Future pathways in diagrammatic mathematics: from Computable Manifolds to Quantum Gravity

<! --- LETTERS ---> $$ \newcommand{\lR}{\mathbb{R}} $$ $$ \newcommand{\lZ}{\mathbb{Z}} $$ $$ \newcommand{\lN}{\mathbb{N}} $$ $$ \newcommand{\lI}{\mathbb{I}} $$ $$ \newcommand{\II}{\mathbb{I}} $$ $$ \newcommand{\bI}{\mathbf{I}} $$ $$ \newcommand{\bC}{\mathbf{C}} $$ $$ \newcommand{\bD}{\mathbf{D}} $$ $$ \newcommand{\bE}{\mathbf{E}} $$ $$ \newcommand{\bF}{\mathbf{F}} $$ $$ \newcommand{\iI}{\mathsf{I}} $$ $$ \newcommand{\iA}{\mathsf{A}} $$ $$ \newcommand{\iB}{\mathsf{B}} $$ $$ \newcommand{\iC}{\mathsf{C}} $$ $$ \newcommand{\iD}{\mathsf{D}} $$ $$ \newcommand{\iE}{\mathsf{E}} $$ $$ \newcommand{\iF}{\mathsf{F}} $$ $$ \newcommand{\iM}{\mathsf{M}} $$ <! --- SYMBOLS ---> $$ \newcommand{\eps}{\epsilon} $$ $$ \newcommand{\SetCat}{\mathrm{Set}} $$ $$ \newcommand{\Bool}{\mathrm{Bool}} $$ $$ \newcommand{\Pos}{\mathrm{Pos}} $$ $$ \newcommand{\lbl}{\mathsf{lbl}} $$ $$ \newcommand{\subdiv}{\mathsf{sd}} $$ $$ \newcommand{\Hom}{\mathrm{Hom}} $$ $$ \newcommand{\Fun}{\mathrm{Fun}} $$ $$ \newcommand{\Psh}{\mathrm{PSh}} $$ <! --- OPERATORS ---> $$ \newcommand{\dim}{\mathrm{dim}} $$ $$ \newcommand{\id}{\mathrm{id}} $$ $$ \newcommand{\src}{\partial^-} $$ $$ \newcommand{\tgt}{\partial^+} $$ $$ \newcommand{\und}[1]{\underline{#1}} $$ $$ \newcommand{\abs}[1]{\left|{#1}\right|} $$ $$ \newcommand{\op}{^{\mathrm{op}}} $$ $$ \newcommand{\inv}{^{-1}} $$ <! --- CATEGORIES, FUNCTORS AND CONSTRUCTIONS ---> $$ \newcommand{\ttr}[1]{\mathfrak{T}^{#1}} $$ $$ \newcommand{\lttr}[2]{\mathfrak{T}^{#1}{(#2)}} $$ $$ \newcommand{\truss}[1]{\mathsf{T}\!\mathsf{rs}_{#1}} $$ $$ \newcommand{\sctruss}[1]{\bar{\mathsf{T}}\!\mathsf{rs}_{#1}} $$ $$ \newcommand{\rotruss}[1]{\mathring{\mathsf{T}}\!\mathsf{rs}_{#1}} $$ $$\newcommand{\trussbun}[1]{\mathsf{TrsBun}_{#1}} $$ $$ \newcommand{\trusslbl}[1]{\mathsf{Lbl}\mathsf{T}\!\mathsf{rs}_{#1}} $$ $$ \newcommand{\blcat}[1]{\mathsf{B}\mathsf{lk}_{#1}} $$ $$ \newcommand{\blset}[1]{\mathsf{Blk}\mathsf{Set}_{#1}} $$ $$ \newcommand{\fcl}[2]{\chi^{#1}_{#2}} $$ $$ \newcommand{\Entr}{\mathsf{Entr}} $$ $$ \newcommand{\Exit}{\mathsf{Exit}} $$ <! --- ARROWS AND RELATIONS ---> $$ \newcommand{\fleq}{\preceq} $$ $$ \newcommand{\fles}{\prec} $$ $$ \newcommand{\xto}[1]{\xrightarrow{#1}} $$ $$ \newcommand{\xot}[1]{\xleftarrow{#1}} $$ $$ \newcommand{\ot}{\leftarrow} $$ $$ \newcommand{\toot}{\leftrightarrow} $$ $$ \newcommand{\imp}{\Rightarrow} $$ $$ \newcommand{\iff}{\Leftrightarrow} $$ $$ \newcommand{\iso}{\cong} $$ $$ \newcommand{\into}{\hookrightarrow} $$ $$ \newcommand{\proto}{\longrightarrow\mkern{-2.8ex}{\raisebox{.3ex}{$\tiny\vert$}}\mkern{2.5ex}} $$ <! --- DIAGRAM ARROWS ---> <! ---... should replace \rlap by \mathrlap, see KaTeX doc ---> $$ \newcommand{\ra}[1]{\xrightarrow{\ \ #1\ \ }\phantom{}\kern-1.5ex} $$ $$ \newcommand{\la}[1]{\xleftarrow{\ \ #1\ \ }\phantom{}\kern-1.5ex} $$ $$ \newcommand{\da}[1]{\downarrow\raisebox{.5ex}{\rlap{$\scriptstyle#1$}}} $$ $$ \newcommand{\dra}[1]{\raisebox{.2ex}{\llap{$\scriptstyle#1$}}\kern-.5ex\searrow}} $$ $$ \newcommand{\dla}[1]{\swarrow\kern-1ex\raisebox{.2ex}{\rlap{$\scriptstyle#1$}}} $$ $$ \newcommand{\ua}[1]{\uparrow\raisebox{.2ex}{\rlap{$\scriptstyle#1$}}} $$ $$ \newcommand{\ura}[1]{\raisebox{.2ex}{\llap{$\scriptstyle#1$}\kern-.5ex\nearrow}} $$ $$ \newcommand{\ula}[1]{\nwarrow\kern-1ex\raisebox{.2ex}{\rlap{$\scriptstyle#1$}}} $$

Abstract. This short note takes a birds-eye perspective on motivations for working as a researcher in mathematics and physics (which are first and foremost motivations personal to me, but I think they may be shared with many others). I briefly round up some (glimmers of) potentially interesting projects that I find fascinating. The note is written on the occasion on moving on to other endeavors.

Introduction

The term ‘grand unified theory’ (GUT) could be used to describe any of many previous developments in mathematics and physics, in which disparate frameworks were discovered to arise from the same underlying ideas. In practice, the term is nowadays mainly used to allude to the hunt for a unification of quantum theory and general relativity which started in the mid 20th century. This ‘hunt’ continues to be a tall order for researchers in the 21st century, bringing them often to the very frontiers of functioning mathematical theory. Yet, even if a satisfying unification is achieved (and I hope it will be) I wouldn’t place any bets on this being the ‘end of the story’ for theoretical physics. Indeed, while we are able to observe the quantum and GR realm (at least with the help of sophisticated tools) one would probably want to be careful in claiming that this is all there is to observe (or, at least, all there is).

While in searching for a GUT of physics we are thus confronted with our physical limitations, it seems without alternative that any formal theory describing how the universe computes itself (or parts of it) will be, in particular, formal. The subject of formal theories (or formal languages, systems, computations, etc.) is usually simply called ‘mathematics’ but that name doesn’t give any further explanation of what ‘formal’ means. I think the essence of the term ‘formal’ has, however, become much clearer over the past century, and with the rise of category theory. Indeed, category theory provides us with a new ‘global’ perspective on the elements of formal languages, and it helps to distill the essential properties of these elements. Moreover, and importantly, the related subject of type theory tells us how category theory itself can be turned into a formal language for mathematical foundations.

From a lens of category theory, a lot unification concerning the interaction of disparate mathematical frameworks such as geometry, topology, algebra, as well as logic becomes possible, but much remains to be understood. During my time at Oxford, I tried and try to address parts of this quest for unification, in particular, attempting to construct formal languages inspired by geometric principles (and, thereby, much less so by the traditions of human language). For more ideas related to this point see e.g. story 9, and maybe also the next note. This work has so far mainly revolved around formalizing and studying the notion of manifold diagrams, which you could call a tool for capturing the ‘geometry of composition’.

On manifold diagrams

Manifold diagrams generalize string and surface diagrams to higher dimensions. They have seen substantial development over the last 5 years and their theory, owing to the natural integration of geometric, topological and algebraic ideas, continues to produce a lot of elegant and interesting mathematics. Nonetheless, a ‘killer’ application of this theory to other areas of contemporary mathematics is still missing (this might be only a matter of time, or it might not, … or maybe we’ll have to wait for GPT-9). Personally, I always really liked working on manifold diagrams as it somehow felt like catching two birds with one stone.

• Firstly, the theory addresses interesting foundational questions spanning across geometry, topology and higher algebra, such as: do we really need topological machinery to easily talk about braids and their higher-dimensional analogs? Do we really need differential machinery to easily talk about manifold singularities? (And further down the line questions such as: what are the higher-dimensional generalizations of Gray-categories, and the geometric semantics of higher category theory?)

• Secondly, the theory studies structures that often appear to be relevant to mathematical physics, via the many interactions of Algebraic Topology and Quantum Field Theory or String theory. Importantly, it does so in (canonically) combinatorial, and often computable, terms. This reduction of geometry in physics to discrete structure is vaguely reminiscent of ideas going by names like ‘it from bit’, ‘computational universe’, and ‘digital physics’, or more peculiar approaches such as ‘causal set theory’ (here, ‘peculiar’ is not meant in any negative way … I would call manifold diagrams peculiar themselves at this point in time).

These two vague points aside, in this note, I wanted to briefly list my aspirations for the ‘manifold-diagrammatic’ line of research at the intersection of geometry and higher algebra. Many of the mentioned projects are still actively worked on by others in the field: in particular, Christopher Douglas, Jamie Vicary, David Reutter, … . If you are interested, you should not hesitate to contact these researchers.

Projects so far

I recently posted the following research summary on the Category Theory Zulip server. I’ll be lazy and just re-post it here.

For a while I’ve been thinking about the “geometry of composition”, or in more technical terms, the higher-dimensional analogs of string diagrams (see manifold diagram). There is a bigger story to be told here, about “geometric”, “topological” and “algebraic” models of higher structures, and I tried to sketch this story in a recent \(n\)-Category Café post. There are two distinct types of interactions in the geometry of composition, and one of my long-term goals is to understand these two parts individually and how they interplay.

• distant interactions: isotopies (also called homotopies, or just coherences) are higher-dimensional analogs of braids. Examples include, for instance, the braid, the Yang-Baxterator, the Zamolodchikov eversion, etc.
• local interactions: singularities are local neighborhoods of the critical points of strata in diagrams (more precisely, in tangle diagrams). Examples include, for instance, the cup/cap, the birth/death of a circle, the saddle, the monkey saddle, the swallowtail, etc. …

In combination, isotopies and singularities make up all of homotopical behaviour (such as the homotopy groups of spheres; e.g. the hopf fibration is a capped-off braid, and the quaterniotic hopf fibration is a ‘capped-off’ Zamolodchikov eversion). In general, homotopical behaviour is very complicated… I’d say much of Algebraic Topology is dedicated to finding patterns in it. However, individually (to the extend that they can be separated) isotopies and singularities don’t seem that complicated! Well, still complicated, but somehow tractably so. Indeed, both can be formalized in a few words in the framework of manifold diagrams, and due to the combinatorial classification of such diagrams, they can be worked with easily ‘by hand’ up to the higher low dimension range. Both appear to have connections to loads of existing math, but the challenge is to understand those connections.

While the above provides my motivation for studying manifold diagrams, here are some more concrete short-term projects I’ve been thinking about.

1. Genericity properties of manifold diagrams: show that, while by default manifold diagrams are weakly globular, generically they are strictly globular. (Project with Lukas Heidemann and Christopher Douglas)

2. Theory of diagrammatic computads: diagrammatic computads and their functors are easily defined, but describing the higher category of diagrammatic computads and its properties is work in progress. (Project with Lukas Heidemann)

3. Complexification: there are a more or less straight-forward complex analogs of meshes (the ‘regular cell structures’ of diagrams). The role of resulting notions of \(\mathbb{C}\)-tangle diagrams should be similar to the role of complex singularities in Arnold’s singularity theory. (Project with Christopher Douglas)

Main problems ahead

We can keep this short as, really, there are only two main problems: the respective classification of the two types of ‘behaviors’ pointed out above. We briefly address them below.

1. Classification of perturbation-stable isotopies. Using the geometry of manifold diagrams to define higher categories naturally yields ‘unbiased’ notions of composition since gluings of manifold diagrams themselves are unbiased in an appropriate sense (you could also say ‘associative’ in place of ‘unbiased’). This similarly affects the description of higher categorical coherences via isotopies of manifold diagrams. A priori there are infinitely many isotopies starting in dimension 3. However, the conjecture (alluded to, for instance, in the manifold diagram paper) is that there is a sub-class of so-called perturbation-stable isotopies. An example note that the ‘triple’ braid can be perturbed into three simple ‘binary’ braids; indeed, the latter is an elementary isotopy, which, as a categorical coherence, appears for instance in the definition of Gray categories.

2. Classification of perturbation-stable singularities. As explained in this blog article, at the root of the connection between geometry and higher algebra is the Thom-Pontryagin construction. In fact, classically, in place of relating higher morphisms and manifold diagrams, this relates invertible morphisms (namely, paths in spaces) and stratified tangles. (Up to putting these tangles into generic position) stratified tangles are closely related to manifold diagrams: in fact, any tangle can be uniquely refined by a manifold diagram by its ‘higher critical points’, also called ‘singularities’, see here and here for details. In order to classify singularities, we must once more restrict our attention to the perturbation-stable case. Similar to the case of isotopies, there is a wealth structure to be explored here: in the linked paper, we made the ‘ADE pattern conjecture’ that perturbation-stable singularities found in smooth tangles (embedded in codimension 1) are governed by the classical ADE classification of Arnold. (The ADE pattern is, in the words of Terry Gannon, a far-reaching ‘meta-pattern’.)

As mentioned in the the previous section, an incredible amount of behaviour ‘emerges’ when combining elementary isotopies and singularities; namely, all homotopical behaviour! (Hopf fibrations, homotopy groups of spheres, Bott periodicity, … you name it). A fun simple example to consider here is the computation of $\pi_3 S^2$ in manifold diagrams. But, given homotopical behaviour is complicated, the following word of warning will not come as a surprise: both points (1.) and (2.) above are not easy. In fact, Chris Douglas and I discussed that we expect (2.) to be closely related to the smooth Poincaré conjecture in dimension 4… one of the big(gest) open problems in geometric topology!

Final thoughts

Who knows where this line of research will go. It would be great if, say, in 20 years from now, we wouldn’t think of Morse and singularity theory as something inherent to differential geometry any more, but understand that it’s simply another facet of a deeper higher-categorical idea. This and other unifications do seem within reach. And, yet more speculative, I’d even hope that this line of research will then ultimately affect our understanding of ‘grand unified physics’, and maybe we are already witnessing the beginning of that.

So long, and thanks for all the fish!