## 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.

# Information about

## Staff

#### 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 Cohenâ€•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 staff:** James Rowe

**Research students:** David Murphy, 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

#### James Rowe : Graduate Teaching Assistant

**Supervisors:** Sira Gratz, Greg Stevenson

#### Dr Greg Stevenson : Lecturer

**Member of other research groups:** Geometry and Topology

**Research staff:** James Rowe

**Research students:** David Murphy, Hao Zhang

#### Prof Christian Voigt : Professor

Noncommutative geometry; K-theory; Quantum groups

**Member of other research groups:** Geometry and Topology, Analysis

**Research student:** 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), Hao Zhang, Samuel (Sam) Lewis

#### Prof Mike Whittaker : Professor of Mathematics

Operator algebras, self-similar groups, and Zappa-Szep products.

**Member of other research groups:** Geometry and Topology, Analysis

**Postgraduate opportunities:** Aperiodic substitution tilings and their C*-algebras., Operator algebras associated to self-similar actions.

#### Prof Joachim Zacharias : Professor

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

## 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

#### Miguel Couto : PhD Student

**Supervisor:** Kenneth A Brown

#### 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

#### 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

#### 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

## Algebra example research projects

#### 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.

#### Operator algebras associated to self-similar actions. (PhD)

**Supervisors:** Mike Whittaker**Relevant research groups:** Algebra, Analysis, Geometry and Topology

This project will focus on self-similar groups and their operator algebras. The primary aim will be to examine a new class of groups that act self-similarly on the path space of a graph and to study the noncommutative geometry of a natural class of operator algebras associated to these self-similar groups.

Self-similar groups are an important and active new area of group theory. The most famous example is the Grigorchuk group, which was the first known example of a group with intermediate growth. This makes investigating C*-algebras associated to them particularly interesting. In particular, these groups are often defined by their action on a graph, and the associated C*-algebra encodes both the group and path space of the graph in a single algebraic object, as well as the interaction between them.