Functional Analysis

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. Exciting new avenues of application are currently being driven by research in Quantum Information theory, Machine Learning and Artificial Intelligence.

Potential 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.

Dynamical Systems

Dynamical Systems are studied to describe, qualitatively, systems which evolve with time. The foundations were laid by Poincaré and by Fatou and Julia over a hundred years ago. Close links have since been established with Analysis, Topology, Probability and Number Theory, while inspiration frequently comes from mathematical physics. Current expertise of the research group lies in thermodynamic formalism, ergodic theory and one-dimensional real and complex dynamics.