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.
More recently, I have investigated a problem in operator theory which concerned the properties of orbits of points under operators.
Some interesting things that do not exist, UCD, October 2012 (general interest talk)
Strictly convex norms and topology, New York, July 2011
A festive look at the Szlenk index, UCD, December 2011