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, 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
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 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
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
Dr Tara Brendle : Professor of Mathematics
Geometric group theory; mapping class groups of surface
Member of other research groups: Geometry and Topology
Research student: 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 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/RA
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
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 Theo Raedschelders : Research Assistant
Member of other research groups: Geometry and Topology
Supervisor: Michael Wemyss
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
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 staff: Theo Raedschelders
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
Postgraduate opportunities: Interactions between von Neumann and C*-algebras, Operator Algebras associated to groups
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
Dr William 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 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
Tomasz Przezdziecki : 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
Quantum spin-chains and exactly solvable lattice models (PhD)
Supervisors: Christian Korff
Relevant research groups: Integrable Systems and Mathematical Physics, Algebra
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).
Operator Algebras associated to groups (PhD)
Supervisors: Stuart White
Relevant research groups: Algebra, Analysis
Operator algebras (both C*-algebras and von Neumann algebras) arise naturally from groups and provide a framework for the study of unitary representations. A driving question is how the structure of the operator algebra reflects that of the original group. This direction of research arguably dates back to the foundational papers of Murray and von Neumann and has repeatedly generated profound new insights over the years. Two possible avenues for PhD research are described below.
Rigidity asks to what extent the group is `remembered' by the operator algebra. There has been dramatic progress in recent years following the discovery of the first examples of von Neumann rigid groups by Ioana, Popa and Vaes in 2012. In the setting of C*-algebras, questions of rigidity are very tantalising; both in the setting of amenable groups, and also finding von Neumann rigid groups which are C*-simple.
A striking new connection to boundary actions developed by Kalentar and Kennedy provides a new framework to examine the situation when a reduced group C*-algebra is simple: in particular how much extra information does one gain from knowing that there is an underlying group? To what extent can one view these reduced group C*-algebras as mirroring behaviour of the associated von Neumann factor?
One particularly nice feature of working with operator algebras associated to groups is that there are concrete examples you can compute with in order to develop intutiution and understanding.
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.
Tree for modular group
