The School of Mathematics and Statistics has a diverse array of research groups.

The actuarial and financial studies research group pursues a wide range of research. Actuaries are professionals who deal with numbers to make judgments about the future. Traditionally actuaries have worked mostly in pensions and life insurance areas as these require significant judgments regarding future financial assessments. However, the group also pursues research in wider areas including risk management, financial and investment markets, economic forecasting, and forestry investment. Professionalism and ethics are also a key research topic.

**Dr Michelle Carey**

**Research Interests**:Functional Data Analysis, Differential Equations, Financial instruments, Biological systems and Climatology

**Mr Colm Fitzgerald**

**Research Interests**: Risk and the psychology of risk, enterprise risk management, economics and banking, investment and financial markets, climate change, forestry, applying classical thought.

**Assoc Prof Russell Higgs**

**Research Interests**: Representation Theory, Coding Theory, Wireless Sensor Network

**Dr Adamaria Perrotta**

**Research Interests:**Pde, Calculus of Variations, Stochastic Analysis, Financial Derivatives, Volatility Surface

**Assoc Prof Shane Whelan**

**Research Interests:**Longevity Modelling and Evaluation of Systems of Pension Provision

Advances in digital communications and storage technology are driven by a range of fundamental areas in algebra and discrete mathematics. The focus of the Applied Algebra Group in UCD is in core topics in algebra with applications to coding, cryptography and computation. We have published widely in topics such as combinatorics over finite rings and fields, algebraic coding theory, coding, graphs & designs, the Discrete Log Problem, elliptic curve cryptography, network coding, APN functions, S-boxes in symmetric cryptosystems, Gröbner bases, finite geometry, number theory, group representations and applications to quantum computing. Beyond this, have strong interests in the application of discrete structures and methods in physics, chemistry, and life sciences.

**Dr Eimear Byrne**

**Research Interests**: Algebraic coding theory, codes over rings, codes and graphs, network coding, finite rings & fields, Groebner bases, decoding algorithms, APN functions, algebra in communications

**Dr Mark Dukes**

**Research Interests**: Discrete Mathematics, Combinatorics, Applications of discrete mathematics and Combinatorics to the study of complex systems

**Emeritus member**

**Assoc Prof Russell Higgs**

**Research Interests**: Representation Theory, Coding Theory, Wireless Sensor Network

**Dr Rupert Levene**

**Research Interests**: Functional Analysis, Operator Algebras, Operator Spaces, Quantum Information Theory

**Prof Gary McGuire**

**Research Interests**: Finite Mathematics, Coding, Cryptography, Computation, Algebraic Number Theory, Algebraic Geometry

**Dr John Sheekey****Research Interests**: Semifields, Nonassociative Algebras, Rank-Metric Coding, Random Network Coding, Tensors

**Post Docs**

**PhD Students**

Emrah Sercan Yilmaz

Stiofáin Fordham

Bayesian statistics is perhaps the oldest branch of statistics, tracing its roots to a paper from 1763 by a Presbyterian minister named Thomas Bayes. The method he came up with allows for the accumulation of information via a simple formula involving probability distributions.

Unfortunately putting the method into practice was hard because, whilst the formula was simple, the calculations involved for real world data sets were too taxing for pen-and-paper mathematicians. With the increase in computer power in the second half of the 20th century it finally became possible to use Bayes’ theorem. It has now become one of the most popular tools for probabilistic analysis, and has found particular favour as part of the big data and artificial intelligence revolution of the early 21st century.

Applications of Bayesian statistics are found in medical statistics and bioinformatics, manufacturing analytics, forecasting, machine learning and artificial intelligence, climate change, social network analysis and many other areas.

The group at UCD contains some of the world’s leading practitioners and proponents of Bayesian statistics with particular expertise in topics such as cluster analysis, model choice, model fitting, stochastic processes, and latent variable modelling. Bayesian statistics forms a core component of the Science Foundation Ireland funded Insight Centre for Data Analytics.

**Prof. Nial Friel**

**Research Interests**: Bayesian Statistics, Statistical Network Analysis, Monte Carlo Methods

**Dr Claire Gormley**

**Research Interests**: Statistical Methodology, Bayesian Methods, Applied Statistics

**Dr Michael Salter-Townshend**

**Research Interests** :Bayesian Statistics, Statistical Modelling, Statistical Genetics.

**Dr Isabella Gollini**

**Research Interests** :Statistical models, bayesian statistics, fast inferential approaches, statistical network analysis

**Assoc Prof Miguel Bustamante** **Research Interests**: Fluid Dynamics and Turbulence, Nonlinear PDEs, Hamiltonian Methods and Integrable Systems

**Dr Michelle Carey**

**Research Interests**: Functional Data Analysis, Differential Equations, Financial instruments, Biological systems and Climatology

**Prof Frederic Dias**

**Research Interests**: Partial Differential Equations, Free-surface Flows, Wave energy

**Prof Gary McGuire**

**Research Interests**: Finite Mathematics, Coding, Cryptography, Computation, Algebraic Number Theory, Algebraic Geometry

**Dr Lennon Ó'Náraigh**

**Research Interests**: Two-Phase Flows, Turbulence, Stability Theory, High-Performance Computing

**Prof Adrian Ottewill**

**Research Interests**: General relativity, Quantum field theory in curved space-time

**Assoc Prof Andrew Parnell**

**Research Interests**: Bayesian Statistics, Stochastic Processes, Climatology

**Dr Barry Wardell****Research Interests**: General Relativity, Gravitational Waves, Black Holes, Cosmic Strings, High-performance Computing

The beginnings of functional analysis go back to the end of the 19th century, and came about in response to questions emerging from other areas of mathematics such as linear algebra, differential equations, calculus of variations, approximation theory and integral equations. Many brilliant mathematicians contributed to its development, but arguably functional analysis emerged as a field in its own right in the 1920s, with the work of Stefan Banach and the Lwów School in Poland. Other major contributors include Hilbert, von Neumann, Grothendieck, and more recently Bourgain and Gowers. Loosely speaking, the subject is the study of linear spaces having infinitely many dimensions. Such things do not exist in reality of course, but intriguingly, it turns out that many mathematical phenomena that are motivated by real world problems, such as solutions of differential equations and wavefunctions in quantum mechanics, are best viewed in the context of these infinite dimensional spaces. The subject itself is an intricate blend of linear and abstract algebra, metric space theory, topology, set theory, combinatorics and probability.

The members of the UCD research group in functional analysis have a diverse range of interests, such as the geometry of Banach spaces, interactions with topology and set theory, operator algebras, infinite dimensional real and complex analysis, bounded symmetric domains and Jordan structures.

Modern applications of functional analysis are many and varied, including the axiomatic foundations of financial mathematics, the existence and the form of solutions to equations of infinitely many variables that arise in physical and engineering and financial models.

**Assoc Prof Pauline Mellon**

**Research Interests**: Complex Analysis, Symmetric Manifolds and Jordan Triple Systems

**Dr Rupert Levene**

**Research Interests**: Functional Analysis, Operator Algebras, Operator Spaces, Quantum Information Theory

**Dr Richard Smith**

**Research Interests**: Banach Space Geometry and Structure, and connections with Set theory and Topology

**Assoc Prof Christopher Boyd**

**Research Interests**: Geometry of Banach Spaces, Analytic Mappings on Infinite Dimensional Banach Spaces, Functional Analysis Methods in Function Spaces

**Assoc Prof Miguel Bustamante**

**Research Interests**: Fluid Dynamics and Turbulence, Nonlinear PDEs, Hamiltonian Methods and Integrable Systems

** Dr Marius Ghergu Research Interests: **Partial Differential Equations, Nonlinear Analysis, Potential Theory

**Dr Rupert Levene** **Research Interests**: Functional Analysis, Operator Algebras, Operator Spaces, Quantum Information Theory

**Prof Adrian Ottewill****Research Interests**: General relativity, Quantum field theory in curved space-time

**Dr Barry Wardell****Research interests**: General Relativity, Gravitational Waves, Black Holes, Cosmic Strings, High-performance Computing

The Mathematics Education Research Group in UCD focuses on mathematics education at the post-primary and university levels. We are interested in the professional development of mathematics educators at these levels, and also in post-primary initial teacher education with a focus on the development of Mathematical Knowledge for Teaching. At the university level, we conduct research into ways to teaching and learning learning in large first year mathematics classes, with an emphasis on the roles that technology, assessment and feedback can play. At post-primary level, we conduct research on mathematics teacher learning through lesson study and on the development of skills for teaching mathematics. Finally, we conduct research into the provision of mathematics support to students in the first two years of their university studies. With a Maths Support Centre that receives an average of 6,000 visits per year we are interested in identifying and recording the main areas that students experience difficulty with and feeding this back to lecturers in a way that is timely and informative.

**Dr Anthony Cronin**

**Research Interests**: Mathematics and statistics support at the university and pre-university (Foundation year) levels.

**Assoc Prof Maria Meehan**

**Research Interests**: Mathematics Education at the university level; supporting students in large first year mathematics courses; noticing as a form of professional development; mathematics support

**Dr Aoibhinn Ní Shúilleabháin**

**Research Interests**: Mathematics Education at post-primary level, initial teacher education, teacher professional development, Lesson Study, developing pedagogical content knowledge

**PhD Students**

Nuala Curley

Emma Howard

Cillian Copeland

Matrix theory is a more advanced version of linear algebra and has strong links with other mathematical disciplines such as graph theory, combinatorics, operator theory, geometry, mathematical biology. Applications of the area range from machine learning and big data, to engineering and biology.

Research in Matrix Analysis in the school covers a range of topics, including combinatorial matrix theory, spectral theory for positive matrices, quantum information theory, algebra lengths, and completion problems.

**Research Interests**: Functional Analysis, Operator Algebras, Operator Spaces, Quantum Information Theory

**Research Interests**: Theory and applications of nonnegative matrices, Combinatorial matrix theory, Algebra lengths, Power series with positive coefficients

**Dr Michelle Carey**

**Research Interests**: Functional Data Analysis, Differential Equations, Financial instruments, Biological systems and Climatology

**Assoc Prof Xuefeng Cui**

**Research Interests**: Climate Change, Land Use Change, Food Security, and Geoengineering

**Dr Conor Sweeeny**

**Research Interests**: Probabilistic Forecasting, Climate Modelling, Statistical Post-Processing

**PhD Students**

Eadaoin Doddy

Seanie Griffin

**Emeritis Staff**

Prof. Peter Lynch

Prof. Ray Bates

Ray McGrath

Number theory is one of the oldest and most vibrant branches of mathematics. It is unique in that it combines problems and easily formulated questions with truly deep and technical methods for addressing these questions. For example, Wiles’ spectacular proof of Fermat’s Last Theorem in 1994 incorporated many powerful techniques which have enriched the whole of mathematics.

The Number Theory Group in UCD has particular expertise in homology theory of special linear groups (Kevin Hutchinson) and q-series and modular forms (Robert Osburn). Our goal is to make UCD a national centre of excellence in this research area.

Number theory possesses an accessibility and applicability, both of which are characteristics of a subject which must play a central role for the future of mathematics in Ireland.

Number theory has enjoyed a long and fruitful interaction with many other areas in mathematics. For example, many number theoretic constructions find an application in cryptography, coding theory, and internet security. Elliptic curve cryptography, the Diffie-Hellman protocal via class groups, integer factorization, and RSA signature schemes are some examples of recent developments. Moreover, multinational technological companies in Ireland such as Zynga, Quest, HCL, Google, Intel, Amgen, Facebook and Dropbox benefit from interactions with Ph.D. graduates, postdocs and faculty in number theory.

Potential theory has its origins in gravitational and electrostatic problems of mathematical physics. Today it is an important field of mathematical research that has rich connections with complex function theory, partial differential equations, fluid flow, stochastic analysis, ergodic theory and dynamical systems, and approximation theory. These varied links are reflected in the research activity of the UCD Potential Theory group.

Stephen Gardiner has written monographs on classical potential theory and harmonic approximation, and also works on a range of problems concerning harmonic functions, free boundaries, and universal series expansions of holomorphic functions.

Hermann Render has published extensively on harmonic and polyharmonic functions, singularities of analytical solutions of PDEs, multivariate spline theory, wavelet analysis and quadrature formulae.

Marius Ghergu works at the interface of partial differential equations and potential theory. He has published monographs on isolated singularities in partial differential inequalities, nonlinear PDEs, and singular elliptic problems.

Neil Dobbs works in real and complex dynamics, examining the rich geometric structures and ergodic properties of such systems, and pursuing questions such as dependence of Hausdorff dimension of Julia sets on parameters.

**Dr Marius Ghergu**

**Research Interests**: Partial Differential Equations, Nonlinear Analysis, Potential Theory

Dr Hermann Render

Dr Hermann Render

**Research Interests**: Computational Harmonic Analysis, Complex Analysis, Geometric Modelling

**Dr Nina Snigireva**

**Research Interests**: Dynamical Systems (Ergodic Theory, Symbolic Dynamics), Fractal Geometry, Multifractal Analysis, Dimension Theory, Fourier Analysis

Within algebra we have researchers in matrix theory and real algebra.

Our in research Real Algebra includes real algebra and some connected areas, such as: quadratic and hermitian forms, algebras with involutions, applications to wireless communications, model theoretical methods, o-minimality.

Research in Matrix theory in the school has emphasis on problems with some positivity aspect, ranging from questions on the spectral properties of positive matrices, to investigating power sums with positive coefficients. Other topics investigated are the inverse eigenvalue problem for graphs, matrix completion problems, integral similarity of matrices**,** the Pascal matrix algebra lengths and stability of switched systems.

**Assoc Prof Thomas Unger**

**Research Interests**: Quadratic and Hermitian Forms,Algebras with Involution, their Applications in Real Algebra and Space-Time Coding

With the ever increasing deluge of genetic data, understanding the structures within the data is increasingly important. Statistical genetics has a broad range of application from quality control and processing of raw data to uncovering association between genes and disease. Population genetics, which studies the shared variation between members of a species, is of particular interest in our group as it can shed light on demographic history and natural selection. To this end, we are involved in building cutting edge statistical models that are informed by the mechanisms of mutation and recombination and can scale to large modern datasets.

The gene regulatory network (GRN) is a complex control system and plays a fundamental role in the physiological and development processes of living cells. Focusing on the ordinary differential equation (ODE) modeling approach, our group implements and develops a novel pipeline for constructing high-dimensional dynamic GRNs from genome-wide time course gene expression data. This pipeline consists of a five-step procedure, i.e., detection of temporally differentially expressed genes, clustering genes into functional modules, identification of network structure, parameter estimate refinement and functional enrichment analysis. Our group has published intensively in this area with applications in influenza infection, HIV and cancer.

**Dr Michelle Carey**

**Research Interests**: Functional Data Analysis, Differential Equations, Financial instruments, Biological systems and Climatology

**Dr Michael Salter-Townshend**** Research Interests** :Bayesian Statistics, Statistical Modelling, Statistical Genetics.

The Statistical Modelling group performs research in both applied and theoretical aspects of statistical modelling. Theoretical interests include spatial statistics, time series, mixed models and survival analysis. Applications include biomedical, clinical, health, environmental, agricultural, forestry and veterinary sciences and the group has published extensively in these areas.

**Dr Michelle Carey**

**Research Interests**: Functional Data Analysis, Differential Equations, Financial instruments, Biological systems and Climatology

**Dr Claire Gormley****Research Interests**: Statistical Methodology, Bayesian Methods, Applied Statistics

**Assoc Prof Gabrielle Kelly****Research Interests**:** **Spatial Modelling, Veterinary Epidemiology, Forest Analytics

**Assoc Prof Patrick Murphy**

**Research Interests**: Multivariate Time Series Analysis, Statistical Issues in Radiological Protection, Official Statistics, Electoral Behaviour, Survey Design and Analysis, Research in Statistics Education

**Dr Michael Salter-Townshend**** Research Interests** :Bayesian Statistics, Statistical Modelling, Statistical Genetics.

**PostDocs**

Mark O'Connell

**PhD Student**

Sen Hu

Fluid mechanics has fascinated researchers for centuries and continues to do so today. Fluid mechanics is a very rich topic, covering a wide variety of applications (meteorology, climate, ocean waves, aeronautics, biomechanics, food industry, ship industry, oil and gas). The problem of turbulence is one of the central problems in theoretical physics. While the theory of fully developed turbulence has been widely studied, the theory of wave turbulence has been less studied, partly because it developed later.

The Waves, Fluids and Turbulence Group in UCD has particular expertise in ocean waves (Frederic Dias), compressible fluid mechanics (Ted Cox), turbulence and dynamical systems (Miguel Bustamante) and multiphase flow (Lennon O’Naraigh). The group is heavily involved not only in fundamental theoretical descriptions of Fluid Mechanics, but also, in Computational Fluid Mechanics, and thereby interacts with major European supercomputing centres. Our goal is to make UCD an international centre of excellence in fluid mechanics.

The group is proud to have hosted two ERC grants, the ERC Advanced Grant MULTIWAVE on rogue waves and the ERC Proof of Concept WAVEMEASUREMENT on the measurement of extreme ocean waves.

Our group plays a key role in European initiatives promoting interactions between mathematics and industry, via membership of Bustamante and O'Naraigh in the management committee of the European COST action MI-NET ("Mathematics for Industry Network", TD-1409). We have developed a number of successful collaborations with industry involving fluid dynamics (RUSAL Aughinish, Wuppermann Steel, Analog Devices) and discrete mathematics (ENGINO).

Fluid mechanics has enjoyed a rich interaction with many other areas in mathematics (partial differential equations, statistics) and many other schools (geophysics, mechanical engineering, biology, physics). The applications covered in the UCD group range from marine renewable energy, a very promising topic in Ireland, to quantum superfluids, with applications ranging from terrestrial superfluidity to astrophysical dynamics.

**Assoc Prof Miguel Bustamante**

**Research Interests**: Fluid Dynamics and Turbulence, Nonlinear PDEs, Hamiltonian Methods and Integrable Systems

**Prof Frederic Dias**

**Research Interests**: Partial Differential Equations, Free-surface Flows, Wave energy

**Dr Lennon Ó'Náraigh**

**Research Interests**: Two-Phase Flows, Turbulence, Stability Theory, High-Performance Computing

**PhD Students**

Brendan Murray

Shane Walsh

Selma Shun

Andrew Gloster

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