PROPOSED RESEARCH PROJECTS
The following projects or project areas are proposed for the consideration of prospective students. If any of these appeals to you, contact Dr. Stefan Bechtluft-Sachs, Postgraduate Admissions Officer. You can also read about current research projects.
- Algebra and Number Theory
- Character Theory of Finite Groups - Dr. John Murray
- Combinatorics and Number Theory - Dr. Pat McCarthy
- Group Theory - Dr. David Redmond
- Analysis
- Classical Analysis - Dr. Pat McCarthy
- Diophantine Approximation and Measure Theory - Dr. Detta Dickinson
- Real and Complex Analysis - Professor Anthony G. O'Farrell
- Geometry and Algebraic Topology
- Differential Geometry - Dr. Stefan Bechtluft-Sachs
- Geometric Analysis - Professor Stephen Buckley
- Algebraic/Differential Geometry - Dr. Anthony Small
- Differential Geometry - Dr. Constantin Vernicos
- Differential Geometry and Algebraic Topology - Dr. David Wraith
- Mathematics Education
- Mathematics Education - Dr. Ciaran Mac an Bhaird & Dr. Ann O'Shea
- Mathematics Education - Dr. Fiacre O'Cairbre
- Statistics
- Statistical Modelling in Environmental and Ecological Science - Dr. Caroline Brophy
- Bayesian methods of Statistical Inference - Dr. Katarina Domijan
- Statistical Computing and Graphics - Dr. Catherine Hurley
Differential Geometry/Algebraic Topology/Variational Methods (MSc/MLitt/PhD)
Potential topics:
- Invariants for Harmonic Maps into Homogeneous Spaces
- Numerical Analysis: Small Dirac-eigenvalues of homogeneous spaces
- Calculus of Variations: Tension field and Index of Energies with polynomial density
- Lie group actions and Curvature: G-manifolds with few orbit types
Statistical Modelling in Environmental and Ecological Science. (MSc/PhD)
My research interests are in the development and application of statistical modelling techniques to non-standard situations in Ecology and Environmental Science. The Statistical topics I am particularly interested in are mixture models, functional relationship models, multinomial models, mixed models, methods for modelling data with large numbers of missing or zero values, methods for predicting the mean response without bias from non-linear models and bootstrapping methods for assessing predictions from non-linear models. The Ecological and Environmental topics I am currently working on are climate change, biodiversity in grassland systems, competition in a range of ecological systems, and genotypic variability in allergenic plant species.
Geometric Analysis (MSc/MLitt/PhD)
Potential topics include:
- Poincaré-type inequalities.
- Quasiconformal mappings and mappings of finite distortion.
- Gromov hyperbolicity.
Diophantine Approximation and Measure Theory (MSc/MLitt/PhD)
- Simultaneous Diophantine Approximation.
- Diophantine Approximation on Manifolds.
- Hausdorff Measure and Dimension.
Bayesian methods of statistical inference: (MSc/MLitt/PhD)
- Dimension reduction methods for classification of micro-array data
- Semi-supervised kernel classification of high-dimensional data
- Fast approximations to posterior distributions that arise in the implementation of Bayesian kernel classifers
- Parameter estimation of large crop models.
Statistical computing and graphics: research projects. (MSc/PhD)
Potential topics include:
- Applications of Grand Tour Methods.
- Regression Analysis: a graphical user interface.
- Interactive display methods for large datasets.
Classical Analysis/Number Theory/Combinatorics(MSc/PhD)
Pat McCarthy is interested in Classical Function Spaces and the inequalities which arise in their study. Examples include HP, LP and Lipschitz spaces.
He has worked on convergence problems for Fourier series and extremal properties of certain orthogonal polynomials. Currently he is examining generalisations of Carleson Interpolation Sequences.
Another interest is Number Theory, cryptography, and the implementation of cryptographic routines and cipher attacks on microprocessors.
Character Theory of Finite Groups (MSc/MLitt/PhD)
Potential topics include:
- Recent conjectures concerning the ordinary characters of finite groups.
- Modular representations of sporadic simple groups.
- Structure of blocks with dihedral or quasidihedral defect group.
- Open problems in Jenning's theory.
- Noncommutative symmetric functions.
- Specht modules for the prime 2.
- Gelfand-Zetlin algebra for Coxeter groups.
- Integral representation theory:lattices and orders.
- Experimentation with GAP: an interactive computer programme for groups and algebras.
Stability Theory / Mathematics Education / History of Mathematics (MSc/MLitt/PhD)
Fiacre Ó Cairbre's research interests are currently in the three areas of stability theory, history of mathematics and mathematics education. He is working on the stability of certain types of switching systems. He is also working on the history of mathematics and resource materials for second level mathematics teachers.
Real and Complex Analysis (MSc/MLitt/PhD)
Potential topics include:
Problems of approximation, capacities, function spaces, geometry, and algebras of functions.
Mathematics Education (MSc/MLitt/PhD)
Dr. Ann O'Shea or Dr. Ciaran Mac an Bhaird
We offer a two year Masters in Mathematics Education by Research. The principle investigations would involve studying the role of the extra Maths support on students' attitudes towards and performances in Mathematics. There are many different avenues of investigation that can be followed including looking at the roles of different teaching techniques, studying the groups of students who actually avail of extra support and considering the students who don't avail of the support and why they don't. If anyone is interested in these topics or other topics in Mathematics Education please contact Ciarán Mac an Bhaird or Ann O'Shea.
Group Theory (MSc/MLitt)
Potential topics include:
- Study of Group theory, Ring theory, Galois theory, ordinary character theory and, in particular, Modular character theory to be able to understand and write an intelligent and intelligible report on some major results in the area of character theory, such as the Brauer-Suzuki theorem on Quaternion 2-sylow subgroups and/or the Glauberman 2* - theorem.
- Study of group theory, ordinary character theory and other algebraic techniques so as to understand and write an intelligent and intelligible report on both the classification of the Frobenius groups and the classification of the Zassenhaus groups.
Algebraic/Differential Geometry (MSc/MLitt/PhD)
Potential topics include:
- MSc/MA problems relating Constant Mean Curvature Surfaces
- MSc/MA problem in Gauge Theory/Complex Geometry
- PhD problems from the differential geometry of surfaces, twistor theory and Yang-Mills.
Differential Geometry (MSc/MLitt/PhD)
Potential topics include:
- Metric Geometry (Finsler, Riemannian, sub-Riemannian)
- Transport Measure Theory and Curvature
- Analysis and Probability on Manifolds
Differential Geometry and Algebraic Topology (MSc/MLitt)
A project would consist of reading a recent landmark paper as well as covering all the relevant background.
Potential topics include:
- The Soul Theorem
(The basic structure theorem for non-compact manifolds of non-negative sectional curvature) - The Splitting Theorem
(A basic structure theorem for non-compact manifolds of non-negative Ricci curvature) - Exotic Structures on 7-Spheres
(An investigation of smooth manifolds which are homeomorphic but not diffeomorphic) - Curvature & Group Growth
(An investigation of how curvature influences the fundamental group of a manifold) - Whitehead Torsion
(An introduction to Algebraic K-Theory and its influence in Topology)