This page lists potential PhD projects in algebra and combinatorics, offered by faculty members of the School of Mathematics. For further details of the School's research in this area, please see the Pure Mathematics Research Group page. For further information about an individual project, please contact the listed supervisor. For information about submitting an application, please see our Research Degrees page.

Homological Methods in Representation Theory Homological Methods in Representation Theory

Supervisor: Dr Vanessa Miemietz

Two central questions of representation theory are the construction of irreducible modules and the construction of larger representations from smaller ones. This project will focus on the latter, which is also called homological algebra. Central questions are: Given an algebra, what extensions do the simple modules have? What is the structure (i.e. composition series) of projective modules? What can we say about the Auslander-Reiten quiver? Its derived category? The PhD project will attack some of these questions, depending on the interests of the candidate for certain algebras appearing in the theory of algebraic groups or categorification, or for abstract  algebras with a given set of homological properties (quasi-heredity, symmetry, cellularity, etc.)

References:

  • M. Auslander, I. Reiten, S. Smalø (1995). Representation Theory of Artin Algebras. Cambridge Studies in Advanced Mathematics, 36. Cambridge University Press.
  • I. Assem, D. Simson, A. Skowroński (2006). Elements of the Representation Theory of Associative Algebras. Vol. 1. Techniques of representation theory. London Mathematical Society Student Texts, 65. Cambridge University Press.

For further information, please contact Dr Vanessa Miemietz.

Topics in Algebraic Combinatorics Topics in Algebraic Combinatorics

Supervisor: Dr Johannes Siemons

Finite combinatorial structures appear in many branches of mathematics, including in graph theory, design theory, coding theory, partially ordered sets and finite geometry. Each gives rise to a finite symmetry group that preserves the structure, and conversely every finite permutation group leads to combinatorial objects that are preserved by it.  In this fashion algebraic techniques are used to analyze combinatorial problems and, conversely, combinatorics becomes a tool in algebra. Today algebraic combinatorics is a fast growing and attractive field of research.

Topics currently on offer include `Reconstruction and permutation groups', `Smith normal forms in combinatorics', `The null space of a graph'  and `Groups defined by invariant relations'. I am happy to discuss other projects in this general area.

References:

  • Jack van Lint and Richard Wilson, A Course in Combinatorics, Cambridge University Press.
  • Richard Stanley, Enumerative Combinatorics, Vols 1 & 2, Cambridge University Press.
  • Bruce Sagan, The Symmetric Group: Representations, Combinatorial Algorithms and Symmetric Functions, Springer GTM 203

For further information please contact Dr Johannes Siemons.

Algorithmic Problems in Algebra Algorithmic Problems in Algebra

Supervisor: Dr Robert Gray.

Algorithmic problems in algebra have their origins in work of Thue, Tietze, and Dehn carried out in the beginning of the 20th century. Their work showed how certain problems in logic and topology turned out to be equivalent to corresponding algebraic problems, namely the word problem for finitely presented semigroups and groups, and the isomorphism and conjugacy problems for finitely presented groups. Even though originally motivated by problems in logic and topology, the investigation of algorithmic problems in algebra is now primarily motivated by the internal needs of algebra itself.  Algorithmic problems often lie at the heart of difficult and important algebraic problems. Most problems are undecidable in general, and so it becomes important to identify and study classes with good algorithmic properties. This point of view has led to a lot of interesting research on topics including hyperbolic groups, word hyperbolic semigroups, automatic groups and semigroups, one-relator groups, finite complete string rewriting systems, and the study of small overlap conditions. For those problems that are decidable there are also interesting questions about how hard these decision problems are, linking the subject with complexity theory.

This PhD project will investigate a range of algorithmic and decision problems in algebra, with a focus on finitely presented semigroups and groups.

References:

  • R. V. Book and F. Otto. String-Rewriting Systems. Texts and Monographs in Computer Science. Springer-Verlag, New York, 1993.
  • M. R. Bridson and A. Haefliger. Metric spaces of non-positive curvature, volume 319 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1999.
  • D. B. A. Epstein, J. W. Cannon, D. F. Holt, S. V. F. Levy, M. S. Paterson, W. P. Thurston, Word processing in groups, Jones and Bartlett Publishers, Boston, MA, 1992.
  • R. C. Lyndon and P. E. Schupp. Combinatorial group theory. Classics in Mathematics. Springer-Verlag, Berlin, 2001.
  • J. Rhodes and B. Steinberg. The q-theory of finite semigroups. Springer Monographs in Mathematics. Springer, New York, 2009.

For further information, please contact Dr Robert Gray.

Representations and Homology Representations and Homology

Supervisor: Dr Joseph Grant.

Rings, or algebras, are often rings of functions, so it makes sense to study them by investigating how they act on other objects.  This is analogous to studying groups by considering group actions.  We often consider algebras acting on vector spaces, as we understand linear algebra relatively well, and it can be easier to work with matrices than elements of algebras.  A vector space with an action of an algebra is called a module, or a representation, and representation theory is the study of these modules.

An arbitrary algebra may have a module which has a submodule but no compliment, i.e., it is not simple but cannot be written as the direct sum of two submodules.  We therefore need to consider quotients.  A module with a submodule L whose quotient by L is isomorphic to M is known as an extension of M by L.  The study of extensions leads us to quivers, which are directed graphs associated to algebra that capture information about which extensions are possible.  In fact, over the complex numbers, the representation theory of every algebra is determined by a unique quiver and a set of relations.  A deep study of extensions leads us to homological algebra, which is closely related to the algebraic parts of topology.  There are connections to many other areas, including Lie theory, (higher) category theory, and knot theory.

A PhD project under my supervision is likely to involve homological algebra, either studying the behaviour of particular examples of algebras, or investigating more general theory for wider classes of algebras.  The focus of the project would depend on the student's interests and strengths.  This is a very active research area with many open and exciting questions to study.

References:

  •  Ralf Schiffler, "Quiver Representations", Springer, 2014
  •  D. J. Benson, "Representations and cohomology, Volume I", Cambridge University Press, 1991

For further information, please contact Dr Joseph Grant.