
Irish Mathematical Society Meeting 2013
NUI Maynooth
26–27 August 2013

Titles and abstracts
Sonia Balagopalan, National University of Ireland
Maynooth
An elementary construction of the Witt design on 22 points
The 22point Witt design is the unique 3(22,6,1) block design,
invariant under an action of the sporadic group $M_{22}$. We present a new
construction of this exceptional combinatorial object, based on wellknown
properties of the complete graph on 6 vertices. We also sketch a
connection with the only known vertexminimal triangulation of real
projective 4dimensional space.
Exponential Riordan arrays and the Toda Chain equations
The Toda Chain equations are a model of dynamical systems that are
integrable. They are intimately associated with timedependent orthogonal
polynomials and Hankel determinants. In this talk we will show how to
construct systems satisfying the Toda equations using exponential Riordan
arrays. The combinatorial origin of some of these arrays will be described.
Cube complexes and 3manifolds
I will outline the recent application of geometric group theory to the
structure of 3manifolds due to Agol and Wise, particularly the virtual
Haken conjecture. This breakthrough essentially finishes the list of
problems proposed by Thurston.
Victor Bible, University College Dublin
Boundaries and geometric renorming theory
Given a geometric property and a Banach Space $X$, we are concerned with
obtaining sufficient conditions for when $X$ has a renorming with the
property. A boundary of $X$ is a subset of the dual unit ball with the
following: Given an element $x$ of $X$, there exists an element of the
boundary which agrees with the norm of $x$ when evaluated at $x$.
Sometimes the existence of a boundary with certain properties will be such a
sufficient condition. I will present some of these results for the property
of polyhedrality. Upcoming areas of focus will then be discussed.
The Fractal Geometry of the Mandelbrot Set
In this lecture we describe several folk theorems concerning
the Mandelbrot set. While this set is extremely complicated
from a geometric point of view, we will show that, as long as
you know how to add and how to count, you can understand this
geometry completely. We will encounter many famous mathematical
objects in the Mandelbrot set, like the Farey tree and the
Fibonacci sequence. And we will find many soontobefamous
objects as well, like the "Devaney" sequence. There might
even be a joke or two in the talk.
Richard Ellard, University College Dublin
Generating New Realisable Lists from Old in the NIEP
Given a list of complex numbers
$\sigma:=(\lambda_1,\lambda_2,\ldots,\lambda_m)$, we say that $\sigma$ is
realisable if $\sigma$ is the spectrum of some (entrywise)
nonnegative matrix. The Nonnegative Inverse Eigenvalue Problem (or NIEP) is
the problem of categorising all realisable lists.
Given a realisable list $(c_1,\lambda_2,\lambda_3,\ldots,\lambda_m)$, where
$c_1$ is the socalled Perron eigenvalue of the list, Šmigoc
(2004) shows that we may replace the eigenvalue $c_1$ with a list
$(\mu_1,\mu_2,\ldots,\mu_n)$, where $\mu_1,\ldots,\mu_n$ satisfy certain
conditions, i.e. we may produce a new realisable list of the form
$(\mu_1,\mu_2,\ldots,\mu_n,\lambda_2,\lambda_3,\ldots,\lambda_m)$.
Subsequently, Šmigoc (2005) gave a construction to replace both the
Perron eigenvalue $c_1$ and another real eigenvalue $c_2$ of a realisable
list.
In this talk, we first expand on the above work by presenting some new lists
which may replace the eigenvalues $c_1$ and $c_2$. We then go on to give a
construction which allows us to replace three eigenvalues of a realisable
list (the Perron eigenvalue and a complex conjugate pair), i.e. from a
realisable list
$(\rho,\alpha+i\beta,\alphai\beta,\lambda_4,\ldots,\lambda_m)$, we generate
a new realisable list
$(\mu_1,\mu_2,\ldots,\mu_n,\lambda_4,\ldots,\lambda_m)$.
Dynamics on complex networks
I will discuss the modelling of the structure of largescale networks, and
dynamics that occur on the nodes (vertices) of such networks. A wide class
of binarystate dynamicsincluding, for example, the voter model, the Bass
diffusion model, and threshold modelscan be described in terms of
transition rates (spinflip probabilities) that depend on the number of
nearest neighbours in each of the two possible states. Highaccuracy
approximations for the emergent dynamics of such models on uncorrelated,
infinite networks are given by recentlydeveloped compartmental models or
approximate master equations (AME). Pair approximations (PA) and meanfield
theories can be systematically derived from the AME; we show that PA and AME
solutions can coincide in certain circumstances. This facilitates
bifurcation analysis, yielding explicit expressions for the critical
(ferromagnetic/paramagnetic transition) point of such dynamics, closely
analogous to the critical temperature of the Ising spin model.
Philipp Hoffmann, University College Dublin
An overview of $(m,p)$isometric operators and operator tuples
Socalled $m$isometric operators were defined in the 1980s on complex
Hilbert spaces, and since then have attracted the interest of many
researchers. In recent years, a generalisation to operator tuples has been
given, as well as a generalisation to operators on general normed spaces. We
give an overview of the history of these objects and present some new
results.
A.N. Other, Trinity College Dublin
Neural encoding of auditory stimuli
The problem of neural encoding is a central issue in theoretical
neuroscience. That is, how are external stimuli encoded by neurons into
sequences of electrical impulses referred to as spike trains? In this talk
an overview of current theory in this area will be presented. Recent results
of modelling the encoding process in auditory response data using a novel
cascade spiking neuron model will be presented and discussed. The model
consists of a linear stimulus filter cascaded with a neuron model from the
class of two dimensional models with adaptation. An approach to fitting this
model to data using a spike train metric, the van Rossum distance, is also
discussed. One advantage of this approaches used here is the generation of
precise spike times as opposed to a rate function. The results will thus be
presented along side those obtained with standard rate models.
Dave McConnell, Trinity College Dublin
The Glimm space of the minimal tensor product of C*algebras
We consider a C*algebra $A$ and its space of Primitive ideals
$\mathrm{Prim}(A)$ in its usual hullkernel topology. The Glimm space of
$A$, $\mathrm{Glimm}(A)$, is defined as the complete regularisation of
$\mathrm{Prim}(A)$  the quotient of $\mathrm{Prim}(A)$ modulo the relation
of inseparability by continuous functions, with topology induced by the
continuous functions on $\mathrm{Prim}(A)$. To each such equivalence class
in $\mathrm{Prim}(A)$ we assign a (closed, twosided) ideal of $A$ given by
the intersection of the primitive ideals contained in it.
For the minimal tensor product $A \otimes B$ of C$^{\ast}$algebras $A$ and
$B$ we describe $\mathrm{Glimm} (A \otimes B )$ in terms of
$\mathrm{Glimm}(A)$ and $\mathrm{Glimm}(B)$, both topologically and as a set
of ideals of $A \otimes B$, generalising a result of Kaniuth (1994). We
apply this to the study of the centre of the multiplier algebra of $A
\otimes B$.
James McTigue, National University of Ireland Galway
Partial matrices whose completions all have the same rank
A partial matrix over a field $\mathbb F$ is a matrix whose entries are
either elements of $\mathbb F$ or independent indeterminates. A completion
of such a partial matrix is obtained by specifying values from $\mathbb F$
for the indeterminates. We determine the maximum possible number of
indeterminates in an $m \times n$ partial matrix ($m \le n$) whose
completions all have a particular rank $r$, and we fully describe those
examples in which this maximum is attained, without any restriction on the
field $\mathbb F$.
We also describe sufficient conditions for an $m \times n$ partial matrix
whose completions all have rank $r$ to contain an $r \times r$ partial
matrix whose every completion is nonsingular.
This is joint work with Rachel Quinlan and it extends some recent results of
R. Brualdi, Z. Huang and X. Zhan.
Stable map spaces for varieties with $\mathbb{C}^*$actions
This talk is based on joint work with Andrei Mustata. The moduli spaces of
stable maps are compactifications of the spaces of smooth curves of a
specific class in a complex projective variety. They are important in
counting curves with given properties in that variety; the numerical answers
to these counts are known as GromovWitten invariants. To provide a count
that is consistent over complex deformations of the target variety, these
moduli spaces have been endowed with a suitable homology class called the
virtual fundamental class. The oldest and one of the most useful tools for
calculating GromovWitten invariants is localization. For target varieties
endowed with a torus $(\mathbb{C}^*)^n$ action, it reduces the
GromovWitten calculations to a study of the fixedpoint locus of the stable
map space under the given action. This has been particularly successful in
the case of toric varieties, i.e. $n$ dimensional varieties with a
$(\mathbb{C}^*)^n$ action. The fixedpoint locus of the stable map space
parametrizes combinations of the following curves: (i) curves mapping to the
fixed loci of the target variety and (ii) curves which are unions of
orbits. In the examples studied so far, the set of curves of the second type
is relatively simple, so that the computations are concentrated on to the
fixed loci of the target variety. In the case of $\mathbb{C}^*$ actions
however, the space of curves which are union of orbits is quite large and
complex. In our project we describe this space as an inverse limit of GIT
quotients of the variety under the $\mathbb{C}^*$ actions, and use this to
give a relatively simple description of the virtual fundamental class.
Special spaces of matrices
A large body of literature considers linear and affine spaces of matrices in
which rank behaves in a controlled way  for example, the ranks of the
nonzero elements are all the same or satisfy specified bounds. This talk
will mention some selected highlights of this subject, including the
RadonHurwitz numbers which give the maximum possible dimensions of spaces
of real square matrices whose nonzero elements are all invertible. A few of
the author's own contributions, arising from (separate) joint works with
R. Gow and J. McTigue, will also be mentioned.
University mathematics teaching: is this the real thing?
Mathematics teaching at the school level can be though to be making various
concessions to the immaturity of school students in its treatment of
mathematical objects and concepts. In particular, the subject is often
represented by reference to various analogies and models, and rigour in
proof is significantly compromised. Mathematics at university can be viewed
as the point when such concessions and compromises can, and perhaps should,
be set aside. So when mathematics is taught in the university, is this the
real thing? My approach to this question will be shaped, in part, by a
theory of mathematical knowledge in teaching  the Knowledge Quartet 
developed in the context of school mathematics.
Topological Quantum Computation
Quantum computation is the use of quantum mechanical processes to manipulate
information. This allows to solve certain types of problem more efficiently
than using classical mechanics (pen and paper, or current
computers). Topological quantum computation is an approach to quantum
computation which uses exchanges of particles to manipulate the
information. The information in the computer and the computations themselves
are then described in terms of the representation theory of certain groups
governing the exchanges, most prominently Artin's braid groups. I will
discuss the types of representations that one may expect to find in physical
systems, as well as some questions about them which emerge from the
potential application to computation.
Patterns of symmetric matrices that allow the characteristic polynomial to
be a square
Let $G$ be an undirected graph on $n$ vertices and let $S(G)$ be the set
of all real symmetric $n \times n$ matrices whose nonzero offdiagonal
entries occur in exactly the positions corresponding to the edges
of $G$. We will discuss the question of for which graphs $G$ does there
exist a matrix $A$ in $S(G)$ whose characteristic polynomial is a square,
i.e., the multiplicities of all the eigenvalues of $A$ are even. This is
joint work with P. Oblak.