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 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.
Staff
Dr Spiros Adams-Florou : Lecturer
Algebraic K-theory, L-Theory
Member of other research groups: Geometry and Topology
Prof Alex Bartel : Professor of Mathematics
- 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.
Member of other research groups: Geometry and Topology
Research student: Ross Paterson
Prof Gwyn Bellamy : Professor of Mathematics
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, Samuel (Sam) Lewis, Kellan Steele, Simone Castellan, Ross Paterson
Dr Tara Brendle : Professor of Mathematics
Geometric group theory; mapping class groups of surface
Member of other research groups: Geometry and Topology
Research students: Tudur Lewis, 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 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, Leo Kaminski, Georgios Antoniou, Johan Wright
Dr Vaibhav Gadre : Lecturer
Teichmuller Dynamics, Mapping Class Groups.
Member of other research groups: Geometry and Topology
Research students: Luke Jeffreys, Tudur Lewis
Dr Sira Gratz : Lecturer
representation theory of algebras, cluster algebras and cluster categories, triangulated categories
Research students: David Murphy, James Rowe, Damian Wierzbicki
Dr Dinakar Muthiah : Lecturer
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.
Dr Matthew Pressland : EPSRC Postdoctoral Fellow
representation theory, homological algebra, cluster algebras, algebraic geometry
Member of other research groups: Geometry and Topology
Dr Franco Rota : Research Associate
Algebraic geometry. In particular, derived categories, moduli spaces of sheaves, the McKay correspondence, Bridgeland stability conditions, the stability manifold and its relation with mirror symmetric questions.
Member of other research groups: Geometry and Topology
Dr Greg Stevenson : Lecturer
Member of other research groups: Geometry and Topology
Research students: David Murphy, James Rowe, Hao Zhang
Dr Christian Voigt : Senior lecturer
Noncommutative geometry; K-theory; Quantum groups
Member of other research groups: Geometry and Topology, Analysis
Research students: Jamie Antoun, Owen Tanner
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, Hao Zhang, Samuel (Sam) Lewis
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 , Kate Gibbins, Mustafa Ozkaraca, Jamie Antoun, Cheng Chen
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: Dimitrios Gerontogiannis
Postgraduates
Simone Castellan : PhD Student
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
Cheng Chen : PhD Student
Research Topic: Semigroup C*-algebras for examples coming from group theory
Supervisors: Brendan Owens, Mike Whittaker
Miguel Couto : PhD Student
Supervisor: Kenneth A Brown
Kate Gibbins : PhD Student
Research Topic: The Ledrappier shift, one-dimensional tilings and continuous
fields of C*-algebras.
Supervisors: Mike Whittaker, Runlian Xia
Niall Hird : PhD Student
Supervisor: Gwyn Bellamy
Luke Jeffreys : PhD Student
Research Topic: Teichmüller dynamics
Member of other research groups: Geometry and Topology
Supervisors: Vaibhav Gadre, Tara Brendle
Leo Kaminski : PhD Student
Research Topic: Special solutions of WDVV equations
Member of other research groups: Integrable Systems and Mathematical Physics, Geometry and Topology
Supervisor: Mikhail Feigin
Sarah Kelleher (Mackie) : PhD Student
Supervisor: Michael Wemyss
Tudur Lewis : PhD Student
Research Topic: Mapping class groups and related structures.
Member of other research groups: Geometry and Topology
Supervisors: Tara Brendle, Vaibhav Gadre
David Murphy : PhD Student
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,
and (tau-)tilting.
Supervisors: Sira Gratz, Greg Stevenson
Ross Paterson : PhD Student
Supervisors: Alex Bartel, Gwyn Bellamy
James Rowe : PhD Student
Supervisors: Sira Gratz, Greg Stevenson
Kellan Steele : PhD Student
Member of other research groups: Geometry and Topology
Supervisor: Gwyn Bellamy
Owen Tanner : PhD Student
Research Topic: Topological full groups and continuous orbit equivalence.
Member of other research groups: Geometry and Topology
Supervisors: Xin Li, Christian Voigt
Ogier Van Garderen : PhD Student
Member of other research groups: Geometry and Topology
Supervisor: Michael Wemyss
Damian Wierzbicki : PhD Student
Research Topic: cluster algebras
Supervisors: Christian Korff, Sira Gratz
Hao Zhang : PhD Student
Research Topic: Bridgeland stability manifolds of Divisor to Curve Contractions
Member of other research groups: Geometry and Topology
Supervisors: Michael Wemyss, Greg Stevenson
Postgraduate opportunities
Quantum spin-chains and exactly solvable lattice models (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 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)
Supervisors: Christian Korff
Relevant research groups: Algebra, Integrable Systems and Mathematical Physics
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.
Tree for modular group
