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Research
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.
Selected Talks
Convergence analysis of the Geometric Thin-Film Equation, UCD, December 2022
Lipschitz-free spaces over unit spheres and the Metric Approximation Property, University of Murcia, March 2022
A topological characterization of dual strict convexity in Asplund spaces, Mathematical Institute, Czech Academy of Sciences, December 2019
Topology and the approximation of norms, CIRM, Luminy, March 2018
Stranger things – the fractal monsters that lurk between dimensions, UCD, October 2017 (general interest talk)
Some interesting things that do not exist, UCD, October 2015 (general interest talk)
Strictly convex norms and topology, CUNY, New York, July 2011
A festive look at the Szlenk index, UCD, December 2011
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