Mathematics, Statistics and applied Mathematics Seminar 2009/10

The seminar usually takes place on Thursdays from 3.45pm to 4.45pm in room C219 of the School of Mathematics, Statistcs and Applied Mathematics, which is located in Áras de Brún (Block C).
   The talks are directed towards a general mathematical audience and everyone interested is very welcome to attend. Tea/coffee and pastries will be available in C108 from 3.15pm.
   Previous seminars are listed here.

Abstracts when available can be obtained by rolling the mouse over a title. Clicking a title sometimes provides a longer version of the talk in PDF format.

Semester I

DateSpeakerTitleContact
Sep 10 Chris Leininger
Urbana-Champaign
Homeomorphisms of surfaces and dynamicsWe will start by describing a standard measure of the dynamics of a homeomorphism of a surface. For a certain generic class of homeomorphisms, we will discuss how this measure relates to algebraic and topological properties. Javier Aramayona
Sep 17 Kevin Jennings
Galway
A dissection of an abelian (341,85,21) difference setA (perfect) difference set is a subset D of a group such that each nonidentity element of the group has exactly L representations as a difference ab^{-1} where a and b are in D. In this talk I will focus on a difference set with 85 elements in a group of order 341. L=21 in this case and this difference set exists, belonging to a classical family.
This group has a subgroup of order 31 and we can show by a few different arguments that there are 15 elements from the normalized difference set in this subgroup. (arguments range from basic counting, to looking at the orbits of the Hall multiplier, to simple applications of more technical combinatoric arguments). Unfortunately, I can not establish that these 15 elements must a priori express each element of this subgroup exactly 7 times i.e. that there is a (31,15,7)-difference set embedded in this larger difference set. This is the problem I will discuss.
In this talk I will give an overview of the known techniques for attacking such a problem, and explain briefly where this particular case might lead and why it is interesting. The talk will be elementary and I will keep it as non-technical as possible. The problem posed is low-hanging fruit.
Sep 21
Monday 9.15am
Ian McLoughlin
Galway
Dihedral Codes
Sep 21
Monday 2.00pm
Bent Jørgensen
Odense
Efficient Estimation for Incomplete Multivariate DataWe compare the Fisher scoring and EM algorithms for incomplete multivariate data, and investigate the corresponding estimating functions under second-moment assumptions. We propose a hybrid algorithm, where Fisher scoring is used for the mean vector and the EM algorithm for the covariance matrix. A bias-corrected estimate for the covariance matrix is obtained. This is joint work with Hans Chr. Petersen. John Hinde
Sep 24 James Gleeson
Limerick
Cascade dynamics on complex networksNetwork models underlie many complex systems, e.g. the Internet, the World Wide Web, gene-regulatory networks, etc. Cascade dynamics can occur when the (binary) state of a node is affected by the states of its neighbours in the network. Such models have been used to aid understanding of the spread of cultural fads and the diffusion of innovations, and can be generalized to include percolation problems, k-core sizes, and the study of (SIR-type) epidemics on networks. For this class of problems, I present recent results on the analytic determination of the expected size of cascades on networks of arbitrary degree distribution, and outline extensions and applications of this research. Petri Piiroinen
Oct 1 Brian Marx
Baton Rouge
Variations on the Varying Coefficient ModelAlthough the literature on varying coefficient models (VCMs) is vast, we believe that there remains room to make these models more widely accessible and provide a unified and practical implementation for a variety of complex data settings. The adaptive nature and strength of P-spline VCMs allow for a full range of models: from simple to additive structures, from standard to generalized linear models, from one-dimensional coefficient curves to two-dimensional (or higher) coefficient surfaces, among others, including bilinear models and signal regression. As P-spline VCMs are grounded in classical or generalized (penalized) regression, fitting is swift and desirable diagnostics are available. We will see that in higher dimensions, tractability is only ensured if efficient array regression approaches are implemented. We also motivate our approaches through several examples to highlight the breadth and utility of our approach. John Hinde
Oct 8 Cyril Lecuire
Toulouse
Convex cores of hyperbolic 3-manifoldsThe study of 3-dimensional hyperbolic manifolds with infinte volume can be reduced to the study of their convex cores. I will explain how the volume of such a convex core is related to its end invariants, in contrast with the 2-dimensional case. Javier Aramayona
Oct 14
Wednesday 3.00pm
Efim Zelmanov
San Diego
Asymptotic properties of finite groups and finite dimensional algebrasThis is a public lecture to be held in the NCBES Seminar Room at 3pm Dane Flannery
Oct 22
Student Autumn GraduationsStudents' graduations will be from 2:30pm till 5:30pm
Oct 29 Natalia Iyudu
Belfast
Quadratic algebras: the Anick conjecture, representation spaces and Novikov structuresI will explain new results on the Anick conjecture (1983) on attaining of the Golod-Shafarevich estimate for the Hilbert series of quadratic algebras. Then I mention some results which describe representation spaces of one quadratic algebra, well-known in non-commutative geometry: Jordan algebra. These include classifcation of irreducible components of the space of n-dimensional representations. As a consequence, the number of irreducible components turned out to be equal to the number of partitions of n. If time permits I also explain a surprising connection between representation theory of quadratic algebras with questions on existance of Novicov structures, prominent by their appearance in the study of Poisson brackets of hydrodynamic type in integrable systems and by their connections to the conformal Lie theory. Alexander Zuevsky
Nov 5 Viacheslav V. Nikulin
Liverpool
On classification of arithmetic hyperbolic reflection groups.I would like to speak about old and new results on finiteness and classification of arithmetic groups generated by reflections (in hyperplanes) of hyperbolic spaces. Alexander Zuevsky
Nov 12 Stephen Wills
Cork
Markov processes and semigroups and their noncommutative counterpartsI will outline the connections between Markov processes (memory-less stochastic processes) and semigroups in the classical theory, before going on to explain their noncommutative counterparts in quantum probability. A large part of the talk will be devoted to discussing how to produce a quantum Markov process from a given quantum Markov semigroup, the problems one typically encounters, and how these have recently been overcome for (some cases of) the quantum exclusion process. Edwin O'Shea
Nov 17
Tuesday 3.00pm
Alexander Rahm
Grenoble
The integral homology of PSL_2 of imaginary quadratic integers with non-trivial class groupWe show that a cellular complex described by Floege allows to determine the integral homology of the Bianchi groups PSL_2(O_{-m}), where O_{-m} is the ring of integers of an imaginary quadratic number field Q[\sqrt{-m}] for a square-free natural number m. We use this to compute the integral homology in the cases m = 5, 6, 10, 13 and 15, which before was known only in the cases of trivial class group m = 1, 2, 3, 7 and 11. Graham Ellis
Nov 19 Jim Anderson
Southampton
Small filling sets of curves on surfacesThis talk is about joint work with Hugo Parlier and Alexandra Pettet. The general problem we are considering is the structure of sets of simple closed curves on a closed surface that fill; that is, sets of curves whose complement is the union of (topological) discs. I will give a brief history of where and why the question arose and discuss our first results, which involve bounding the minimal number of curves in a set of filling curves. Javier Aramayona
Nov 26 Mathieu Dutour
Zagreb
Lattice Packings and CoveringsA family of balls in Euclidean space is called a packing if for any two balls B and B' their interior do not self-intersect. It is called a covering if every point belong to at least one ball.
We focus here on packings and coverings for which the calls are of the form x + B(0,R) with x belonging to a lattice L. If L is fixed then we adjust the value of R to a value R0 to find the best packing. Alternatively we can adjust the value of R to a value R1 to find the best covering. This allow us to define the packing density pack(L) and covering density cov(L) of L.
The geometry of the function pack on the space of lattices has been elucidated by Minkovski, Voronoi and Ash. They showed that the function pack has no local minimum, that it is a Morse function and they give a characterization of the local maximum in terms of the algebraic notions of perfection and eutaxy.
The covering function cov is much more complex. It has local minimum and local maximum and it is not a Morse function. We also characterize the local maximum of the covering density in terms of the corresponding notions of perfection and eutaxy this time for Delaunay polytope.
Graham Ellis
Dec 3 Jochen Einbeck
Durham
Data compression and regression based on local principal curves and surfaces. In a multivariate regression problem with p-dimensional predictor space, the intrinsic dimensionality of the latter is often far smaller than p, sometimes even just one or two. Usual modelling attempts such as the additive model, which try to reduce the complexity of the regression problem by making additional structural assumptions, are then inefficient as they ignore the inherent structure of the predictor space and involve complicated model and variable selection stages.

In a fundamentally different approach, one may consider first approximating the predictor space by a (usually nonlinear) curve, and then regressing the response only against the one-dimensional projections onto this curve. This entails the reduction from a p- to a one- dimensional regression problem. As a tool for the compression of the predictor space we apply local principal curves. We show how local principal curves can be parametrized and how the projections are obtained. The actual regression step can then be carried out using any nonparametric smoother. If the intrinsic dimension of the predictor space is 2 (or more), one needs to replace the local principal curve by an adequate structure of higher dimension. We demonstrate how the idea of local principal curves can be extended towards local principal surfaces (or manifolds) through relatively simple means. The proposed techniques are illustrated using astronomical and oceanographic data examples.
John Newell

Semester II

DateSpeakerTitleContact
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 rubber-like 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 one-dimensional 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
Right-angled Artin groups and their AutomorphismsAssociated to any Coxeter group is an infinite group known as an Artin group. The Artin groups associated to right-angled 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 Procesi-Schacher conjecture and Hilbert's 17th problem for algebras with involutionIn 1976 Procesi and Schacher developed an Artin-Schreier 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
Gauss-Jacobi 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 Gauss-Wilson theorem, INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY, 8, (2008), #A39 (http://www.integers-ejcnt.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 Navier-Stokes 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 layer-adapted 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 non-linear codes defined over finite fields. One viewpoint in establishing the LP-bound 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 age-dependent 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 Random-effects Approach to Modeling Repeated, Overdispersed Count DataNon-Gaussian outcomes are often modeled using members of the so-called 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 Dem├ętrio 1998, Computational Statistics and Data Analysis 27, 151-170), in the sense that the variability in the data is not adequately captured by the model's prescribed mean-variance 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 negative-binomial model. Hierarchies are often accommodated through the inclusion of subject-specific, 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 Dem├ętrio 2007, LIDA, 13, 513-531). This is in line with the proposal by Booth, Casella, Friedl and Hobert (2003, Statistical Modelling 3, 179-181). The model extends both classical overdispersion models for count data (Breslow 1984, Applied Statistics 33, 38-44), in particular the negative binomial model, as well as the generalized linear mixed model (Breslow and Clayton 1993, JASA 88, 9-25). 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 four-space? 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\,884-dimensional Conway--Griess--Norton algebra (simply called the Monster algebra). There is a remarkable correspondance between the so-called 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 Knuth-Bendix 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