### Irish Mathematical Society

- Cumann Matamaitice na hÃ‰irinn

# DCU mathematical sciences

## Irish Mathematical Society/Cumann Matamaitice na hEireann

### September Meeting 2005 - Dublin City University

The September meeting of the Irish Mathematical Society was held this year at Dublin City University on 1st-2nd September 2005. The lectures were in room XG19 on the ground floor of the building on the DCU campus marked Chemical Sciences, Biotechnology, Mathematical Sciences.

### Confirmed Speakers

SPEAKER | AFFILIATION | TITLE OF LECTURE |
---|---|---|

Dr. J. A. D. Appleby | DCU | Harmless and dangerous stochastic perturbations |

Prof. J. Berndt | UC Cork | Transformation groups of cohomogeneity one |

Prof. E. Buffet | DCU | Parrondo's Paradox: how many sow's ears does it take to make a silk purse? |

Prof. T. Burton | Northwest Research Institute, USA | Introduction to stability by fixed point theory |

Dr. G. Crawley | Frontiers Engineering & Scientific Research Directorate, SFI | The SFI Mathematics Initiative |

Dr. J. Cruickshank | NUI, Galway | The rearrangement inequality |

Prof. Bernard Hanzon | UC Cork | Filtering and estimation in stochastic volatility models with rationally distributed disturbances |

Dr. N. Kopteva | UL | TBA |

Prof. D. Lewis | UC Dublin | Quaternion algebras |

Dr. D. Saunders | Open University/EBS Trust | MathTutor - a teaching resource for the school/university transition |

### Registration

The conference was free to all IMS members. It was possible to join the IMS on the day and new members are always welcome.

For further information, please contact Dr. Niamh O'Sullivan.

### Dinner

A dinner was held on Thursday, 1st September in the 1838 Club on campus.

### Campus Accommodation

Accommodation on DCU campus in student residences was available.

## IMS September 2005 Meeting: Abstracts for Lectures

### Jurgen Berndt: Transformation groups of cohomogeneity one

Transformation groups appear in the context of understanding symmetries of mathematical and physical structures. The cohomogeneity of a transformation group, or of its action, is the dimension of its orbit space. In recent years methods based on cohomogeneity one actions have been successfully used for the construction and investigation of special structures on manifolds. These developments have been of particular interest for geometry and theoretical physics. The classification of cohomogeneity one actions is a problem that exhibits interesting algebraic and geometric phenomena. I plan to give a survey about this topic.

### Ted Burton: Introduction to stability by fixed point theory

For more than 100 years Liapunov's direct method has been the main tool for studying stability of differential equations. It is a complete theory in that it includes converse theorems. Yet, numerous difficulties persist and it seems that other avenues need to be explored. A few years ago several investigators began a study of stability based on fixed point theory. In no manner does this theory replace Liapunov's direct method, but it does offer solutions to some of the aforementioned difficulties. Moreover, the conditions derived for stability using fixed point theory are frequently more consistent with real-world problems than conditions derived in the direct method. Recently, John Appleby has shown that the fixed point methods also show stability under stochastic perturbations which frequently occur in applied problems. Many of the results can be obtained quickly and cleanly with nothing more than a variation of parameters formula and a contraction mapping theorem. Thus, the method is well-suited to students and investigators who are not specialists in differential equations.

### Bernard Hanzon: Filtering and estimation in stochastic volatility models with rationally distributed disturbances

This presentation is based on joint work with Dr Wolfgang Scherrer (Technical University Vienna). In the area of mathematical finance, the Black-Scholes model is often used for modeling the behaviour of the price of stocks, exchange rates etc. This is also the basis for much of the literature on pricing of derivative financial instruments such as options. However it is considered to be a well-known fact that, although the volatility is assumed to be constant in the Black-Scholes model, in practice it is varying. This has led to various types of models in which the volatility is allowed to vary. One is the type of model in which the volatility is varying over time and its dynamic behaviour is described by some stochastic process, the so-called stochastic volatility models. A problem with such models is that it is generally difficult to solve the volatility estimation problem: calculation of the conditional probability density function of the volatility at some point in time, given the observations up till that same point in time. Here we deal with this problem, the so-called filtering problem, for a class of discrete-time stochastic volatility models in which the disturbances have rational probability density functions. Using state-space realizations to represent the rational probability density functions we are able to solve the filtering problem exactly. However the size of the matrices involved tends to grow very quickly with each time step. Therefore we use a reduction technique called stochastically balanced truncation (SBT) to approximate the high degree rational functions involved by lower degree rational functions. An error bound is available for the reduction step which is very well-suited for application in this setting.

### David Lewis: Quaternion algebras

In the first part of the talk we will describe the basic definitions and fundamental properties of quaternion algebras over fields. In the second part we attempt to show that Hamilton's 1843 discovery of the quarternions was a major turning point in the subject of algebra. Non-commutative algebra started here! We will emphasise especially the theory of division algebras and other kinds of algebra which emanated from this discovery.

### David Saunders: MathTutor — a teaching resource for the school/university transition

MathTutor — a teaching resource for the school/university transition “Making Mathematics Count”, the report of the Smith enquiry into post-14 mathematics education in the UK, was published in February 2004. In his introduction, Adrian Smith says: “The Inquiry has therefore found it deeply disturbing that so many important stakeholders believe there to be a crisis in the teaching and learning of mathematics in England.” To help address part of this problem, HEFCE and the Gatsby Charitable Foundation have funded a project to provide learning materials for the school/university transition level. “MathTutor” is a set of seven DVDs containing 40 hours of video tutorials, text files, diagnostics and over 1200 interactive exercises; “MathCentre” is a website providing online access to the MathTutor files together with additional support materials. This talk provides some background to these projects and includes demonstrations of the material.

Ollscoil Chathair Bhaile Átha Cliath 9, Éire / Dublin City University, Dublin 9, Ireland.

Tel. +353 (0) 1 700 5000, Fax. +353 (0) 1 836 0830.