Notes

On pattern matching, computation and (geometric) dependent type theory

Comments first published 8 Apr 2023

Pattern matching is an ubiquitous tool in theoretical computer science and type-theoretical mathematics, as well as the formalization of formal languages themselves. We briefly review its role in these subjects, giving examples along the way. We then motivate a deeper connection between pattern matching and higher-dimensional geometry, and suggest how geometric type theory, while not a real type theory yet, tries to provide a general formal framework in which mathematics can be built from higher-dimensional patterns. → read note

Unification in Mathematics: from Diagrams to Quantum Gravity

Comments first published 7 Apr 2023

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. → read note

An invitation to geometric higher categories (external link) Permalink

first published 13 Mar 2023

Fast and mathematically self-contained introduction to basic ideas in geometric higher category theory, written up as an article for the n-Category Café. → go to link

Mazur manifold as tangle diagram

Comments first published 21 Feb 2023

Mazur manifolds are intimately related to homology spheres, and are part of the puzzling world of 4-manifolds. Here we construct the simplest example of a Mazur manifold as a tangle diagram. → read note

Principles of higher-dimensional logic

Comments first published 26 Oct 2022

We allow ourselves to speculate a bit about how the 1-dimensional logical constructs that one frequently encounters in practical approaches to mathematical foundations may in fact emerge from a set of “more fundamental” principles of compositionality in higher-dimensional logic. → read note

Towards computable manifolds

Comments first published 26 Oct 2022

In this (exceedingly brief) note, we discuss computability questions in the context of manifold theory. The main observation is that several computabilty issues in classical “set-point (differential) topological” foundations may potentially be overcome when working with manifolds from the higher-categorical perspective of manifold diagrams and tangles. → read note

The $D_4$ dualizability law

Comments first published 26 Jun 2022

This note gives a brief visual guide to how one can understand the classical \(D_4\) singularity in terms of a categorical pasting diagram (aka manifold diagram, or more specifically, a tame tangle). → read note

The categorical Pontryagin-Thom construction

Comments first published 8 Mar 2022

The Pontryagin-Thom construction is a deep relation between smooth manifold theory and homotopy theory. We use the language of manifold diagrams and computads to give a purely combinatorial perspective on the construction. On our way, we’ll discuss computadic functors and transformations, and how the combinatorics of manifold diagrams may provide a “combinatorialization” of smooth structures on manifolds altogether. → read note

From trusses to geometric computads to combinatorial manifold diagrams

Comments first published 24 Feb 2022

This note gives a fast introduction to the rich combinatorial theory of trusses. We also discuss how presheafs of truss blocks give rise to geometric computads, and how this can be used to understand manifold diagrams in purely combinatorial terms. → read note

Tame tangles, singularities, and Morse theory

Comments first published 29 Jan 2022

Classical singularity theory and higher category theory have tantalizing connections: via the generalized tangle hypothesis we can understand singularities as expressing categorical laws satisfied by dualizable objects. However, classical singularity theory, based in differential-topological foundations, encounters technical problem in higher dimensions, which stalls this connection. Using our technology of manifold diagrams we outline steps towards remedying the problem, paving the way towards “higher” Morse theory. → read note

Paradigms of higher category theory

Comments first published 21 Jan 2022

We explore two fundamental paradigms underlying models of higher categories. Most models for higher categories are based on the paradigm of contraction; this requires spaces of evaluations of a single pasting diagram to be contractible. The paradigm, however, generally fails to yield easy descriptions of interesting non-contractible behaviour in spaces of composites. This is remedied by working instead with the paradigm of isotopy as we will explain. → read note

Manifold diagrams: a geometric approach

Comments first published 16 Jan 2022

We describe the notion of manifold diagrams. Manifold diagrams generalize string diagrams to higher dimensions. Our focus will lie on giving an intuitively clear geometric description of these diagrams in terms of conical stratifications in a framed background space. We will also provide further details on the dual notion of pasting diagrams of computadic cells. → read note

The class of computadic cells

Comments first published 14 Jan 2022

We describe the problem of defining a “universal” class of cell shapes for higher categories, the central obstruction in the classical approach to the question, and how to overcome that obstruction in the context of framed combinatorial topology. → read note

Framed Combinatorial Topology Permalink

`Framed combinatorial topology’ is the title of a book we just uploaded to the arXiv. On the linked page you’ll find more information! → go to link

An invitation to geometric higher categories (external link) Permalink

first published 13 Mar 2023

Fast and mathematically self-contained introduction to basic ideas in geometric higher category theory, written up as an article for the n-Category Café. → go to link

Mazur manifold as tangle diagram

Comments first published 21 Feb 2023

Mazur manifolds are intimately related to homology spheres, and are part of the puzzling world of 4-manifolds. Here we construct the simplest example of a Mazur manifold as a tangle diagram. → read note

Towards computable manifolds

Comments first published 26 Oct 2022

In this (exceedingly brief) note, we discuss computability questions in the context of manifold theory. The main observation is that several computabilty issues in classical “set-point (differential) topological” foundations may potentially be overcome when working with manifolds from the higher-categorical perspective of manifold diagrams and tangles. → read note

The $D_4$ dualizability law

Comments first published 26 Jun 2022

This note gives a brief visual guide to how one can understand the classical \(D_4\) singularity in terms of a categorical pasting diagram (aka manifold diagram, or more specifically, a tame tangle). → read note

The categorical Pontryagin-Thom construction

Comments first published 8 Mar 2022

The Pontryagin-Thom construction is a deep relation between smooth manifold theory and homotopy theory. We use the language of manifold diagrams and computads to give a purely combinatorial perspective on the construction. On our way, we’ll discuss computadic functors and transformations, and how the combinatorics of manifold diagrams may provide a “combinatorialization” of smooth structures on manifolds altogether. → read note

From trusses to geometric computads to combinatorial manifold diagrams

Comments first published 24 Feb 2022

This note gives a fast introduction to the rich combinatorial theory of trusses. We also discuss how presheafs of truss blocks give rise to geometric computads, and how this can be used to understand manifold diagrams in purely combinatorial terms. → read note

Tame tangles, singularities, and Morse theory

Comments first published 29 Jan 2022

Classical singularity theory and higher category theory have tantalizing connections: via the generalized tangle hypothesis we can understand singularities as expressing categorical laws satisfied by dualizable objects. However, classical singularity theory, based in differential-topological foundations, encounters technical problem in higher dimensions, which stalls this connection. Using our technology of manifold diagrams we outline steps towards remedying the problem, paving the way towards “higher” Morse theory. → read note

Manifold diagrams: a geometric approach

Comments first published 16 Jan 2022

We describe the notion of manifold diagrams. Manifold diagrams generalize string diagrams to higher dimensions. Our focus will lie on giving an intuitively clear geometric description of these diagrams in terms of conical stratifications in a framed background space. We will also provide further details on the dual notion of pasting diagrams of computadic cells. → read note

The class of computadic cells

Comments first published 14 Jan 2022

We describe the problem of defining a “universal” class of cell shapes for higher categories, the central obstruction in the classical approach to the question, and how to overcome that obstruction in the context of framed combinatorial topology. → read note

Framed Combinatorial Topology Permalink

`Framed combinatorial topology’ is the title of a book we just uploaded to the arXiv. On the linked page you’ll find more information! → go to link

Enriched cohesive infinity toposes

Comments first published 30 Nov 2021

I gave a talk in the Advanced Topology Class at Oxford about “Generalized Differential Cohomology” based on this book. Part of the talk required setting up an “enriched cohesive \(\infty\)-topos”. Here’s an outline of what that object could be and what it does. → read note

Factorization homology

Comments first published 2 Oct 2019

I gave a talk at CUNY about factorization homology, and have written up some notes which I’m posting here. → read note

On pattern matching, computation and (geometric) dependent type theory

Comments first published 8 Apr 2023

Pattern matching is an ubiquitous tool in theoretical computer science and type-theoretical mathematics, as well as the formalization of formal languages themselves. We briefly review its role in these subjects, giving examples along the way. We then motivate a deeper connection between pattern matching and higher-dimensional geometry, and suggest how geometric type theory, while not a real type theory yet, tries to provide a general formal framework in which mathematics can be built from higher-dimensional patterns. → read note

Unification in Mathematics: from Diagrams to Quantum Gravity

Comments first published 7 Apr 2023

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. → read note

An invitation to geometric higher categories (external link) Permalink

first published 13 Mar 2023

Fast and mathematically self-contained introduction to basic ideas in geometric higher category theory, written up as an article for the n-Category Café. → go to link

Principles of higher-dimensional logic

Comments first published 26 Oct 2022

We allow ourselves to speculate a bit about how the 1-dimensional logical constructs that one frequently encounters in practical approaches to mathematical foundations may in fact emerge from a set of “more fundamental” principles of compositionality in higher-dimensional logic. → read note

Towards computable manifolds

Comments first published 26 Oct 2022

In this (exceedingly brief) note, we discuss computability questions in the context of manifold theory. The main observation is that several computabilty issues in classical “set-point (differential) topological” foundations may potentially be overcome when working with manifolds from the higher-categorical perspective of manifold diagrams and tangles. → read note

The categorical Pontryagin-Thom construction

Comments first published 8 Mar 2022

The Pontryagin-Thom construction is a deep relation between smooth manifold theory and homotopy theory. We use the language of manifold diagrams and computads to give a purely combinatorial perspective on the construction. On our way, we’ll discuss computadic functors and transformations, and how the combinatorics of manifold diagrams may provide a “combinatorialization” of smooth structures on manifolds altogether. → read note

From trusses to geometric computads to combinatorial manifold diagrams

Comments first published 24 Feb 2022

This note gives a fast introduction to the rich combinatorial theory of trusses. We also discuss how presheafs of truss blocks give rise to geometric computads, and how this can be used to understand manifold diagrams in purely combinatorial terms. → read note

Paradigms of higher category theory

Comments first published 21 Jan 2022

We explore two fundamental paradigms underlying models of higher categories. Most models for higher categories are based on the paradigm of contraction; this requires spaces of evaluations of a single pasting diagram to be contractible. The paradigm, however, generally fails to yield easy descriptions of interesting non-contractible behaviour in spaces of composites. This is remedied by working instead with the paradigm of isotopy as we will explain. → read note

Mazur manifold as tangle diagram

Comments first published 21 Feb 2023

Mazur manifolds are intimately related to homology spheres, and are part of the puzzling world of 4-manifolds. Here we construct the simplest example of a Mazur manifold as a tangle diagram. → read note

The $D_4$ dualizability law

Comments first published 26 Jun 2022

This note gives a brief visual guide to how one can understand the classical \(D_4\) singularity in terms of a categorical pasting diagram (aka manifold diagram, or more specifically, a tame tangle). → read note

Tame tangles, singularities, and Morse theory

Comments first published 29 Jan 2022

Classical singularity theory and higher category theory have tantalizing connections: via the generalized tangle hypothesis we can understand singularities as expressing categorical laws satisfied by dualizable objects. However, classical singularity theory, based in differential-topological foundations, encounters technical problem in higher dimensions, which stalls this connection. Using our technology of manifold diagrams we outline steps towards remedying the problem, paving the way towards “higher” Morse theory. → read note

Manifold diagrams: a geometric approach

Comments first published 16 Jan 2022

We describe the notion of manifold diagrams. Manifold diagrams generalize string diagrams to higher dimensions. Our focus will lie on giving an intuitively clear geometric description of these diagrams in terms of conical stratifications in a framed background space. We will also provide further details on the dual notion of pasting diagrams of computadic cells. → read note

The class of computadic cells

Comments first published 14 Jan 2022

We describe the problem of defining a “universal” class of cell shapes for higher categories, the central obstruction in the classical approach to the question, and how to overcome that obstruction in the context of framed combinatorial topology. → read note

Github discussions and writing mathematics (deprecated)

Comments first published 30 Jan 2022

Github markdown does not allow for easily writing mathematical formulaes. We discuss some work-arounds, pairable with the comment plugin giscus. → read note

Enriched cohesive infinity toposes

Comments first published 30 Nov 2021

I gave a talk in the Advanced Topology Class at Oxford about “Generalized Differential Cohomology” based on this book. Part of the talk required setting up an “enriched cohesive \(\infty\)-topos”. Here’s an outline of what that object could be and what it does. → read note

Factorization homology

Comments first published 2 Oct 2019

I gave a talk at CUNY about factorization homology, and have written up some notes which I’m posting here. → read note

Unification in Mathematics: from Diagrams to Quantum Gravity

Comments first published 7 Apr 2023

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. → read note

Principles of higher-dimensional logic

Comments first published 26 Oct 2022

We allow ourselves to speculate a bit about how the 1-dimensional logical constructs that one frequently encounters in practical approaches to mathematical foundations may in fact emerge from a set of “more fundamental” principles of compositionality in higher-dimensional logic. → read note

Paradigms of higher category theory

Comments first published 21 Jan 2022

We explore two fundamental paradigms underlying models of higher categories. Most models for higher categories are based on the paradigm of contraction; this requires spaces of evaluations of a single pasting diagram to be contractible. The paradigm, however, generally fails to yield easy descriptions of interesting non-contractible behaviour in spaces of composites. This is remedied by working instead with the paradigm of isotopy as we will explain. → read note

On pattern matching, computation and (geometric) dependent type theory

Comments first published 8 Apr 2023

Pattern matching is an ubiquitous tool in theoretical computer science and type-theoretical mathematics, as well as the formalization of formal languages themselves. We briefly review its role in these subjects, giving examples along the way. We then motivate a deeper connection between pattern matching and higher-dimensional geometry, and suggest how geometric type theory, while not a real type theory yet, tries to provide a general formal framework in which mathematics can be built from higher-dimensional patterns. → read note

Github discussions and writing mathematics (deprecated)

Comments first published 30 Jan 2022

Github markdown does not allow for easily writing mathematical formulaes. We discuss some work-arounds, pairable with the comment plugin giscus. → read note

On pattern matching, computation and (geometric) dependent type theory

Comments first published 8 Apr 2023

Pattern matching is an ubiquitous tool in theoretical computer science and type-theoretical mathematics, as well as the formalization of formal languages themselves. We briefly review its role in these subjects, giving examples along the way. We then motivate a deeper connection between pattern matching and higher-dimensional geometry, and suggest how geometric type theory, while not a real type theory yet, tries to provide a general formal framework in which mathematics can be built from higher-dimensional patterns. → read note