Twist Points of Planar Domains
Speaker: Fausto Di Biase
We establish a potential theoretic approach to the study of twist points in the boundary of simply connected planar domains. A joint work with Nicola Arcozzi, Enrico Casadio Tarabusi, and Massimo Picardello
Factorizing Maps into Involutions and Reversibles
Speaker: Anthony O'Farrell
An involution g in a group G has g2=1. An element g ∈ G is reversible if there exists h ∈ G with h-1gh=g-1. I'm interested in expressing general elements as products of involutions, or as products of reversibles.
I'll describe recent progress on this in a particular group: the group of invertible formal maps of (C,0) to itself, i.e. formally-invertible maps whose components are formal series in n variables with complex coefficients and no constant term.
Cluster Analysis in Educational Testing
Speaker: Nema Dean
It is often the case that clustering takes place in supbspaces of the whole Euclidean space and that there may be shape restrictions on these subspaces.
Examples of these include the simplex, spherical subspaces and hypercubes. In our case we look at clustering on the unit hypercube (of arbitrary dimension). The standard methods for clustering often have implicit spherical/elliptical shape constraints on the clusters they can discover. Examples of such methods include: k means, complete hierarchical clustering, ward's method, model-based clustering (with finite Gaussian mixtures) as well as many others. Clearly while these should perform well for discovering groups in the centre of the hypercube, they may perform poorly when searching for groups at the corners. We look to compare a standard method for clustering, k means, with alternative methods tailored to the shape of the space. The results of a finite mixture of beta distributions (estimated by EM algorithm) and also k means and model-based clustering applied to arcsine transformed data will be presented, giving the various pros and cons of all methods. Results from both simulated data from various models and a dataset in the cognitive diagnosis field, looking at grouping students based on skill estimates (from 0 to 1) of various skills used in an electronic test, will be presented.
Triangulating Real Projective Spaces
Speaker: Sonia Balagopalan
The problem of finding vertex-minimal triangulations (as simplicial complexes) of compact manifolds has long been of interest to geometers and topologists. There are interesting insights to be gained from considering these problems from a combinatorial point of view. We shall summarise the current state of knowledge on the problem in the case of RPn, the n-dimensional projective space over the real numbers, and describe combinatorial constructions of one or two minimal complexes. This talk will be more or less elementary and accessible to undergraduate mathematics students.
Fragmentation of braids and diffeomorphisms
Speaker: Frédéric Le Roux
The braid group on n strands is a group that has an intuitive geometrical representation. I will define the size of a braid and discuss the so-called fragmentation problem: Is there a bound in the number of pieces we need to reconstruct any given braid from small ones?
A partial answer is obtained as a consequence of the work of Calegari and Fujiwara (2009). Then I will consider the similar problem in the group of area-preserving diffeomorphisms of the plane. Here again, a partial answer is provided by Entov and Polterovich (2003). The general problem, which is still open, is equivalent to the algebraic simplicity of some area-preserving homeomorphism group.
Considerations on jury size and composition using lower probabilities
Speaker: Brett Houlding
The use of lower probabilities is considered for inferences in basic jury scenarios to study aspects of the size of juries and their composition if society consists of subpopulations. The use of lower probability seems natural in law, as it leads to robust inference in the sense of providing a defendant with the benefit of the doubt. The method presented in this paper focuses on how representative a jury is for the whole population, using a novel concept of a second 'imaginary' jury together with exchangeability assumptions. It has the advantage that there is an explicit absence of any assumption with regard to guilt of a defendant. Although the concept of a jury in law is central in the presentation, the novel approach and the conclusions hold for representative decision making processes in many fields, and it also provides a new perspective to stratified sampling.
Propositional Logic and Type-Theory, the Curry-Howard correspondence
Speaker: J. Roger Hindley
The Curry-Howard correspondence is a curious parallel between between the notations of type-theory and propositional logic. It was first noticed in the 1930s and seemed then to be merely a shallow coincidence. But 40 years later, people studying both systems began to realise that this link allowed non-trivial results proved about one system to be transferred to the other. The correspondence was then extended to stronger systems, and today it is an important topic in logic and the theory of computing. (Try Googling "Curry Howard".) This talk will explain some of its background and history. It will include an account of basic type-theory and versions of propositional logic for which the correspondence holds. Incidentally, one of the pioneers involved was Carew Meredith, of Trinity College, Dublin, in the 1950s.
Speaker: Tugba Aysel
Both Ireland and Turkey have high-stakes examinations at the end of second level schooling that determine entry to third level education. I aimed to explore the effect of such examinations on the teaching and learning of mathematics at second level in both countries. Various research reports, for example Lyons, Lynch, Close, Sheerin & Boland (2003) and Hourigan & O’Donoghue (2007), have expressed concern in relation to the teaching and learning of mathematics in Irish post-primary schools in terms of an undue focus being placed on the attainment of examination results and a tendency to teach (and learn) to the examination rather than to the aims of the curriculum. However, there is less evidence in Turkey of such an effect from the examinations that mark the end of second-level schooling and entry to the third level system. This talk will report on a comparison of the mathematics examinations in the two countries and on data from a series of interviews with Irish and Turkish mathematics teachers.
Classification Challenges in Chemistry
Speaker: Deirdre Toher
As chemists aim to develop new, higher throughput analysis techniques the resulting datasets are often significantly more complex than previous approaches providing scope for interesting challenges for classification of observations when n << p. I will discuss how to approach modelling such data, where issues with the data collection process mean that variables are not well defined with machine “slippage” being an issue with High Performance Liquid Chromatography (HPLC) and broad ill-defined peaks being a feature of Near Infrared Spectroscopy (NIR). Particular applications to be examined include the analysis of food samples (NIR) and also the diagnosis of C.difficile and Noravirus using volatile compounds released from samples of stool collected from in-patients suffering from diarrhoea (HPLC). The relative advantages and disadvantages of using feature selection prior to classification compared to an “all-in-one” classification approach where feature selection is embedded within the classification technique will also be discussed.
Foliations and submanifolds of symmetric spaces
Speaker: Tommy Murphy
The aim of the talk is to outline some central problems in submanifold geometry which arise when one tries to generalize the classical theory of surfaces. Many are simple to state, but surprisingly difficult to answer. I will disuss the natural interplay between submanifold geometry and foliation theory for symmetric spaces. For simplicity, we will discuss submanifolds of spheres, Euclidean spaces, and complex projective and hyperbolic spaces.
Clustering Time-Course Microarray Data Using the Linear Mixed Effects Model
Speaker: Norma Coffey
Time-course microarray analyses involve measuring the expression levels of thousands of genes repeatedly through time, resulting in extremely high-dimensional data. Multivariate clustering methods such as k-means clustering, self-organizing maps, hierarchical clustering, finite mixture models and mixtures of linear mixed effects models have been useful to reduce the dimensionality of gene expression data and provide some insight into the number and types of characteristic behaviour evident in the set of expression profiles. However, gene expression data exhibits problems such as missing values, large amounts of measurement error, correlation between measurements made over time on the same gene and high dimensionality. Many of the techniques mentioned above have difficulties handling missing values, require uniform sampling for all genes, fail to account for the correlation between measurements made on the same gene or do not facilitate the removal of noise from the measured data thus ignoring any smoothness that may be evident in the expression profiles. More recently, curve-based clustering methods have been employed to cluster time-course gene expression data. Such methods remove measurement error using smoothing techniques, account for the correlation between measurements made over time on the same gene and can handle missing values and irregularly sampled data. This paper implements curve-based clustering and uses penalized spline smoothing to estimate the gene expression curves. The penalized smoothing problem is represented as a linear mixed effects model which has several advantages. Writing the smoothing problem as a mixed effects model provides a framework for simultaneously determining a smooth estimate of the mean expression profile in each cluster, determining estimates of the gene-specific expression profiles within a cluster through the use of additional random effects (e.g. a random intercept for each gene) and clustering expression profiles using mixtures of mixed effects models. In addition, an optimal value for the smoothing parameter λ is automatically chosen via restricted maximum likelihood (REML) and all modelling can be carried out using standard software. This is joint work with Emma Holian and John Hinde.
Some Open Problems in Classical Fourier Analysis
Speaker: Joaquim Bruna
In the lecture I will survey several open problems in Fourier analysis, some dating back to N. Wiener and A. Beurling, that have attracted renewed interest in connection with the development of wavelet theory. I will present the formulation of these problems both in terms of Complex Analysis and Functional Analysis, and explain some connections with the modern Compressed Sensing theory. The only prerequisite to understand the lecture is a basic knowledge of Fourier series and one variable complex analysis.
Bayesian source separation of Cosmic Microwave Background
Speaker: Simon Wilson
Source separation is one of the initial data processing tasks for multi-channel image data, particularly in astronomy where satellites such as COBE, WMAP and more recently Planck have obtained multi-channel images at microwave frequencies. In this work, we look at a factor analysis approach to source separation. First I discuss how prior information, available from understanding of the physics of the sources, can be incorporated into the analysis. Priors for the images of each source are modelled as Gaussian Markov random fields. Then I show that an analytical approximation to the posterior is possible which allows for practical separation of large images; Planck data consists of 9 images at different microwave frequencies, each of about 10 million pixels. The work is applied to reconstruct the Cosmic Microwave Background (CMB) signal from satellite observation, by separating it from other sources, using WMAP 7 year data. The performance and limitations of the approximation are also discussed.
Conditions implying ring commutativity
Speaker: Stephen Buckley
It is well known that Boolean rings, characterized by the polynomial condition x2=x, are commutative. In this talk we discuss (mostly one-variable) polynomial-type conditions that imply ring commutativity, beginning with the important work of N. Jacobson and I.N. Herstein.
Among the topics we will discuss are:
(1) Elementary proofs, for certain values of n, of the commutativity of rings such that for xn=x for all x in R.
(2) A classification of all polynomials f such that a ring with unity R is necessarily commutative if f(x)=0 for all x in R. (The non-unital variant was previously characterized by Laffey and MacHale.)
(3) Variants of the above where equality is replaced by the difference of the quantities involved being central (thus giving necessary and sufficient conditions for commutativity).
Along the way, we discuss problems in this area that are suitable for undergraduates. Much of the talk will be accessible to undergraduates.
This is based on joint work with Des MacHale.
Ultrafilter semigroups and related questions
Speaker: Yuliya Zelenyuk
The operation of a discrete semigroup S extends naturally to the Stone-Cech compactification βS of S so that for each p ∈ βS, the right translation βS ∋ x ↦ xp ∈ βS is continuous, and for each a ∈ S, the left translation βS ∋ x ↦ ax ∈ βS is continuous. We take the points of βS to be the ultrafilters on S, identifying the principal ultrafilters with the points of S, and S* = βS\S. The semigroup βS has been actively studied for the last 50 years, and is a powerful tool in combinatorial number theory and topological dynamics [2, 3].
I shall speak about some old and new questions related to βS, in particular, about the radical of l1(N*) [1].
References
[1] G. Dales, D. Strauss, Ye. Zelenyuk, Yu. Zelenyuk, Radicals of some semigroup algebras,
2012, manuscript.
[2] N. Hindman, D. Strauss, Algebra in the Stone-Cech compactification, De Gruyter,
Berlin, 2nd edition, 2011.
[3] Ye. Zelenyuk, Ultrafilters and topologies on groups, De Gruyter, Berlin, 2011.
Vibrational Spectroscopy and Classification for Medical Diagnosis
Speaker: Claudia Beleites
Spectroscopic techniques are routinely used in analytical chemistry for sample characterization. Nowadays, chemically more and more complex samples, such as biological samples, are studied. Multivariate data analysis is needed to interpret the biochemical and medical meaning of such extremely complex data.
Vibrational spectra like mid-infrared absorbance and Raman spectra have two important characteristics for biomedical research. On the one hand, the spectra can easily be
interpreted in terms of biochemical substance groups like proteins, lipids, carbohydrates etc. On the other hand, tissues have a clearly defined composition and thus can be classified using the so-called fingerprint property of the vibrational spectra.
As example applications, ongoing work on an automatic cell sorting system aimed at tumour cells circulating in the blood and Raman-based grading of astrocytoma tissues (brain tumours) for intra-surgery diagnosis will be discussed.
Alongside the example applications, general particularities of the chemometric data analysis of spectroscopic data sets will be illustrated.
Discovering clustering structure and influential variables in high-dimensional metabolomic data
Speaker: Claire Gormley
Metabolomics is the quantitative study of metabolites in a biofluid and is widely used in many areas including nutritional biochemistry. Metabolomic data typically take the form of high-dimensional spectra. The spectral peaks relate to specific metabolites and the height of a peak details metabolite abundance. Metabolite patterns provide insight to underlying molecular mechanisms of disease. Typically metabolomic scientists are interested in using metabolomic spectra to diagnose and understand metabolomic disease. Further, they are interested in studying the influence of covariates jointly with the spectral data.
The high dimensionality of the metabolomic spectra provides statistical challenges; latent factor models can be employed in such situations to represent high-dimensional
data in lower dimensional space. Here such models are extended to facilitate joint modeling of metabolomic spectra and covariate data. Additionally a mixture modeling framework is employed to provide clustering capabilities, aiding the identification of subtypes of metabolomic disease.
Very many of the spectral peaks in a metabolomic spectrum tend to be noise, which can lead to spurious inference being drawn. Additionally, metabolomic scientists are interested in discovering which metabolites are influential. Inference is therefore performed within the Bayesian paradigm where sparse priors are employed to appropriately deal with the high dimensionality of the spectral data and to facilitate variable (i.e. metabolite) selection. The methodology is illustrated through the analysis of a real metabolomic data set.
Speaker: Fiona Boland
As milk yield increases milk somatic cell count (SCC) is diluted and as a consequence, estimates of SCC from high yields are lower than estimates of SCC from low yields in dairy cows. Additionally, high SCC is usually indicative of subclinical mastitis. In Ireland, costs related to mastitis are extensive but the cost of this problem may have been over-estimated in the past. Estimates of reduced milk yield associated with high SCC have not been adjusted for any dilution of SCC and ignoring this dilution is therefore likely to lead to an overestimate of the reduction in milk yield with increasing SCC. However, it is not known how to estimate the presence of a potential dilution effect. In this study various approaches to adjust milk loss for dilution were examined and the possible impact of this dilution on the association between milk yield and SCC investigated. The data used consisted of 100 herds extracted from the Irish Cattle Breeding Federation (ICBF) database which contained lactation records for 8,229 milk recorded cows between 2008 and 2010. There was an inherent hierarchical structure in the data with lactations nested within cows and cows within herds, thus a linear mixed model with two random effects was used. The various approaches to adjust milk loss for dilution were investigated and compared with the unadjusted model.
Graph regularity and removal lemmas
Speaker: David Conlon
Szemerédi's regularity lemma states that every large graph may be partitioned into a small number of parts so that the bipartite graph between almost all pairs of parts is random-like. One of the most important applications of this theorem is the graph removal lemma, which roughly says that every graph with few copies of a fixed graph H can be made H-free by removing few edges. In this talk, we will discuss recent progress on bounds for these theorems and for several important variants. This is joint work with Jacob Fox.
Difference sets and Hadamard matrices
Speaker: Pádraig O'Catháin
The study of difference sets involves ideas and methods from combinatorics, algebraic number theory, group theory and finite geometry. In this talk, I will give an introduction to difference sets, and show how difference sets with parameters (4t-1,2t-1,t-1) can be used to construct Hadamard matrices. I will show how deep results from the theory of permutation groups can be used to classify algebraically interesting classes of Hadamard matrices and difference sets.
Rough CAT(0) spaces and groups
Speaker: Kurt Falk
In the talk I will be discussing rough CAT(0) spaces, or rCAT(0) in short, introduced by Buckley and myself, in an attempt to unify CAT(0)-theory and Gromov hyperbolicity, with special emphasis on boundary theory. When defining the boundary of a rCAT(0) space, bouquets of short segments, a concept related to Vaisala's roads, replace the classical geodesic rays to infinity known from the definition of the ideal boundary of a metric space. However, rCAT(0) spaces come in three flavours of which only the strongest allows a meaningful boundary theory, and the weakest turns out to be already known as so-called bolic spaces. As of now, we do not know whether these three conditions are equivalent or not. In the last part of the talk I will concentrate on rCAT(0) groups.
The Euler equation for analysts (and similar people)
Speaker: Joan Verdera
I will talk about the Euler equation in the plane from an elementary perspective. The speaker himself is new at this topic and was prompted to look at it by a result in singular integrals, which will be mentioned.
Light and temperature effects on the circadian clock
Speaker: Mirela Domijan
The circadian clock is endogenous 24h timer driving numerous metabolic, physiological, biochemical and developmental processes. The clock has a complex interaction with its environment as it responds to light and temperature cues. It can be entrained to daily cycles of light and temperature, yet it also remains very robust to their stochastic fluctuations. Another key striking feature of the clock is that it can maintain nearly constant period over a broad range of physiological temperatures (a feature called temperature compensation). These properties enable the clock to do a variety of functions: it can be used to predict transitions at dusk and dawn, measure day length, and it allows an organism to respond accurately to seasonal rhythms. Elucidating the interaction of the clock with its environment can help us gain greater understanding of the design principles of this important mechanism.
Here I will present some recent work in this direction [1, 2].
References
[1] M. Domijan and D.A. Rand, Balance equations can buffer noisy and sustained environmental perturbations of circadian clocks Interface Focus 1 177–186.
[2] P.D. Gould , N. Ugarte, M. Domijan, M. Costa, J. Foreman, D. McGregor, S. Penfield, D.A. Rand, A. Hall, K. Halliday, A.J. Millar, Photoreceptors contribute temperature-specific regulation to the biological clock in Arabidopsis, preprint.
Will computers ever be able to do mathematical research?
Speaker: Professor W.T. Gowers
There is a widespread view that computers, while extremely helpful to mathematicians in a number of ways, could never do interesting research. The rough reason often given is that computers can only ever do what you tell them to do, so if they solve mathematics problems it will be because a programmer has had all the interesting ideas in advance. If this view is correct, then it suggests that there is a boundary between the routine tasks that computers are good at and the sort of research that is typical of what humans undertake. I shall look at a number of candidates for where this boundary might lie, arguing in each case that it does not in fact lie there. The conclusion I draw from this is that there is no boundary and that computers will in due course be better than humans at solving mathematics problems. However, this view is controversial and I do not expect to persuade everybody of its correctness.
Constructivism Made Visible in Contingency: Learning to Teach Mathematics in a Community of Practice
Speaker: Dolores Corcoran
Constructivism has become to be widely accepted as a grand theory of human learning which various scholars have interpreted and applied differently (Tobias & Duffy, 2009). While student teachers are routinely exposed to various propositions of constructivism in their studies, they rarely get an opportunity to ‘see’ constructivism in action. Yet such opportunities frequently occur in primary classrooms and constitute invaluable learning opportunities. The study on which this paper draws uses an analytical framework, known as the Knowledge Quartet (KQ) to highlight the various dimensions along which the mathematical knowledge of the teacher impacts the lesson (Rowland, Huckstep and Thwaites, 2005). The four dimensions of the KQ, grounded in the study of mathematics teaching, were devised in a linear fashion with the foundation dimension - that which is learned in the teacher education institution - believed to form the basis for the next two dimensions; transformation, the choices by which the mathematics to be taught is made available to learners, and connection, the manner by which mathematical ideas are synthesized into a coherent, curricular whole. Fourth among the KQ dimensions is contingency, which demonstrates the practitioner’s ability to think on his/her feet when lessons do not go according to plan. In this talk, I will develop the notion that contingency moments during mathematics lessons are important sites of learning for student teachers, which are to be welcomed for the insights they give into pupils’ construction of mathematical ideas. I first became aware of the ‘unveiling’ of constructivism in action through engagement in a community of practice with third year student teachers who took an education elective course called Learning to Teach Mathematics Using Lesson Study. Lesson study is a very simple protocol that has been found to influence personal and communal learning in sophisticated ways (Corcoran, 2011). I will argue in this talk that foundation knowledge expands with participation in lesson study with focus on contingency opportunities.
References
Corcoran, D. (2011) Learning from Lesson Study: Power Distribution in a Community of Practice. In L. Hart, A. Alston & A. Murata (Eds.). Lesson Study Research and Practice in Mathematics Education: Learning Together. USA: Springer.
Rowland, T., Huckstep, P. and Thwaites, A. (2005) Elementary Teachers’ Mathematics Subject Knowledge: The Knowledge Quartet and the Case of Naomi, Journal of Mathematics Teacher Education, 8 (3), 255-281.
Tobias, S. and Duffy, T. (Eds.) (2009). Constructivist instruction: success or failure? London: Routledge.