# Welcome

Welcome to my personal website. Here, I mainly collect ideas related to my research and, more generally, interesting bits of mathematics and computer science and life.

## Research overview

My research is centered around the question of how “diagrams of manifolds” can be described in combinatorial and computable terms, and how this provides a new approach to the study of manifolds, spaces, and more generally higher categories. Such diagrams have provided useful conceptual tools in topological quantum field theory, quantum algebra, knot theory and related fields, but have resisted previous attempts of mathematical formalization. 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 combine classical notions of combinatorial topology with a new combinatorial approach to framings, to obtain a well-behaved and combinatorializable notion of “framed stratified space”. We show how this can be used to overcome a set of fundamental classical obstructions to the computable combinatorial representation of topological phenomena. We then formally introduced (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 programme

Framed combinatorial topology (FCT) is a novel combinatorial-topological theory describing phenomena arising at the intersection of higher algebra, stratified topology, and singularity theory (or “higher Morse theory”). Both shorter and longer term aims of the theory can be outlined as follows.

• FCT provides a brigde between combinatorial and geometric semantics for higher category theory (here, “geometric semantics” 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. This is meant to shine new light on open problems in differential topology, such as the smooth Poincare conjecture in dimension 4.
• Via FCT one obtains a combinatorial approach to singularity theory, and to “higher Morse theory”, and this overcomes difficulties in the differential foundations of both subjects. The classification of singularities in all dimensions is an open problem, the importance of which derives from its connection to the laws of dualizable objects.
• Eventually, FCT is meant to provide computable foundations to higher directed spaces, on one hand making the internal language of $$n$$-categories accessible, and on the other hand overcoming several obstructions to computable foundations present in the classical theory of undirected spaces and manifolds.

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 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.

### Open challenges

A list of current open problems is maintained here.

## News

### The Past

• 2 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”

## Comment policy

Comments are enabled for most online notes (and some other pages on this website). Relevant and respectful comments are most welcome. You can also write mathematics in the comments.