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 and computational semigroup theory, both commutative and noncommutative ring theory, as well as topics in representation theory and homological algebra.

Staff

Dr Spiros Adams-Florou  Lecturer

Algebraic K-theory, L-Theory

Member of other research groups: Geometry and Topology

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  • Dr Andrew J Baker  Reader

    Algebraic Topology, especially stable homotopy theory, operations in periodic cohomology theories, structured ring spectra including Galois theory and other applications of algebra, number theory and algebraic geometry.

    Member of other research groups: Geometry and Topology

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  • Dr Gwyn Bellamy  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 student: Tomasz Przezdziecki
    Postgraduate opportunities: Deformation-Quantization algebras on K3 surfaces, Homological properties of Deformation-Quantization algebras, Resolutions of symplectic singularities, Sheaves of Cherednik algebras, Rational Cherednik algebras and geometric Langlands

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  • Dr Tara Brendle  Professor of Mathematics

    Geometric group theory; mapping class groups of surface

    Member of other research groups: Geometry and Topology
    Research students: Alan McLeay, Luke Jeffreys

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  • 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

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  • Dr Ben Davison  Lecturer

    Algebraic geometry, geometric representation theory, cluster algebras, Higgs bundles, Nakajima quiver varieties

    Member of other research groups: Geometry and Topology

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  • 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

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  • Dr Vaibhav Gadre  Lecturer

    Teichmuller Dynamics, Mapping Class Groups. 

    Member of other research groups: Geometry and Topology
    Research student: Luke Jeffreys

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  • Dr Sira Gratz  Lecturer

    representation theory of algebras, cluster algebras and cluster categories, triangulated categories

  • Dr Alan Logan  Jack Fellowship

    Geometric and combinatorial group theory

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  • Dr Ciaran Meachan  Lecturer

    Member of other research groups: Geometry and Topology

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  • Lucia Rotheray  MacLaurin Lecturer

  • Dr Christian Voigt  Senior lecturer

    Noncommutative geometry; K-theory; Quantum groups

    Member of other research groups: Geometry and Topology, Analysis
    Research students: Andrew Monk, Samuel Evington, Lucia Rotheray

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  • 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

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  • Dr Stuart White  Professor of Mathematics

    Rigidity properties for groups; noncommutative geometry

    Member of other research groups: Geometry and Topology, Analysis
    Research student: Samuel Evington
    Postgraduate opportunities: Interactions between von Neumann and C*-algebras, Operator Algebras associated to groups

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

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  • 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 staff: Joan Bosa
    Research students: Luke Hamblin, Dimitrios Gerontogiannis

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  • Postgraduate opportunities

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

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

    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. 

     

    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.

     

    Rational Cherednik algebras and geometric Langlands (PhD)

    Supervisors: Gwyn Bellamy
    Relevant research groups: Algebra, Geometry and Topology

    The geometric Langlands program (google it!) is one of the most profound (conjectural) relations in mathematics, and is often viewed as the very pinnacle of the subject. It turns out thet there is a connection between rational Cherednik algebras and the affine Lie algebra glb n at the critical level (which is the key object of study in geometric Langlands). Very little is known about this relationship and virtually nothing written. The goal of this project would be to remedy this situation.

    All projects are subject to the availability of funding. 

     

    Sheaves of Cherednik algebras (PhD)

    Supervisors: Gwyn Bellamy
    Relevant research groups: Algebra, Geometry and Topology

    Soon after Etingof and Giznburg introduced rational Cherednik algebras, Etingof showed that they belong to a much large family of sheaves of Cherednik algebras that can be defined for any smooth variety and finite group acting on this variety. In this generality, very little is know about the algebras. The goal of this project would be to develop tools, as exist in the theory of D-module, to study these algebras. Hopefully one can say something non-trivial about the representation theory of these algebras then.

    All projects are subject to the availability of funding. 

     

    Resolutions of symplectic singularities (PhD)

    Supervisors: Gwyn Bellamy
    Relevant research groups: Algebra, Geometry and Topology

    Symplectic singularities appear naturally in representation theory of non-commutative algebras. As such their geometry tells us a lot about the representation theory of these algebras. Through work of Namikawa we now know a great deal about symplectic singularities and their resolutions. However, there are still many open problems. One of the most basic is to count, and explicitly construct, these resolutions. The goal of this project would be to do so for some concrete, but interesting examples.

    All projects are subject to the availability of funding. 

     

    Homological properties of Deformation-Quantization algebras (PhD)

    Supervisors: Gwyn Bellamy
    Relevant research groups: Algebra, Geometry and Topology

    In the study of the representation theory of deformation-quantization algebras, several natural categories of sheaves play an important role. Most of these categories should be “smooth” in an appropriate sense. In particular, they should have finite global dimension. The goal of this project would be to understand the homological properties, in particular global dimension, of these categories. The idea here would be to generalize classical (but difficult) results in the theory of D-modules on the global dimension of sheaves of differential algebras.

    All projects are subject to the availability of funding. 

     

    Deformation-Quantization algebras on K3 surfaces (PhD)

    Supervisors: Gwyn Bellamy
    Relevant research groups: Algebra, Geometry and Topology

    Smooth K3 surfaces are a natural sources of symplectic manifolds. They have been studied by algebraic geometers for over a century now. One can also study deformation-quantization algebras on them. The goal then would be to 1 relate the properties of these non-commutative algebras, and their representation theory, to the rich underlying geometric properties of the surfaces. 

    All projects are subject to the availability of funding.