Christopher Boyd   Associate Professor

Functional Analysis,Infinite Dimensional Holomorphy, Geometry of Banach spaces, Analytic semigroups, Operator Ideals, Locally convex spaces,




Neil Dobbs   Assistant Professor

Dynamical systems is the qualitative study of long-term behaviour of systems which evolve with time. My research is primarily in the field of real and complex one-dimensional dynamics, examining the rich geometric structures and ergodic properties of such systems. Questions such as dependence of Hausdorff dimension of Julia sets on parameters, existence of equlibrium states, dependence of equilibrium states on parameters are current topics of research.

Marius Ghergu   Associate Professor

My research lies at the interface between Nonlinear Analysis, Potential Theory and Partial Differential Equations. Particularly, I am interested in the study of classical solutions for elliptic and parabolic problems.

Here are several directions of research I have investigated in the past few years:
- Isolated singularities of solutions for differential equations and systems;
- Isolated singularities in Choquard type equations and inequalities;
- Bergman analytic content and isoperimetric inequalities;
- Reaction-Diffusion systems;
- Applications of Dynamical Systems methods in the study of asymptotic of differential equations with gradient terms;
- Elliptic problems involving the polyharmonic operator.

Rupert Levene   Assistant Professor

My research involves the study of operator algebras, operator spaces and quantum information theory. In particular, I am interested in: Reflexivity and hyperreflexivity: a (non-selfadjoint) operator algebra is reflexive if, loosely speaking, it has a lot of invariant subspaces. It is hyperreflexive if the distance to the algebra can be estimated using its invariant subspaces; this is stronger than reflexivity. These properties have natural generalisations from operator algebras to operator spaces, and a key question is to try to determine which operator spaces are reflexive and which are hyperreflexive. Completely bounded mappings between operator spaces and their norms: completely bounded mappings are at the heart of the theory of operator spaces. A linear map between operator spaces comes equipped with two natural norms: the operator norm, and the completely bounded norm. One fundamental problem is to determine when one of these two norms may be estimated using the other, or when they are equal. Schur multipliers: these completely bounded mappings have many attractive properties; for example, their norm and completely bounded norm always agree. However, there are many interesting open questions about this class of mappings, such as: what are the possible values of the norm of an idempotent Schur multiplier? Quantum information theory tackles problems of fundamental importance to the success of quantum computing, a technology still very much in its infancy. There turn out to be intimate connections with the theory of completely bounded maps. My work to date has focussed on privacy and correctability in the infinite dimensional setting.

Michael Mackey   Assistant Professor

My main area of research is in JB*-triples. These are the complex Banach spaces whose unit ball has a high degree of holomorphic symmetry. They include Hilbert spaces and C*-algebras which are familiar structures but usually considered to be very different from one another. Every JB*-triple possesses a "triple product" meaning we can multiply elements "three at a time". The theory of JB*-triples showcases an amazing interplay of complex and functional analysis, operator algebras and non-associative algebra.

Myrto Manolaki   Assistant Professor

Complex Analysis, Potential Theory, Approximation, Universality, Spaces of Analytic Functions, Several Complex Variables



Hermann Render   Assistant Professor

My research area refers to several different subjects in Pure and Applied Mathematics. Generally speaking, I am interested in applications of mathematics which are based on methods developed in the theory of Partial Differential Equations (PDE). My interests focus on: Harmonic and polyharmonic functions Potential theory Singularities of analytical solutions of PDE's Multivariate Spline Theory Wavelet Analysis and Sampling Theory Quadratures, cubatures and moment problems Geometric Modeling The Theory of PDE's is a vast area which has undergone several changes in the last decades. My interests ly in analytical aspects in this theory, in particular a better understanding of the analytical behaviour of solutions for analytical data functions. In this connection I solved positively in 2008 the Khavinson-Shapiro conjecture for a large class of domains saying that ellipsoids are the the only bounded domains in Euclidean space in this class allowing entire solutions for the Dirichlet problem for entire data functions. This result was published in 2008 in a top mathematical journal (Duke Math. Journal, rank 10 of 280). A cooperation with Prof. P. Ebenfelt (University of California San Diego) has lead to two more important papers on a related subject. Quadrature and cubature formula are simple rules for approximating integrals, a basic and important theme in numerical analysis. In cooperation with Prof. Kounchev (Bulgarian Academy of Sciences) we developed a new method based on Fourier-Laplace Expansion and univariate quadrature formula for Chebyshev systems. A new direction of research are so-called polyharmonic (multivariate) subdivision schemes which are based on nonstationary subdivision schemes for exponential polynomials. In cooperation with two leading experts, Prof. Nira Dyn and Prof. David Levin (Tel Aviv University) and Prof. Kounchev (Bulgarian Academy of Sciences) we have completed two papers on this subject (supported by the Seed Funding Scheme).

Richard Smith   Assistant Professor

I have interests in functional analysis and topology, principally the geometry of Banach spaces. The natural norm on a general Banach space need not have any good geometrical properties. However, sometimes it is possible to introduce a new norm, topologically equivalent to the original norm, but with, for example, some property of smoothness or strict convexity. In this way, we can improve the geometry of the space. In general, it is very difficult to decide whether or not such norms can be found, so we try to characterise their existence in terms of more easily determined linear and topological properties of the given space. The techniques involved in this endeavour are wide ranging, and as well as requiring deep results from functional analysis and topology, they bring to bear elements of set theory, combinatorics, smooth function theory, variational methods and much more. I have also published papers on linear dynamics, tensor products, topological properties and Lipschitz-free spaces, and have done recent work applying functional analytic techniques to the theory of partial differential equations. More details of my research interests can be found at http://www.maths.ucd.ie/~rsmith .

Stephen Gardiner   Emeritus Full Professor

Stephen Gardiner's research interests are in potential theory, function theory and approximation by harmonic functions. He has authored over 100 scientific publications in leading journals and books and has written the research monograph Harmonic Approximation (Cambridge University Press, 1995). His second book, Classical Potential Theory, written jointly with Professor David Armitage, is in the Springer Monographs in Mathematics Series (352pp., Springer, London, 2001). In 1995 he was awarded the degree of Doctor of Science by Queen's University on the basis of published research. In 2000 he was elected a Member of the Royal Irish Academy. He spent the year 1992 on sabbatical leave at McGill University, Montreal, and visited Charles University, Prague for the autumn term in 2005. Other assignments outside the university include lecture series in Sweden (1996), Japan (1997), the Czech Republic (1998), Canada (2000), Russia (2001) and Hungary (2009), and collaborative research projects at the Oberwolfach Mathematical Research Institute (1995, 1997, 1999) and the Mittag-Leffler Institute (1999). Other awards include a UCD President's Research Fellowship (2005/06), and an Erskine Fellowship at the University of Canterbury, New Zealand (2006). He served as the Irish co-ordinator of the European Union Research Training Network 'Classical Analysis, Operator Theory, Geometry of Banach Spaces, their Interplay and Applications' (2000-2004) and as a member of the Steering Committee of the European Science Foundation Research Network 'Harmonic and Complex Analysis and its Applications' (2007-2012). He was Principal Investigator of the Science Foundation Ireland Research Frontiers projects 'Potential theory and quadrature domains' (2006-2010) and 'Universality, approximation and potential theory' (2009-2013). He served from 2001-2010 as an Editorial Adviser for the Proceedings, Journal and Bulletin of the London Mathematical Society and is at present on the Editorial Board of the Springer journal Potential Analysis .

Pauline Mellon   Emeritus Professor

There are two very different strands to Prof Mellon's research, the long-standing one is related to the study of symmetric spaces and involves complex analysis of (possibly infinite dimensional) Banach spaces. The other, and more recent strand is in Evolution Algebras. Surprisingly, there is a common thread between these two strands, in that both involve (different) underlying non-associative algebraic structures.

Prof Mellon's research interests in complex analysis primarily involves spaces with symmetry. These correspond to the Banach spaces whose open unit balls have a transitive group of biholomorphic automorphisms, known as the JB*-triples. As such, the area straddles complex analysis, functional analysis and differential geometry. Prof. Mellon has studied these manifolds from several different perspectives: their curvature, the class of holomorphic maps that they admit and differences between the finite and infinite dimensional cases. She has published extensively on JB*-triples. Recent work includes versions of Wolff's classical theorem on these spaces (concerning invariant domains of holomorphic mappings) and work on composition operators. She has several collaborations in this area with C.H. Chu (University College London), and with S. Dineen and M. Mackey of UCD.

More recently, Prof Mellon has also worked in the area of Evolution algebras. Mendel’s Laws (1856) describing genetic inheritance have been given a mathematical formulation in terms of non-associative algebras and these so-called ‘Genetic Algebras’ have provided a framework for studying various types of inheritance. Certain genetic phenomena, however, for example incomplete dominance, do not follow Mendel's laws and evolution algebras were introduced relatively recently in 2006 as an attempt to study such non-Mendelian behaviour. Dr Mellon has collaborated with M.V. Velasco-Collado of the University of Granada and M. Bustamante of UCD on this topic.