Department of Mathematics
Postgraduate Research Topics in Pure Mathematics
Considering your postgraduate research options? Why not study in the field of Pure Mathematics?
Current honours topics
Nilpotent subgroups of classical groups over commutative rings  Associate Professor Jianbei An  MATHS 320 
Central series of nilpotent subgroups of classical groups over commutative rings  Associate Professor Jianbei An 
MATHS 720 
Search for patterns in the spectrum of regular maps of small genera  Professor Marston Conder 
Knowledge and understanding of a certain amount of group theory, graph theory and number theory, and experience in computing 
Construction of families of finite images of finitelypresented groups  Professor Marston Conder  Knowledge and understanding of group theory (e.g. from MATHS 320 or 720), and experience in computing 
Graphs (networks) with prescribed properties  Professor Marston Conder  Knowledge and understanding of graph theory (e.g. from MATHS 326 or 715), and experience in computing 
Algorithms for the discrete logarithm problem on elliptic curves  Associate Professor Steven Galbraith 

Efficient implementation of latticebased cryptography  Associate Professor Steven Galbraith 

Differentiability as continuity: Characterisations of differentiability of functions in terms of continuity under judiciously chosen topological structures.  Professor David Gauld 

Knot theory: Different aspects of the topic.  Professor David Gauld  
The geometry of surfaces, including special surfaces: e.g. minimal, Willmore  Professor Rod Gover  
Connections and their applications to PDE theory  Professor Rod Gover  
Lie representation theory and applications to invariant operators  Professor Rod Gover  
General topology  Dr Sina Greenwood  
Generalised inverse limits. This is a new area of research involving spaces of infinite dimension, and there are loads of open questions.  Dr Sina Greenwood  
Invariants of matrices  Dr Igor Klep 

Positive polynomials and sums of squares  Dr Igor Klep  
Complex potential theory  Dr Sione Ma'u  
Applications of topology to analysis  Associate Professor Warren Moors  
Investigating finitelypresented groups  Professor Eamonn O'Brien 

Automorphisms of free, abelian, soluble and pgroups  Professor Eamonn O'Brien  
Geometric group theory: The goal is to understand a deep theorem in this area, e.g. Gromov's polynomial growth theorem or Tits' alternative theorem. 
Dr Jeroen Schillewaert  
Tits buildings: The goal is to understand the classification of spherical buildings, which are a geometrical description of algebraic groups  Dr Jeroen Schillewaert  
An introduction to BruhatTits theory:The project could be to study the onedimensional version of the theory as described in Serre's book on Trees, which is a very rich subject in itself.  Dr Jeroen Schillewaert  
Groups acting on CAT(0) spaces: The goal is to learn the basics from the book by Bridson and Haefliger and to elaborate on a topic of your choice.  Dr Jeroen Schillewaert  
Classification of small Condorcet domains  Professor Arkadii Slinko 

Voting manipulation games  Professor Arkadii Slinko 

Linear secret sharing schemes  Professor Arkadii Slinko 

An interesting topic in (analytic) semigroup theory, elliptic operators with applications to PDE.  Professor Tom ter Elst  
Local action in arctransitive digraphs  Dr Gabriel Verret  
Eigenspaces over finite fields for arctransitive graphs  Dr Gabriel Verret 

The relationship between (convex) complex polytopes and harmonic frames, and the construction of tight frames from pseduo reflection groups  Dr Shayne Waldron  Requires basic linear algebra and a little analysis, involves hand, symbolic and numerical calculations 
The construction and properties (such as the spectral structure) of generalised Bernstein operators (based on the recently introduced canonical barycentric coordinates for an affinely independent set of points  Dr Shayne Waldron  Requires basic linear algebra and group theory, involves magma computations 
Current masters topics
Topic  Supervisor  Special prerequisites 

Nilpotent subgroups of classical groups over commutative rings  Associate Professor Jianbei An  MATHS 320 
Central series of nilpotent subgroups of classical groups over commutative rings  Associate Professor Jianbei An 
MATHS 720 
Search for patterns in the spectrum of regular maps of small genera  Professor Marston Conder  Knowledge and understanding of a certain amount of group theory, graph theory and number theory, and experience in computing 
Construction of families of finite images of finitelypresented groups  Professor Marston Conder  Knowledge and understanding of group theory (e.g. from MATHS 720), and experience in computing 
Graphs (networks) with prescribed properties  Professor Marston Conder  Knowledge and understanding of graph theory (e.g. from MATHS 715), and experience in computing 
Determining the spectrum of genera of faithful actions of given groups on Riemann surfaces  Professor Marston Conder  Knowledge and understanding of group theory (e.g. from MATHS 720), and experience in computing 
Geometry of numbers and applications  Professor Steven Galbraith 

Quadratic forms and supersingular elliptic curves  Professor Steven Galbraith 

Differentiability as continuity: Characterisations of differentiability of functions in terms of continuity under judiciously chosen topological structures.  Professor David Gauld  
Volterra spaces: These spaces arose from a study of a paper by Volterra which shows that there is no function from the reals to the reals which is continuous precisely at all rational points.  Professor David Gauld  
Knot theory: Different aspects of the topic  Professor David Gauld  
Riemannian geometry and special submanifolds, including minimal, Wilmore, and their generalisations  Professor Rod Gover 

Conformal geometry, invariants, invariant operators and applications  Professor Rod Gover  
Projective differential geometry, invariant operators and applications  Professor Rod Gover  
Generalised inverse limits. This is a new area of research involving spaces of infinite dimension, and there are loads of open questions.  Dr Sina Greenwood 

Brunnian links. Investigate questions on generalised Brunnian links. The project combines some knowledge of knots, braids and free groups.  Dr Sina Greenwood  
Invariants of matrices  Dr Igor Klep 

Positive polynomials and Hilbert's 17th Problem  Dr Igor Klep  
Quaternion algebras and their subfields  Dr Igor Klep 

Potential theory on complex manifolds and varieties  Dr Sione Ma'u  
Applications of topology to analysis  Associate Professor Warren Moors  
Algorithms to construct automorphism groups  Professor Eamonn O'Brien  
Use of CauchyFrobenius theorem to count groups  Professor Eamonn O'Brien  
Algorithms to decide various properties for linear groups  Professor Eamonn O'Brien  
Geometric group theory: The goal is to understand a deep theorem in this area, e.g. Gromov's polynomial growth theorem or Tits' alternative theorem. 
Dr Jeroen Schillewaert  
Tits buildings: The goal is to understand the classification of spherical buildings, which are a geometrical description of algebraic groups  Dr Jeroen Schillewaert  
An introduction to BruhatTits theory:The project could be to study the onedimensional version of the theory as described in Serre's book on Trees, which is a very rich subject in itself.  Dr Jeroen Schillewaert  
Groups acting on CAT(0) spaces: The goal is to learn the basics from the book by Bridson and Haefliger and to elaborate on a topic of your choice.  Dr Jeroen Schillewaert  
Classification of small Condorcet domains  Professor Arkadii Slinko 

Voting manipulation games  Professor Arkadii Slinko 

Linear secret sharing schemes  Professor Arkadii Slinko 

An interesting topic in (analytic) semigroup theory, elliptic operators with applications to PDE.  Professor Tom ter Elst  
Local action in arctransitive digraphs  Dr Gabriel Verret  
Eigenspaces over finite fields for arctransitive graphs 
Dr Gabriel Verret 

Three term recurrence relations and zeros of multivariate orthogonal polynomials (on a ball or simplex). To obtain a 'natural' threeterm recurrence for the based on symmetry conditions, expressed in terms of the zonal polynomials. Possible applications include quadrature rules and points for optimal multivariate polynomial interpolation  Dr Shayne Waldron  Basic linear algebra and analysis, and will involve multivariate hypergeometric functions, and perhaps potential theory 
Constructing complex equiangular tight frames from graphs, as has been done for real frames. To find classes of graphs which play the same role as the strongly regular graphs in the construction of tight frames.  Dr Shayne Waldron  Basic linear algebra, some graph theory, particularly about 'incidence matrices' whose entries are roots of unity 
Current PhD topics
Topic  Supervisor  Special prerequisites 

Controlled plocal ranks of some finite groups  Associate Professor Jianbei An  
Determining the symmetric genus of particular classes of groups  Professor Marston Conder  Knowledge and understanding of group theory (e.g. from MATHS 720), and experience in computing 
Investigating the prevalence of chirality among regular maps on orientable surfaces  Professor Marston Conder  Knowledge and understanding of a certain amount of group theory, graph theory and number theory, and experience in computing 
Distinguishing finitelypresented groups by their finite quotients  Professor Marston Conder  Knowledge and understanding of group theory (e.g. from MATHS 720), and experience in computing 
Deriving conditions for the extendability of group actions on nonorientable surfaces  Professor Marston Conder  Knowledge and understanding of group theory (e.g. from MATHS 720), and experience in computing 
Computational problems in lattices and applications to cryptanalysis  Professor Steven Galbraith  
Mathematical foundations of program obfuscation  Professor Steven Galbraith  
The geometry of partial differential equations  Professor Rod Gover 

The geometry associated with PDE solutions  Professor Rod Gover  
Applications of conformal and projective geometry in general relativity  Professor Rod Gover  
CR geometry, invariant operators and applications  Professor Rod Gover  
Geometric aspects of higher dimensional complex analysis  Professor Rod Gover  
Submanifolds in parabolic geometries, invariants invariant operators and applications  Professor Rod Gover  
Geometric compactifications and the linking of submanifold to ambient manifold structure  Professor Rod Gover  
PoincareEinstein and related structures  Professor Rod Gover  
Conformal and projective ideas in almost complex and Kaehler geometry  Professor Rod Gover 

Abstract dynamical systems. The investigation of various questions relating to the question: given a set X and a function from X to itself, how does the action of the function impact on the properties that any topology on X may have, and with respect to which X is continuous  Dr Sina Greenwood  
Generalised inverse limits investigating open questions on inverse limits with upper semicontinuous bonding functions  Dr Sina Greenwood  
Invariants of matrices  Dr Igor Klep  
Positive polynomials and Hilbert's 17th Problem  Dr Igor Klep  
Central simple algebras and their subfields  Dr Igor Klep  
Weighted pluripotential theory and complex analytic geometry  Dr Sione Ma'u  
Applications of topology to analysis (thesis project)  Associate Professor Warren Moors 
A solid background in general topology, real and functional analysis. 
Algorithms to decide isomorphism among groups  Professor Eamonn O'Brien  
Deciding finiteness of matrix groups  Professor Eamonn O'Brien  
Algorithms to study finite and infinite matrix groups  Professor Eamonn O'Brien  
Parameterized complexity in game theory and social choice  Professor Arkadii Slinko 

Combinatorial problems in simple games and comparative probability orders  Professor Arkadii Slinko 

Multiwinner voting rules and their axiomatic characterisation  Professor Arkadii Slinko 

Simple games and secret sharing schemes  Professor Arkadii Slinko 

Condorcet domains associated with finite groups and median graphs  Professor Arkadii Slinko  
Axiomatic and algorithmic aspects of multiwinner voting rules  Professor Arkadii Slinko  
An interesting topic in (analytic) semigroup theory, elliptic operators with applications to PDE  Professor Tom ter Elst 

Local action in arctransitive digraphs  Dr Gabriel Verret  
Eigenspaces over finite fields for arctransitive graphs  Dr Gabriel Verret 

An analytic construction of Heisenberg frames (equivalently, a solution Problem 23 in quantum information theory SIC POVM's and Zauner's conjecture, or the construction of d2 equiangular lines in Cd)  Dr Shayne Waldron 
Requires some geometry, basic multilinear algebra and the representation theory of finite groups. Some background in Physics and projective representations/geometry may help. 
Classification of harmonic frames. To give a simple proof of the asymptotic estimate of the number of harmonic frames. This is closely tied to the 'zero sums problem' for roots of unity  Dr Shayne Waldron  Requires deep knowledge of finite abelian groups and their characters, and some number theory and asymtotic analysis. 