Date  Speaker  Title  Contact 
Jan 14 
Edwin O'Shea Galway 
Algebraic methods in discrete optimizationWe will discuss recent developments in the use of Groebner
bases of toric varieties and the Rees algebras of monomial ideals in
discrete optimisation. We will place particular emphasis on how each
method provides new results and new ways of thinking about central
problems in discrete optimisation. We will close with an application
to statistical disclosure limitation. Rather than describing the algebraic
content head on we will instead provide geometric and combinatorial
descriptions of these techniques. No former background in computational
algebra or discrete optimisation will be assumed. 

Jan 21 
Thomas Waters Galway 
Chaotic geodesics on a certain class of surfacesThe equations defining geodesic curves on surfaces admit a Hamiltonian formulation. A natural
question which arises is that of integrability. By combining numerical and analytical techniques we are
able to prove that the geodesic equations of a class of surfaces defined in terms of the spherical
harmonics are not integrable, and that there are chaotic regions of phase space. This talk will combine
elements of differential geometry, dynamical systems, and group theory, and will not assume any previous knowledge. 

Jan 28 
Niall Horgan Virginia 
Nonlinear elasticity theory: a rich source of interesting mathematical problemsContinuum mechanics theories for solids and fluids have played a significant role
in the development of many areas of applied mathematics. For example, the classical theory of
linear elasticity, applicable to the mechanical behaviour of metals, has been an important
motivation for the study of important linear partial differential equations and systems of
such equations. Here we describe how the nonlinear theory of hyperelastic materials,
applicable to the mechanical behaviour of rubber and soft biological tissues, plays
an analogous role. The basic theory of continuum mechanics of such materials is first
reviewed. Some classical models relating stress and strain in rubberlike materials are
described as well as more recent models applicable to biomaterials. Some particular
deformation modes are then described leading to the governing quasilinear partial
differential equations and systems of such equations. This lecture is of expository
nature and assumes no prior familiarity with continuum mechanics theories. The main
objective is to demonstrate that nonlinear elasticity theory, in addition to being
a fundamental continuum mechanics theory applicable to a wide variety of materials,
continues to provide a rich source of interesting and challenging mathematical problems. 
John Newell 
Feb 1 Monday 9.10am

David Quinn Galway 
Three Problems in Algebraic Combinatorics 

Feb 1 Monday 3.10pm

Volkmar Welker Marburg 
Random to Random Shuffles and Commuting Families of MatricesWe describe a family of matrices with rows and columns indexed by permutations. The entries generalize the inversion statistics on the symmetric group. These matrices are not only related to the inversion statistics but also are scaled versions of the transition matrices of Markov chains generalizing the random to random shuffle and can be factored into projections on a polytope generalizing the linear ordering polytope. We show some of the beautiful properties of these matrices. In particular, we study the algebra generated by the matrices, which can be seen as a subalgebra of the group algebra of the symmetric group. Finally we describe a generalization of the matrices within the symmetric group and for general finite Coxeter groups. (this is joint work with Franco Saliola and Vic Reiner). 
Graham Ellis 
Feb 4 
Michael McGettrick Galway 
Quantum Random Walks with MemoryWe investigate the quantum versions of a onedimensional random walk, whose corresponding Markov Chain is of order 2 (this corresponds to the walk having a memory of its previous step). We derive the amplitudes and probabilities for these walks, and point out how they differ from both Classical Random Walks, and Quantum Walks without memory. We prove localization for a particular example of the walk with memory. The talk will involve aspects of mathematics, physics and computer science. 

Feb 11 
Ruth Charney Boston 
Rightangled Artin groups and their AutomorphismsAssociated to any Coxeter group is an infinite group known as an Artin group. The Artin groups associated to rightangled Coxeter groups have proved to be particularly interesting from both an algebraic and geometric viewpoint. We will discuss these groups, their automorphism groups, and their associated geometries. 
Graham Ellis 
Feb 18 
Thomas Unger Dublin 
The ProcesiSchacher conjecture and Hilbert's 17th problem for algebras with involutionIn 1976 Procesi and Schacher developed an ArtinSchreier type theory for central simple algebras with involution and conjectured that in such an algebra a totally positive element is always a sum of hermitian squares. I will present elementary counterexamples to this conjecture and some cases where the conjecture does hold. This is joint work with Igor Klep. 
Rachel Quinlan 
Feb 25 
John Cosgrave Dublin 
GaussJacobi advancesIn [1], introducing the notion of a Gauss factorial, we gave the first extension of Gauss's generalisation of Wilson's theorem (Disquisitiones Arithmeticae, 1801); [1] extended, to composite moduli, work which began with Lagrange in 1777 and ended with Mordell in 1961.
In 1828 Gauss proved his beautiful mod p binomial coefficient congruence, and in 1837 Jacobi proved his closely related mod p congruence. The former concerns primes (5, 13, 17, 29...) that are 1 (mod 4), the latter primes (7, 13, 19, 31 ... ) that are 1 (mod 3).
In a 1983 Paris seminar Frits Beukers conjectured a mod p^{2} extension of Gauss' congruence, which was settled by S. Chowla, B. Dwork and R. Evans (1986). R. Evans and K. M. Yeung independently proved (late 1980's) a mod p^{2} extension of Jacobi's congruence.
No mod p^{3} extension of either Gauss's or Jacobi's congruences had been conjectured until 2007, when  as an outcome of our investigations of one very special class of Gauss factorials  we formulated and proved mod p^{3} extensions of both the Gauss and Jacobi congruences.
In this entirely elementary talk I shall treat all of this.
References.
[1] John B. Cosgrave and Karl Dilcher, Extensions of the GaussWilson theorem, INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY, 8, (2008), #A39 (http://www.integersejcnt.org/vol8.html)
[2] John B. Cosgrave and Karl Dilcher, MOD ^{3}$ ANALOGUES OF THEOREMS OF GAUSS AND JACOBI ON BINOMIAL COEFFICIENTS. Acta Arithmetica (to appear) 
Ted Hurley 
Mar 4 
Sebastian Franz Limerick 
The Capriciousness of Numerical Methods for Singular PerturbationsSingularly perturbed problems occur in a number of different models in fluid
dynamics, such as linearised NavierStokes and Oseen equations, in
semiconductor device simulation and flows in chemical reactions. The
understanding of numerical methods applied to these models is important in
order to solve them numerically.
In this talk we will see a collection of typical examples showing the
unexpected behaviour of numerical methods when applied to singular
perturbation problems.
Not only on standard equidistant meshes, but even on layeradapted meshes
several surprising phenomena are shown to occur. We will discuss and analyse
them. 
Niall Madden 
Mar 11 
Eimear Byrne Dublin 
The Linear Programming Bound for Codes Over RingsIn classical algebraic coding theory the linear programming bound
is one of the most powerful and restrictive bounds for the existence of
both linear and nonlinear codes defined over finite fields.
One viewpoint in establishing the LPbound comes from MacWilliams'
theorem relating the weight enumerator of a code with its formal dual.
In this talk we discuss generalizations of this bound for block codes over
finite Frobenius rings. 
Edwin O'Shea 
Mar 22 Monday 8.00pm

Des MacHale Cork 
Some of my favourite mathematics puzzles 
Edwin O'Shea 
Apr 22 

School Research Day 

May 6 
Ken Duffy Maynooth 
The Cyton Model: a stochastic analysis of the adaptive immune responseIn this talk, the basic mechanics of the adaptive immune response
will be introduced. We explain the nature of the experimental data
that is currently available and how deductions from it naturally
lead to a stochastic description of the adaptive immune response.
We introduce the Cyton Model hypotheses (Hawkins et al., PNAS, 2007)
and describe an analytic framework for studying it based on
agedependent branching processes, which enables us to predict
expected variability in immune response. We compare model predictions
with experimentally observed lymphocyte population sizes. The
significant biological conclusion is that the immune response is
robust and predictable despite the potential for great variability
in the experience of each individual cell.
This talk is based on joint work with Vijay Subramanian at the
Hamilton Institute, NUIM, and members of the Hodgkin Laboratory at
the Hall Institute for Medical Research, Australia. 
Cathal Seoighe 
May 13 
Clarice Demetrio Sao Paulo 
An Extended Randomeffects Approach to Modeling Repeated,
Overdispersed Count DataNonGaussian outcomes are often modeled using members of the
socalled exponential family. The Poisson model for count data falls within
this tradition. The family in general, and the Poisson model in particular,
are at the same time convenient since mathematically elegant, but in need of
extension since often somewhat restrictive. Two of the main rationales for
existing extensions are (1) the occurrence of overdispersion (Hinde and
Demétrio 1998, Computational Statistics and Data Analysis 27, 151170), in
the sense that the variability in the data is not adequately captured by the
model's prescribed meanvariance link, and (2) the accommodation of data
hierarchies owing to, for example, repeatedly measuring the outcome on the
same subject (Molenberghs and Verbeke 2005, Models for Discrete Longitudinal
Data, Springer), recording information from various members of the same
family, etc. There is a variety of overdispersion models for count data,
such as, for example, the negativebinomial model. Hierarchies are often
accommodated through the inclusion of subjectspecific, random effects.
Though not always, one conventionally assumes such random effects to be
normally distributed. While both of these issues may occur simultaneously,
models accommodating them at once are less than common. This paper proposes
a generalized linear model, accommodating overdispersion and clustering
through two separate sets of random effects, of gamma and normal type,
respectively ( Molenberghs, Verbeke and Demétrio 2007, LIDA, 13, 513531).
This is in line with the proposal by Booth, Casella, Friedl and Hobert
(2003, Statistical Modelling 3, 179181). The model extends both classical
overdispersion models for count data (Breslow 1984, Applied Statistics 33,
3844), in particular the negative binomial model, as well as the
generalized linear mixed model (Breslow and Clayton 1993, JASA 88, 925).
Apart from model formulation, we briefly discuss several estimation options,
and then settle for maximum likelihood estimation with both fully analytic
integration as well as hybrid between analytic and numerical integration.
The latter is implemented in the SAS procedure NLMIXED. The methodology is
applied to data from a study in epileptic seizures. 
John Hinde 
May 14 Friday 2pm

David Chillingworth Southampton 
Gravitational lensing and Galois Theory 
Petri Piiroinen 
May 18 Tuesday 2pm

Thomas Banchoff Brown University 
Visualizing the Fourth Dimension: From Flatland to HypergraphicsWhat does is mean to see phenomena beyond the third dimension? How can interactive computer graphics make it possible to see and manipulate objects in fourspace? This presentation will feature new film and text versions of the 1884 classic Flatland. 
Javier Aramayona 
May 20 10am

John Butcher Auckland 
A beginner's guide to numerical ODEsOrdinary differential equations are at the heart of applied mathematics and arise in models of almost every scientific phenomenon. An analytical solution is not usually available and numerical approximations therefore become necessary.
This beginners guide starts with the classical numerical method associated
with the name of Euler and shows how this can be generalized to obtain more
accurate and efficient numerical procedures. It is hoped to demonstrate
that this is not only a useful subject but that it is also a subject which
contains some beautiful mathematics. 
Niall Madden 
May 20

Alexander Ivanov Imperial College, London 
Majorana TheoryThe Monster group $, which is the largest among the 26 sporadic simple groups is the automorphism group of 196\,884dimensional ConwayGriessNorton algebra (simply called the Monster algebra). There is a remarkable correspondance between the socalled AhBinvolutions in $ and certain idempotents in the Monster algebra (we refer to these idempotents as Majorana axes). The isomorphism types of the subalgebras in the Monster algebra generated by pairs of Majorana axes were calculated by S.Norton a while ago (there are precisely nine isomorphism types). More recently these nine algebras were characterized by S.Sakuma in the context of Vertex Operator Algebras, relying on earlier work by M.Miyamoto. The properties of Monster algebras used in the proof of Sakuma's theorem are rather elementary and they have been axiomatized under the name of Majorana representations. In this terminology Sakuma's theorem amounts to classification of the Majorana representations of the dihedral groups together with a remark that all the representations are based on embeddings into the Monster. In the present paper it is shown that the alternating group $ of degree 5 possesses precisely two Majorana representations, both based on embeddings into the Monster. The dimensions of the representations are 20 and 26; the scalar squares of the identities are 0$ and $\frac{72}{7}$, respectively (in the Vertex Operator Algebra context these numbers are doubled central charges). 
Alexander Zuevsky 
May 27 
George Havas Queensland, Australia 
On Coxeter's families of group presentationsIn 1939 Coxeter published three infinite families of group presentations. He studied their properties, in particular determining when groups defined by members of the families are infinite and the structure of
finite ones. Eight presentations remained for which the finiteness question was unsettled. We show that two of these eight presentations define finite groups (for which we give comprehensive proofs and provide detailed structural information) and that two of the presentations define infinite groups. Our results rely on substantial amounts of computer calculations, in particular on coset enumeration to prove finiteness and on computation of automatic structures using KnuthBendix rewriting to prove infiniteness. This is joint work with Derek Holt. 
Alla Detinko 
Jul 22 2.15pm

Jessica O'Shaughnessy NUI Galway 
Convolutional Codes from Group Rings 
Ted Hurley 