Mathematical sciences

Research Interests and Activity

John Appleby

Deterministic and stochastic functional differential equations, Stochastic Analysis, Mathematical Finance.

John Carroll

Mathematics, Numerical Analysis, Ordinary differential equations, Partial differential equations, Finite difference methods, Adaptive schemes

Turlough Downes

Astrophysics : Star formation, High energy astrophysics, Computational fluid dynamics

Olaf Menkens

Mathematical finance, more specific: - Crash hedging strategies and optimal portfolios under the threat of a crash - Value at Risk and self-similarity - Insider trading - Liquidity risk

Angela Murphy

Public-Key cryptography/ Elliptic Curve Cryptography

Mathematics education research: Mathematics support; Mathematics issues at the transition from second-level to third-level; Mathematical diagnostic testing of incoming university students; Mathematics for engineers; Assessment in mathematics

Brien Nolan

General relativity and gravitation; mathematics education.

Eugene O'Riordan

Singularly perturbed differential equations. Numerical analysis. Shishkin meshes.

Niamh O'Sullivan

Localizations of soluble groups. Finiteness conditions in soluble groups.

Jason Quinn

I work on numerical methods for nonlinear singularly perturbed differential equations. A short explanation: singularly perturbed D.E's are equations in which the highest derivative is multiplied by an arbitrary small quantity (perturbation parameter). Singularly perturbed problems arise in various disciplines such as option pricing in financial mathematics, Computational fluid dynamics and Semiconductor design for example. Solutions typically exhibit steep gradients, inversely proportional to the perturbation parameter. The steep gradient may appear on the boundary or in the interior of the problem domain (boundary\interior layers). Software packages have `difficulties' with numerically solving these problems. They regularly return oscillatory solutions, exacerbated as the perturbation parameter vanishes. Thus we must construct numerical methods which converge independently of the perturbation parameter (parameter uniform methods). I am currently working on nonlinear singularly perturbed interior layer problems. For problems where the layer location is known e.g. at the boundary or in the interior (in the case of linear problems say), a usual approach is to apply a fine Shiskin mesh on the layer region in a numerical algorithm. However, in general, the location of the interior layer for nonlinear problems is unknown. Although approximations of the location may exist, these approximations are not immediately justifiable in a numerical algorithm. My current interest is in investigating the requirements for such approximations to be successfully used in proposed numerical algorithms.