Most of my work is organised around a single question: what is the right (geo)metric structure in which to place a given analytic object — and what survives when that structure is allowed to degenerate? The topics below are different incarnations of it.
Gromov's work on the convergence of metric structures, and the theory of Lott, Sturm and Villani, made it possible to do geometry on spaces that are not smooth. Recent developments in general relativity suggest that spacetime need not be smooth either, which raises a concrete question: what is the correct notion of convergence for Lorentzian spaces, and what is the analogue of a compactness theorem in that setting? My work on the recently developed directed completion and on Lorentz \(p\)-harmonic functions is aimed at the objects one would need in order to state such a theorem.
Fractional Sobolev seminorms interpolate between \(L^p\) and the classical Sobolev scale, and their behaviour at the endpoints encodes geometric information: the Maz'ya–Shaposhnikova formula describes the limit \(s \to 0\), where fractional perimeters degenerate under long-range interactions. I am interested in this limit for very general classes of functions, and in its unexpected relation to the Gromov compactification of metric spaces on a logarithmic scale.
Linear elliptic equations with measurable coefficients live naturally in Sobolev spaces, and the qualitative behaviour of their solutions is governed by the regularity of the data. The same patterns recur in nonlinear systems, in parabolic problems and in \(X\)-elliptic equations — which is what makes the low-regularity setting worth understanding on its own terms.
I also work on the analysis of cardiac magnetic signals recorded by SQUID arrays. Reconstructing current sources from the measured field is an ill-posed inverse problem, and making it well-posed requires choosing the right space in which the source is assumed to live — a question of the same nature as the ones above, asked about a heart instead of a spacetime. This line of work also involves heart rate variability and autonomic modulation indices.