**Dr. Stefan Bechtluft-Sachs [1]**

Stefan Bechtluft-Sachs’ research deals with the relation of (algebraic) topology on one side and differential geometry and (global) analysis on the other. Specifically Stefan works on the role variational calculus, in particular natural functionals, plays in homotopy theory. Moreover he is interested in topological obstructions to certain curvature properties of manifolds.

**Dr. Caroline Brophy [2]**

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.

My interests span various areas of algebra, geometry, and analysis. In algebra, I have mainly worked in ring theory, particularly ring commutativity conditions and various aspects of combinatorial ring theory. One of my current projects involves the investigation of a notion of isoclinism in a universal algebra context, with applications to combinatorial ring theory. I am also interested in group and semigroup theory.

My geometry interests lie mostly in the area of metric geometry. I am interested in various weak notions of negative or nonpositive curvature such as Gromov hyperbolicity, CAT(0), Busemann convexity, and the Ptolemaic inequality. I am currently investigating a notion that I call *Rough CAT(0)*.

My interests in analysis are mostly in the area of geometric analysis, and specifically quasiconformal mappings, potential theory, metric measure spaces, geometric function theory, and applications of Gromov hyperbolicity in analysis. I have studied various types of Poincaré, Sobolev, and Hardy inequalities over Euclidean and non-Euclidean spaces, especially the connection between such analytic inequalities and geometry.

Detta Dickinson's research interests lie in the areas of measure theory and metric Diophantine approximation. In particular, Diophantine approximation on manifolds.

Classically, Diophantine approximation is the study of how well real numbers can be approximated by rationals. This can be extended to higher dimensions by asking how well real points in n-dimensional Euclidean space can be approximated by rational points or by rational hyperplanes. Results in this area are very delicate as shown in Khintchine's theorem, where the set of well approximable points has either zero or full measure depending on the convergence or divergence of a certain volume sum. This leads to further questions - those of Hausdorff dimension in the case of measure zero and those of asymptotic number of solutions in the case of full measure.

Both of the above questions become more difficult when the set under investigation is restricted to a manifold embedded in Euclidean space and this is Detta's current area of interest.

My research interests lie in applying Bayesian methods of statistical inference to analyze data of complex structure that arise in a variety of applications. In particular, I am interested in classification problems in data with large feature spaces. One of the most important aspects of modelling high-dimensional data is feature selection and I am interested in developing novel methods for this challenging problem.

Catherine Hurley's research interests are in statistical computing, graphics and data analysis. At present the focus of these interests is the design of software for interactive statistical graphics. This work has resulted in new software for statistical graphics, which is part of the QUAIL (for Quantitative Analysis In Lisp) system, available from the University of Waterloo Statistical Computing Laboratory.

Ciarán Mac an Bhaird's current areas of research focus on Mathematics Education and Algebraic Number Theory. In Algebraic Number Theory he is working on Gauss' Sums and Cyclotomic Numbers. He is currently working with the computer package Singular in order to investigate these topics further. In Mathematics Education he is interested in the impact that additional interventions and new teaching initiatives can have on students. He is also interested in the role of the History of Mathematics as both an aid to teaching and a method to increase student understanding.

**Dr. John Murray [9]**

John Murray works on the modular representation theory of groups. Representation theory is the study of concrete realisations of the axiomatic systems of abstract algebra. It originated in the study of permutation groups and algebras of matrices. The representation theory of finite groups was developed by G. Frobenius in the last decade of the nineteenth century. Major applications were quickly found by W. Burnside and I. Schur. R. Brauer began his investigations into the modular representations of finite groups in 1939. His work was the genesis of the programme to classify the finite simple groups, which reached fruition in the early 1980's. Other landmarks in the subject include the 1956 paper of J. Green on the general linear group, the work of P. Deligne and G. Lusztig in the 1970's on algebraic groups, and the (still open) conjectures of J. Alperin, G. Robinson and E. Dade from the late 1980's on the p-defects of characters.

Dr. Murray is fascinated by all aspects of this rapidly changing subject. He is particularly interested in the structure of the centres of modular group algebras, the block theory of finite groups, properties of involutions, and generally in the connections between ring theoretic and group theoretic invariants of algebras. He has written a number of papers on these topics. John has developed and implemented algorithms for the computer package GAP to facilitate his investigation into the structure of finite algebras. At the moment he is working on the proof of a result that concerns the involution classes of symmetric groups, using the ring of symmetric functions and the class algebra of I. MacDonald.

**Dr. Fiacre Ó Cairbre [10]**

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.

**Professor Anthony G. O'Farrell [11]**

Pairs of noncommuting involutive maps play central roles in a wide range of superficially-unconnected applications. An involution is a map which equals its own inverse. The theory of single involutions is not simple, but is well-developed. In abstract algebra, group theorists have studied involutions since the beginning, and they play a basic structural role. Groups generated by exactly two involutions are called dihedral (because they include the classical symmetry groups of the regular polygons). They are very much more special than the groups generated by three involutions. At the same time, the group generated by two involutive maps displays rather complex and varied behaviour, and there is a rich field of possibilities. Examples occur naturally in problems as diverse as classical Hamiltonian dynamical systems (such as the n-body problem), complex dynamics in one and several variables, the theory of biholomorphic classification of surfaces, the conformal mapping of quartic lemniscates, multivariate uniform approximation theory, complex polynomial approximation, one-dimensional real dynamical systems (with applications in electronics, laser physics, population biology, etc.).

It is also interesting to study reversibility in general, even when the reversing map is not involutive. Applications of such reversible systems occur in the study of the so-called reciprocal geodesics in hyperbolic geometry, and in the (related) study of the representation of integers by binary quadratic forms. They also occur in the group of biholomorphic germs, and in problems about the classification of 3-manifold foliations.

For some years now, Professor O'Farrell and others have been studying reversibility systematically, with a view to classifying the possibilities and elucidating the general structure. A good deal of work has been completed on reversibilty in various groups of maps. The difficult thing in each case is to understand the interaction between the algebraic reversibility condition and the topological (or differential, or analytic, or formal) properties of the maps. There are many groups that are, as yet, unexplored, and some of these provide reasonable, interesting and challenging projects. A monograph, written jointly with Ian Short, has been drafted, setting out the state of current knowledge about reversible discrete systems, and this should prove a useful guide to new entrants to the field.

Professor O'Farrell remains interested in other areas, notably approximation problems involving functions of one or several complex variables. More specifically, the functions of interest are analytic (or holomorphic) on some open set, and one is interested in approximating them by analytic polynomials or rational functions, or even just by functions analytic on a slightly larger open set. Such problems are connected with geometric, measure-theoretic, potential-theoretic and functional-analytic questions, and especially with capacities, plurisubharmonic functions, polynomial, rational and and complex analysis. It is sometimes advantageous to exploit additional algebraic structure, such as algebra or lattice structures, present in the situation.

**Dr. Ann O'Shea [12]**

Ann O'Shea is interested in Mathematics Education. Current research projects include: investigating concept formation; investigating the effects of beliefs and attitudes on learning; measuring the effectiveness of mathematics support. She has also worked in the area of Value Distribution Theory in Several Complex Variables.

**Dr. David Redmond [13]**

David Redmond's area of specialization is Group Theory and Permutation Groups. He has been working with Professor Quinn (Maynooth) and Dr. P.W. Fowler (Exeter) on the application of group theory in Chemistry and in particular on the recent developments in the chemistry and geometry of Fullerenes.

**Dr. Anthony Small [14]**

Anthony Small is working on problems in algebraic/differential geometry. In particular the construction and study of differential geometric objects of variational origin, via 'transforms' that convert the data into more tractable algebro-geometric objects, e.g. Minimal surfaces (soap films), Constant mean curvature surfaces (soap bubbles), and Monopoles.

**Dr. David Wraith [15]**

David Wraith's research interests encompass Differential Geometry and Algebraic Topology, and focus primarily on the topological implications of positive curvature. Most of his work to date explores the effects of surgery on Ricci positive manifolds.