UK and Republic of Ireland SIAM Section Meets in Cork

April 1, 2005

At University College Cork for the 2005 annual meeting of the UK and Republic of Ireland SIAM Section are (front row, from left) Gabriel Lord (UKIE Section secretary/treasurer), invited speaker Neil O’Connell, and Martin Stynes (UKIE Section president); (back row, from left) invited speaker Ivan Graham, Jim Grannell (chair of the School of Mathematics, Applied Mathematics and Statistics, UCC), invited speaker Russell Davies, Joyce Aitchison (UKIE Section vice-president), invited speaker Alexii Pokrovskii, Maurice O’Reilly (president of the Irish Mathematical Society), and invited speaker Niall Ó Murchadha.

By James Gleeson and Gabriel Lord

The UK and Republic of Ireland SIAM Section met this year for the first time in the Republic of Ireland. The annual meeting, held January 7, 2005, at University College Cork, coincided with the opening of Cork as “European City of Culture.” Despite the rain and wind, the turnout for the meeting was excellent.

Neil O’Connell (Warwick) opened the meeting with an informative survey talk on random matrices. Starting with some applications of random matrices, including telephone encryption, he examined the distribution of eigenvalues of random (orthogonal and unitary) matrices. By analogy with electrostatics, he explained why the eigenvalues are spaced more regularly than might be expected. An open puzzle is a phase transition in the distribution of eigenvalues of powers of unitary matrices. Finally, O’Con-nell explained how eigenvalues of random unitary matrices are closely related, in a statistical sense, to the zeros of the Riemann zeta function.

Niall Ó Murchadha (UCC) gave an entertaining talk on the degrees of freedom of the gravitational field as specified by the equations of general relativity—an appropriate topic in the centenary year of three of Einstein’s seminal publications. He made extensive use of the analogy with the Maxwell equations of the electromagnetic field to show the necessity of splitting the equations into constraint and evolution parts. The talk placed the research in historical context, and showed how it underlies current efforts in numerical relativity. A question regarding the well-posedness of the evolution equations gave the speaker further opportunity to demonstrate his enthusiasm for this fascinating subject, and to highlight the difficulty of outstanding research challenges.

In a talk on efficient boundary element methods for PDEs, Ivan Graham (Bath) considered typical applications, such as computing electrostatic capacitance and acoustic obstacle scattering. This clear and interesting talk contrasted boundary element methods with standard finite element/finite difference methods and surveyed recent results on complexity issues. Mathematical challenges presented included those arising from the treatment of irregular boundaries. Graham showed how, with an appropriate fast matrix vector multiplication method, the complexity of boundary element methods is competitive with that of finite element methods. He also summarised recent research extending the methods and analysis to degenerate meshes.

Russell Davies (Aberystwyth), in an interesting talk on wobble, creep, and relaxation, described his work in modelling materials with memory. As he enthusiastically explained, what constitutes a fluid depends on the time scale of the observations, which means that sagging tombstones and flowing mountains can be included. He considered the corkscrew instability that arises in the extrusion of plastics or similar viscoelastic flows. He also asked why a jelly wobbles, giving the answer, in terms of mathematics, as a Sturm–Liouville problem. An open question, of relevance to the industrial processing of all polymeric materials, is to determine the dominant rate of decay of the transient.

The final speaker, Alexei Pokrovskii (UCC), introduced equations with hysteresis nonlinearities. Using computer applets, he illustrated the important rate-independent property of hysteresis, which cannot be easily described in terms of standard mathematical formulas. He outlined the Krasnoselskii program for investigating hysteresis, emphasising the Identification Theorem, which links many types of hysteresis behaviour to the Preisach operator. The lecture concluded with an overview of some recent results on the complicated behaviour of a pendulum subject to hysteresis, including a rigorous proof of the existence of chaos.

James Gleeson is at University College Cork and Gabriel Lord is at Heriot–Watt University.

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