: The Structure of Infinite Dimensional Hopf Algebras
Supervisor: Kenneth A Brown

Jake Goodman PhD Student

Research Topic: Homological algebra
Supervisor: Ulrich Kraehmer

Astrid Jahn PhD Student

Research Topic: Hopf algebras
Member of other research groups: Geometry and Topology
Supervisor: Kenneth A Brown

Alan Logan PhD Student

Research Topic: Semigroup theory
Supervisors: Stephen J Pride, Tara Brendle

Ana Rovi PhD Student

Research Topic: Lie algebroids
Member of other research groups: Geometry and Topology
Supervisor: Ulrich Kraehmer

Paul Slevin PhD Student

Research Topic: Homological Algebra in Monoidal Categories
Supervisor: Ulrich Kraehmer

Charalampos Stylianakis PhD Student

Research Topic: Mapping Class Groups and Related Groups
Member of other research groups: Geometry and Topology
Supervisor: Tara Brendle

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

Supervisors: Christian Korff
Relevant research groups: Algebra, Integrable Systems and Mathematical Physics

Quantum spin-chains and 2-dimensional statistical lattice models, such as the Heisenberg spin-chain and the six and eight vertex model 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).

The sl(n)-WZNW fusion ring: a combinatorial construction and a realisation as quotient of quantum cohomology. with Catharina Stroppel. Adv Math 225, 1 (2010) 200-268; arXiv:0909.2347

 

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.

 

Whittaker Prize of the Edinburgh Maths Society

Friday 7th June, 2013
Dr Stuart White has been awarded the Whittaker Prize of the Edinburgh Maths Society. It is a very prestigious award and very hard to win,...
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