Welcome to my personal website, which is permanently “under construction”, and where I collect some things that interest me, some things about me, and some completely unrelated things.
My research is centered around the diagrammatic calculus of embedded (stratified) manifolds, and the question of how this calculus can be described in combinatorial and computable terms. Such diagrams generalize, for instance, string diagrams or knot diagrams. My research aims to provide new approaches to many questions in the study of manifolds, spaces, and more generally higher categories.
My thesis introduced an inductive combinatorial data structure, intended to work as a uniform combinatorial description of “manifold diagrams” across all dimensions. In recent work with Christopher Douglas this ambition was made precise as follows. In our book on “framed combinatorial topology” we introduced a (combinatorializable) notion of “framed stratified space”. Using this, we then defined combinatorial-topological notions of “manifold diagrams and tame tangles” in a recent same-named paper. The work posits that manifold diagrams provide a language linking combinatorial and smooth topological phenomena. It sets out a series of “smooth combinatorialization conjectures”, that will guide further research.
Overview and aims
Framed combinatorial topology (apart from being the title of a book) is a research program concerned with finding the “right” foundations for many phenomena arising at the intersection of higher algebra, stratified topology, and singularity theory. Both shorter and longer term aims of the theory can be outlined as follows.
- FCT provides a brigde between combinatorial and geometric models for higher category theory (here, “geometric model” means interpreting higher categories as directed spaces). In doing so, FCT provides a novel approach to the construction of higher categorical coherences: namely, via a (computable) notion of isotopy in manifold diagrams.
- The setting of FCT is meant to link “tame” topology, PL topology and smooth topology, by providing a language that can describe the essential phenomena of all three of these areas. In the long run, this is meant to shine new light on open problems in differential topology, such as the smooth Poincare conjecture in dimension 4.
- More generally, via FCT one obtains a novel, fully combinatorial approach to singularity theory (or, to “higher Morse theory”) and this overcomes certain difficulties in the differential foundations of the subject. The importance of this approach derives from its connection to the laws of dualizable objects: I hope that understanding higher-parameter families of singularities better will eventually lead to a constructive resolution of the cobordism hypothesis.
Further reading on FCT and its aims can be found here.
Some research outcomes
FCT addresses several problems of previous approaches to describing the combinatorial interplay between higher algebra and the geometry of (stratified) manifolds.
- A(nother) resolution to the problem of undefinability of computadic cell shapes, formalized using the combinatorics of trusses.
- A formalization of the notion of manifold diagrams, both geometrically and combinatorially, which generalizes string diagrams to higher dimension and provides a Poincaré dual to pasting diagrams of computadic cells.
- A formalization of the idea of constructing higher categorical coherences via isotopies.
- A combinatorial toolset for studying tangles, singularities and higher Morse theory.
Next in line: a combinatorialization of smooth structure and diffeomorphism.
- 10 January 2023. Talk at the QTCat Seminar in Hamburg on “Geometric Higher Categories”. handwritten notes
- 09 Nov 2022. Oxford mathematics research case study and corresponding tweet (with video) about our work
- 04 Nov 2022. Talk at Oxford Maths Fridays@4 on “Illustrating Mathematics”. Follow-along document Example Files
- 25 Oct 2022. Drafted two (fun) new notes: “towards computable manifolds” and “principles of higher-dimensional logic”.
- 13 Sep 2022. New paper draft: “A brief introduction to framed combinatorial topology”.
- 30 Aug 2022. Chris Douglas and I uploaded the “Manifold diagrams and tame tangles” preprint. It is accompanied by a note on the D4 singularity.
- 02 Jun 2022. I gave a talk at the Geometric Structures Lab at the Fields Institute, University of Toronto. Here’s the video (password: #d8=Wm40vV where # is the lowest dimension in which exotic smooth spheres have been constructed).
- 16 Mar 2022. I gave a talk at the TQFT Club at the University of Lisbon. Here’s the video.
- 29 Dec 2021. Chris Douglas and I uploaded the FCT book preprint
- 22 Nov 2021. I gave a talk in the Advanced topology class on “Generalized differential cohomology theories”.
- Rest of 2022. preparing one preprint with Lukas Heidemann and Christopher Douglas