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, both commutative and noncommutative ring theory, as well as topics in representation theory and homological algebra. All information about our group, our members, our activities, and a full list of our expertise, can be found at our Core Structures webpage.
Our group has an active PhD student community, and every year we admit new PhD students. We welcome applications from across the world, and we encourage you to browse our available supervisors, and also to consult our general advice on how to navigate the application process.
The non-exhaustive list under Postgraduate Opportunities contains a sample of the types of PhD projects that our group offers.
- 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 Cohen―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.
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.
Geometric group theory; mapping class groups of surface
Noncommutative algebra; Hopf algebras; homological algebra
Representation theory of algebras and orders; modular representation theory; orders over discrete valuation rings
Representations of Cherednik algebras; rings of quasi-invariants
Teichmuller Dynamics, Mapping Class Groups.
representation theory of algebras, cluster algebras and cluster categories, triangulated categories
Markov bases of toric ideals; graphs; matroids; factoriality of threefolds.
Geometric and combinatorial group theory
Member of other research groups: Geometry and Topology
Noncommutative geometry; K-theory; Quantum groups
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.
Rigidity properties for groups; noncommutative geometry
Operator algebras, self-similar groups, and Zappa-Szep products.
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.
Supervisor: Kenneth A Brown
Supervisor: Gwyn Bellamy
Supervisor: Michael Wemyss
Research Topic: Representations of finite dimensional algebras
This project studies representations of finite dimensional
algebras via a homological approach. The central idea is to
study structures and properties of categories coming from
representations of finite dimensional algebras, with possible
focus on thick subcategories, Serre functors, lattice theory,
Supervisors: Sira Gratz, Greg Stevenson
Quantum spin-chains and exactly solvable lattice models (PhD)
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 and algebraic combinatorics. 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)
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.
Topological full groups and continuous orbit equivalence (PhD)
This proposed PhD project is part of a research programme whose aim is to develop connections between C*-algebras, topological dynamics and geometric group theory which emerged recently.
More specifically, the main goal of this project is to study topological full groups, which are in many cases complete invariants for topological dynamical systems up to continuous orbit equivalence. Topological full groups have been the basis for spectacular developments recently since they led to first examples of groups with certain approximation properties, solving long-standing open questions in group theory. The goal of this project would be to systematically study algebraic and analytic properties of topological full groups. This is related to algebraic and analytic properties of topological groupoids, the latter being a unifying theme in topological dynamics and operator algebras. A better understanding of the general construction of topological full group -- which has the potential of solving deep open questions in group theory and dynamics -- goes hand in hand with the study of concrete examples, which arise from a rich variety of sources, for instance from symbolic dynamics, group theory, semigroup theory or number theory.
Another goal of this project is to develop a better understanding of the closely related 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, the concept of continuous orbit equivalence is also closely related to Cartan subalgebras in C*-algebras and the notion of quasi-isometry in geometric group theory. Hence we expect that progress made in the context of this project will have an important impact on establishing a fruitful interplay between C*-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.