The following projects or project areas are proposed for the consideration of prospective postgraduate research students.

If any of these appeal to you, please contact Dr. David Redmond, Postgraduate Coordinator.

### Algebra and Number Theory

**Professor Stephen Buckley - Ring Theory **

Potential topics include:

- Combinatorial ring theory.
- Notions of isoclinism and isologism for rings.
- More details.

**Dr. Ollie Mason - Algebra (Applied Linear Algebra) **

Potential topics include:

- The matrix Riccati equation and applications to the stability of time-delay systems
- Matrix theory over the max algebra and general semirings
- Combinatorial and nonnegative matrix theory: applications to data privacy

**Dr. Pat McCarthy - Combinatorics and Number Theory **

One of Dr. McCarthy's interests is Number Theory, cryptography, and the implementation of cryptographic routines and cipher attacks on microprocessors.

**Dr. John Murray - Character Theory of Finite Groups**

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.

**Professor Anthony O'Farrell - Real and Complex Analysis **

Potential topics include:

Problems of approximation, capacities, function spaces, geometry, groups, algebras of functions, Functional Analysis, Dynamical Systems, Potential Theory and Partial Differential Equations.

**Dr. David Redmond - Group Theory **

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.

### Analysis

**Dr. Detta Dickinson - Diophantine Approximation and Measure Theory **

Potential topics include:

- Simultaneous Diophantine Approximation.
- Diophantine Approximation on Manifolds.
- Hausdorff Measure and Dimension.

**Dr. Ollie Mason - Analysis (Stability Theory) **

Potential topics include:

- Joint spectral radius for sets of positive operators
- Nonautonomous dynamical systems arising in population dynamics
- Stability theory of differential inclusions and functional differential equations

**Dr. Pat McCarthy - Classical Analysis **

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.

**Professor Anthony O'Farrell - Real and Complex Analysis **

Potential topics include:

Problems of approximation, capacities, function spaces, geometry, groups, algebras of functions, Functional Analysis, Dynamical Systems, Potential Theory and Partial Differential Equations.

### Geometry and Algebraic Topology

**Dr Stefan Bechtluft-Sachs - Differential Geometry/Algebraic Topology/Variational Methods **

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
- More details

Potential topics in geometric analysis include:

- Poincaré-type inequalities.
- Curvature and convexity on metric spaces.
- Quasiconformal mappings and mappings of finite distortion.
- More details.

**Professor Anthony O'Farrell - Real and Complex Analysis **

Potential topics include:

Problems of approximation, capacities, function spaces, geometry, groups, algebras of functions, Functional Analysis, Dynamical Systems, Potential Theory and Partial Differential Equations.

**Dr Anthony Small - Algebraic/Differential Geometry **

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.

**Professor David Wraith - Differential Geometry **

Projects related to the geometry and topology of Riemannian manifolds with curvature bounds.

### Mathematics Education

**Dr. Fiacre Ó Cairbre - Maths Education / History of Mathematics **

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.

**Dr. Ann O'Shea - Maths Education **

Possible topics include:

- Task Design and Implementation
- Advanced Mathematical Thinking

**Dr. Ciarán Mac an Bhaird - Maths Education / History of Mathematics**

Potential topics include:

- Student Engagement and Motivation
- Evaluation of Teaching Initiatives
- Various topics from the History of Mathematics, including mathematical biography, a history of mathematical teaching and history resources for teaching

### Statistics

**Dr. Caroline Brophy - Statistical Modelling in Environmental and Ecological Science**

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.

**Dr. Katarina Domijan - Bayesian methods of Statistical Inference **

Potential topics include:

- 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.

**Dr. Catherine Hurley - Statistical Computing and Graphics **

Potential topics include:

- Applications of Grand Tour Methods.
- Regression Analysis: a graphical user interface.
- Interactive display methods for large datasets.