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 number theory, 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, number theory (algebraic and analytic), 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 K-theory, L-Theory
Member of other research groups: Geometry and Topology
- 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
Representations of Cherednik algebras; rings of quasi-invariants
Teichmuller Dynamics, Mapping Class Groups.
representation theory of algebras, cluster algebras and cluster categories, triangulated categories
I work in Representation Theory and Algebraic Geometry. I am interested in geometric representation theory inspired by the Langlands program and mathematical physics. A few topics that particularly interest me: affine Grassmannians, p-adic groups, loop and double loop groups, Hecke algebras, intersection cohomology, Coulomb branches, symplectic varieties.
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.
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.
Research Topic: Short star products in (Poisson) vertex algebras
Member of other research groups: Integrable Systems and Mathematical Physics, Geometry and Topology
Supervisors: Daniele Valeri, Gwyn Bellamy
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