Welcome

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.

Research interests

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.

FCT program

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.

  1. A(nother) resolution to the problem of undefinability of computadic cell shapes, formalized using the combinatorics of trusses.
  2. 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.
  3. A formalization of the idea of constructing higher categorical coherences via isotopies.
  4. A combinatorial toolset for studying tangles, singularities and higher Morse theory.

Next in line: a combinatorialization of smooth structure and diffeomorphism.

News

The Past

The future

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