Professor Anthony O'Farrell
Subject Professor
Room: 30 Top Logic
Phone: +353 1 708 3914
Fax: +353 1 708 3913
Email: admin@maths.nuim.ie
Research Interests
Analysis. Singularities, Extensions, and Approximations. Dynamical Systems, Algebraic structures in Analysis and Geometry.
Mathematics is densely interconnected. Temperamentally, the centre of my interest is in analysis. I particularly like the amazing things that happen in Complex Analysis, which still seems magic to me, after all these years. I also like qualitative approximation problems. Most of the things I've studied grow out of these two areas, in some way. The interplay between algebraic structure and concepts from analysis is a theme that has proven enormously fruitful, and has a lot of energy left. The structures I have found most useful are algebras over the field of complex numbers. In the past few years, I've been exploring groups of maps. Geometric insight can also be brought to bear on problems in analysis (and algebra). I've seen that probability can provide the key to solving problems that don't seem to have any uncertainty about them. Intuition derived from the physics of Hamiltonian dynamical systems or of the Brownian motion have been valuable to me, even in situations which are very un-physical (e.g. involving two-dimensional time).
In recent years, my friends and I completed investigations into algebra-related problems involving real and complex analysis. These involved Nachbin's problem about the approximation of infinitely-differentiable functions, De Paepe's problem about the polynomial hulls of a certain surface in the space of two complex variables and the phenomenon of pervasiveness in real and complex function spaces, and problems about conjugacy in groups of maps, particulary the reversibility problem of when a map is conjugate to its own inverse.
At present I am learning some new things, writing some expositions, and thinking about reversible elements in groups of maps (related to reversible dynamical systems), rational approximation, and geometrical stuff. At most times I am happy to talk to anyone about analysis at the drop of a hat. Proposals for regular periodic bilateral conversations about analysis are normally accepted. I have discovered that it is usually better to say yes.
For potential research students, have a look at my list of project proposals (in pdf).