Research
My research is in algebraic and geometric topology. I like to answer questions about natural objects arising in geometry and topology, usually using ideas and tools from homotopy theory. The objects of interest are often moduli spaces---like configuration spaces, or the space of all submanifolds of R∞ diffeomorphic to some fixed manifold M, or the space of all positive scalar curvature metrics on some manifold M, or the space of all embeddings of one manifold into another---or symmetry groups like diffeomorphism groups, general linear groups, symplectic groups, or braid groups. These examples all have a lot to do with each other. My overriding interest is the common structural features that these kinds of examples have: for example, their homology often stabilises as the "complexity" grows, and we often understand the limiting homology very well. See the survey Moduli spaces of manifolds: a user’s guide for more on this in the case of diffeomorphism groups.
Most recently I have been working to understand low-complexity examples, namely: what is the topology of the group Diff(Dd) of diffeomorphisms of the disc Dd? This turns out to be more or less the same question as: what is the topology of the group Top(d) of homeomorphisms of Rd? This seems like a basic question in topology, and has a surprisingly rich structure with connections to algebraic K-theory, graph complexes, the little disc operads, topological cyclic homology, and much else. See the survey Diffeomorphisms of discs for more on this.
What you work on as a PhD student with me will probably be adjacent to my own research. I don't fix PhD projects in advance: typically my students spend the first term reading background material, and we develop a direction together during that time, shaped by their interests and what seems tractable.
Background
A solid grounding in algebraic topology, for example at the level of Hatcher’s book, is sufficient preparation. Students coming from Cambridge’s Part III Algebraic Topology course are well-prepared, though in that case I would expect you to also write your Part III essay with me.
The quality I look for most is not a taste for developing abstract theory, but rather the ability to take the standard tools of algebraic topology and apply them flexibly to new and concrete examples in order to answer specific questions. If your tastes run more towards theory-building then another advisor would most likely be more appropriate.
Application timeline
For up to date details of the application process see here. The typical timeline is:
- Early January: general application deadline for October admission and funding consideration
- Late January: interviews
- Mid-February onwards: decisions communicated
I am always interested in taking new PhD students, and if some of the topics described above interest you then I encourage you to apply. If you have specific questions about applying please ask, but note that there is no point in e-mailing me your application materials to me as I will not look at them: I look at all applications straight after the deadline in early January.
Current and past PhD students
The best guide as to the kind of mathematics that you might work on under my supervision is what my previous students have done.Matteo Poletto.
Samuel Muñoz-Echániz, Spaces of diffeomorphisms and embeddings via algebraic K-theory, University of Cambridge, 2025.
Ismael Sierra, Homological stability of spaces of manifolds via Ek-algebras, University of Cambridge, 2023.
Andreas Stavrou, Homology of Configuration Spaces of Surfaces as Mapping Class Group Representations, University of Cambridge, 2023.
Nils Prigge, On tautological classes of fibre bundles and self-embedding calculus, University of Cambridge, 2020.
Nina Friedrich, Automorphism Groups of Quadratic Modules and Manifolds, University of Cambridge, 2017.
Mauricio Gómez López, Spaces of piecewise linear manifolds, University of Copenhagen, 2014.
Federico Cantero Morán, Homology Stability for Spaces of Surfaces, Universitat de Barcelona, 2013.