Mathematics

 

Mathematics Research

Research interests within the Mathematics group cover a wide spectrum including non linear phenomena, applied analysis and differential equations, geometric integration, integrable systems, Lie groups and moving frames, mathematical physics and quantum groups, representation theory and invariant theory, computational algebra, discrete mathematics and functional analysis.

Publications by members of the Mathematics group.

We can provide PhD supervision in a wide range of areas, including the following topics and projects:

A∞-algebras PDF, 160 KB

Supervisor: Dr Constanze Roitzheim

Classical and quantum behaviour of Skyrmions PDF, 306 KB

Supervisor: Dr Steffen Krusch

Cluster algebras with periodicity and discrete dynamics over finite fields

Supervisor: Prof Andy Hone
This PhD project may involve techniques from algebra, mathematical physics, and/or number theory, linking to the 5 year EPSRC fellowship of the same title. The PhD student would most likely focus on one particular problem area, involving algebraic combinatorics (cluster algebras), mathematical physics (classical/quantum dynamics), or number theory (rational maps over finite fields). Candidates would need to have a strong background and interest in at least one of these areas.

The complete intersection property in Modular Invariant theory

Supervisor: Dr Jim Shank
Invariant Theory is the study of polynomial functions with specified symmetry. The subject involves aspects of commutative algebra, representation theory, and algebraic geometry. The primary object of study is the ring of invariants of a representation. The representation is said to be modular if the characteristic of the field divides the order of the group. The ring of invariants for a modular representation often fails to be Cohen-Macaulay. However, when the ring of invariants is Cohen-Macaulay, it is often also a Complete Intersection, i.e. has a presentation with n+k generators and k relations, where n is the dimension of the representation. The goal of this project is to identify those modular representations of p-groups whose ring of invariants is a Complete Intersection.

Discrete variational methods

Supervisor: Prof Elizabeth Mansfield
The project concerns embedding physically important conservation laws and geometrical properties into approximation schemes for variational problems. The mathematical methods employed include the Calculus of Variations, Lie groups and their actions, and approximation theory. The project will have both a theoretical and a computational component.

Fast numerical methods for PDE-constrained optimization problems arising from scientific processes

Supervisor: Dr John Pearson
A vast number of important and challenging applications in mathematics and engineering are governed by optimization problems. One crucial class of these problems, with significant applicability to real-world processes, is that of PDE-constrained optimization. Generating accurate numerical solutions on the discrete level involves complex matrix systems and tackling practical problems involves devising strategies for storing and working with systems of huge dimensions. These result from fine discretizations of the PDEs in space and time variables.

Kronecker and plethysm coefficients

Supervisor:Dr Chris Bowman
"Perhaps the most challenging, deep and mysterious objects in algebraic combinatorics", the Kronecker and plethysm coefficients describe the decomposition of important representations of symmetric groups into their simple constituents. Despite 80 years of study “frustratingly little is known” about these coefficients. These coefficients also form the centrepiece of a new approach that seeks to settle the celebrated P versus NP problem. This project makes use of combinatorial and representation theoretic methods to calculate families of Kronecker and plethysm coefficients. Certain diagrammatic algebras provide the most important tools for studying these coefficients. Another strand of this project is devoted to developing the representation theory of these algebras for their own sake. (Quotes from Pak--Panova 2014, Burgisser 2009.)

Mirror symmetry and homogeneous spaces PDF, 31 KB

Supervisor: Dr Clelia Pech

Model categories and rigidity PDF, 160 KB

Supervisor: Dr Constanze Roitzheim
Investigating "rigidity", algebraic models, exotic models, uniqueness of underlying model structures and their relations.

Modular Invariants and Galois Algebras PDF, 112 KB

Supervisor: Prof Peter Fleischmann
The theory of groups and their invariants is the mathematical language for analyzing symmetries. The project investigates open questions about the structure of modular rings of invariants of finite groups. This is a particularly active field of current research, where one uses methods from algebra, arithmetic and representation theory.
Keywords include:

  • Polynomial property and depth of invariant rings;
  • localisations and their properties (Galois property, regularity);
  • local cohomology of modular invariant rings and their modules.

Non-standard orthogonal polynomials: computation and applications PDF, 82 KB

Supervisor: Dr Alfredo Deaño

On multiple orthogonal polynomials PDF, 48 KB

Supervisor: Dr Ana Loureiro

Noncommutative algebra, Poisson geometry and combinatorics  PDF, 30 KB

Supervisor:Prof Stéphane Launois

Polynomial modular rings of invariants of finite ρ-groups PDF, 110 KB

Supervisor: Prof Peter Fleischmann

Positivity in dynamics and analysis PDF, 52 KB

Supervisor: Dr Bas Lemmens

Representation theory of symmetric groups, wreath products and related algebras PDF, 285 KB

Supervisor: Dr Rowena Paget

Spectral Theory of Non-selfadjoint Operators PDF, 79 KB

Supervisor: Dr Ian Wood

Symmetry Integrability of discrete systems

Supervisor: Prof Jing Ping Wang
This project is devoted to the study of non-linear partial difference equations possessing continuous symmetries. Our main focus is on integrable difference equations, i.e. equations with an infinite hierarchy of symmetries. We plan to study their exact solutions and rich algebraic and geometric properties. We would also like to give a classification of integrable systems for a given family of discrete equations.

Topological solitons and their moduli spaces PDF, 62 KB

Supervisor: Dr Steffen Krusch

 

 

 

The Mathematics group has the long term aim of being recognised internationally as a centre of excellence in our field of research.

In the 2014 Research Excellence Framework (REF) 72% of the School's research output was assessed as International Quality or above. The School is in the top 25 in the UK when considered on weighted GPA.

Our active seminar programme involves a wide range of speakers.

Mathematical analysis of MRI scan of a human brain

Image, courtesy of Iulia Cimpan, shows an analysis of an MRI scan of a human brain. Iulia's research is into segmentation of MRI scans in dementia using a discrete differential geometry approach.

 

School of Mathematics, Statistics and Actuarial Science (SMSAS), Sibson Building, Parkwood Road, Canterbury, CT2 7FS

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Last Updated: 25/05/2017