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 finitely-presented 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 lattice-based 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 finitely-presented groups Professor Eamonn O'Brien
 
Automorphisms of free, abelian, soluble and p-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 Bruhat-Tits theory:The project could be to study the one-dimensional 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 arc-transitive digraphs Dr Gabriel Verret  
Eigenspaces over finite fields for arc-transitive 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
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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 finitely-presented 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 Cauchy-Frobenius 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 Bruhat-Tits theory:The project could be to study the one-dimensional 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 arc-transitive digraphs Dr Gabriel Verret  
Eigenspaces over finite fields for arc-transitive graphs
Dr Gabriel Verret
 
Three term recurrence relations and zeros of multivariate orthogonal polynomials (on a ball or simplex). To obtain a 'natural' three-term 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
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Current PhD topics


Topic Supervisor Special prerequisites
Controlled p-local 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 finitely-presented 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 non-orientable 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  
Poincare-Einstein 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 semi-continuous 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 multi-winner 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 arc-transitive digraphs Dr Gabriel Verret  
Eigenspaces over finite fields for arc-transitive 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.
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