Listen pal, you can't just waltz in here, use
my toaster and spout universal truths without qualification!
Hal Hartley 
Surviving Desire
I am a pure mathematician interested in various aspects of geometry,
topology, category theory, metric spaces and quantum algebra. I am a
Catster and a host of the nCategory Café.
Some graphical experiments
The Catsters
The Catsters consists of my colleague Eugenia Cheng and me. We have put
up instructional YouTube videos on Category Theory.
You know that I write slowly.
This is chiefly because I am never satisfied until I have said as much
as possible in a few words, and writing briefly takes far more time
than writing at length.
Karl Friedrich Gauss
Research
Research Publications

The
magnitude of odd balls via Hankel determinants of reverse Bessel
polynomials
Discrete Analysis 2020:5, 42 pp.
arxiv:1708.03227

Categorifying the magnitude of a graph
(with Richard Hepworth)
Homotopy, Homology and Applications 19 (2017) 3160.
arxiv:1505.04125
[Blogpost]

The LegendreFenchel transform from a category theoretic perspective
Theory and Applications of Categories (to appear). arxiv:1501.03791.
[Blogposts I
and
II]

Spread: a measure of
the size of metric spaces
International Journal of Computational Geometry and Applications,
25 (2015) p. 207.
arxiv:1209.2300
[Blogpost]

Tight spans,
Isbell completions and semitropical modules
Theory and Applications of Categories 28 (2013) 696732.
arxiv:1302.4370
[Blogpost]

On the
magnitude of spheres, surfaces and other homogeneous spaces
Geometriae Dedicata (2013).
arxiv:1005.4041

On the
asymptotic magnitude of subsets of Euclidean space
(with Tom Leinster)
Geometriae Dedicata 164 (2013) 287310.
arxiv:0908.1582
[Blogpost]

The Mukai pairing, I: a
categorical approach
(with A.Caldararu),
New York Journal of Mathematics 16 (2010) 6198.
arXiv:0707.2052

On
the RozanskyWitten weight systems
(with J.Roberts),
Algebraic & Geometric Topology 10 (2010) 14551519.
math.DG/0602653.
See also Justin Roberts'talk.

A
diagrammatic approach to Hopf monads
[For less wide typesetting style see arXiv version.]
Arabian Journal of Science and Engineering C  Theme Issue
"Interactions of algebraic and coalgebraic structures (theory and
applications)" December 2008; Vol. 33, Number 2C math/0807.0658

The
twisted Drinfeld double of a finite group via gerbes and finite
groupoids
Algebraic & Geometric Topology 8 (2008) 14191457
math.QA/0503266

Gerbes
and Homotopy Quantum Field Theories (with U.Bunke and
P.Turner)
Algebr. Geom. Topol. 4 (2004) 407437 math/0201116

Homotopy quantum
field theories and related ideas (with M.Brightwell and P.Turner)
Int. J. Modern Phys. A 18 Supplement (2003) 115122.

An
almostintegral universal Vassiliev invariant of knots
Algebraic and Geometric Topology 2 (2002) paper no. 29, 649664
math/0105190

On
the first two Vassiliev invariants,
Experimental Mathematics (2002).

Free
groups and finite type invariants of pure braids (with
J.Mostovoy)
Math. Proc. Camb. Phil. Soc. 132 (2001) 117130.

The
Kontsevich integral and algebraic structures on the space of
diagrams,
Knots in Hellas '98, Series on Knots and Everything vol 24, World
Scientific, 2000, 530546.
 Vassiliev
invariants as polynomials,
Knot Theory, Banach Centre Publications 42 (1998) 457463.

A combinatorial
halfintegration from weight system to Vassiliev knot invariant,
J. Knot Theory Ramifications, 7 no. 4 (1998) 519526.

Vassiliev invariants
and the Hopf algebra of chord diagrams,
Math. Proc. Camb. Phil. Soc., 119 (1996) 5565.
Unpublished
 On the Vassiliev invariants for knots and for pure braids,
(official abstract) Edinburgh
University PhD Thesis, July 1997. This contains material from the above
papers together with ideas on the relationship between Vassiliev
invariants for pure braids and de Rham homotopy theory.
 Vassiliev invariants for knots,
Essay for Part III of the Cambridge Mathematical Tripos, May 1993. A
survey of BarNatan's paper On the Vassiliev knot invariants,
which contained some additional work of my own.
Selected writings at the nCategory Café

Optimal Transport and Enriched Categories III: Duality Within Pricing
(Aug 7, 2021)

Learn how conjugation between optimal prices comes from an
enriched adjunction.

Optimal Transport and Enriched Categories II
(Jun 19, 2021)

Read about how the dual transport problem arises from general
linear programming duality (and how that arises from general
Lagrangian duality).

Optimal Transport and Enriched Categories I
(Jun 6, 2021)

Read about the optimal transport problem and its dual.

Torsion:
Graph Magnitude Homology Meets Combinatorial Topology
(Apr 11, 2018)
 Work through an example of torsion in magnitude homology based on the work of Kaneta and Yoshinaga.
 On the Magnitude Function of Domains in Euclidean Space, II
(Mar 25, 2018)
 Continue reading about the work of Gimperlein and Goffeng
 Magnitude Homology Reading Seminar, I
(Mar 19, 2018)
 See the first in a Sheffield seminar series on the LeinsterShulman paper on magnitude homology.
 SageMath and 3D Models in Webpages
(Dec 19, 2017)
 Emded 3d models of surfaces into your websites using SageMath.
 Lattice Paths and Continued Fractions II
(Sep 22, 2017)
 Contemplate further continued fractions.
 Lattice Paths and Continued Fractions I
(Sep 18, 2017)
 Feel flabbergasted by Flajolet’s Fundamental Lemma
 Schröder Paths and Reverse Bessel Polynomials
(Aug 28, 2017)
 See how the reverse Bessel polynomials (which make an appearance in the magnitude of balls) have a combinatorial interpretation.
 Instantaneous Dimension of Finite Metric Spaces via Magnitude and Spread
(Aug 5, 2017)
 Download a talk on instantaneous dimension.
 Barceló and Carbery on the Magnitude of Odd Balls
(Sep 9, 2016)
 Read about Barcel'o and Carbery’s calculation of the magnitude of odd dimensional balls, utilizing the potential theory developed by Meckes.
 Categorifying the Magnitude of a Graph
(May 13, 2015)
 See how there’s a homology theory for graphs with magnitude as its Euler characteristic.
 A ScaleDependent Notion of Dimension for Metric Spaces (Part 1)
(Mar 11, 2015)
 Try to understand how dimension can depend on scale

Mathematics and Magic: the de Bruijn Card Trick
(Jan 5, 2015)
 Perform a magic trick using the power of maths.

Enrichment and the LegendreFenchel Transform II
(May 22, 2014)
 Get the second installment of how LegendreFenchel duality is an example of the profunctor nucleus construction.

Enrichment and the LegendreFenchel Transform I
(Apr 16, 2014)
 Remind yourself about basics of the Legendre–Fenchel transform.

Fuzzy Logic and Enriching Over the Category [0,1]
(Mar 15, 2014)
 Watch me try to understand fuzzy logic from an enriched category theory perspective

Galois Correspondences and Enriched Adjunctions
(Feb 5, 2014)
 Translate from category theory to order theory
 Ends
(Jan 5, 2014)
 End your ignorance of ends!
 Classical Dualities and Formal Concept Analysis
(Sep 12, 2013)
 Find out what algebraic varieties, convex sets, linear subspaces, real numbers, logical theories and extension fields have in common with formal concepts.
 Formal
Concept Analysis (Sep 2, 2013)
 Have a peek at the notion of formal concept analysis
 The
Nucleus of a Profunctor: Some Categorified Linear Algebra (Aug 19,
2013)
 Watch some linear algebra being categorified.
 Torsors
and enriched categories (Jun 3, 2013)
 Read about a different take on torsors
 Project
Scheduling and Copresheaves (Mar 24, 2013)
 Find out what PERT graphs have to do with enriched categories
 Tight
spans, Isbell completions and semitropical modules (Jan 20, 2013)
 See how these three things are related.
 The
Spread of a Metric Space (Sep 5, 2012)
 Read how this notion of size for metric spaces has some
interesting properties.

Integral
Transforms and the PullPush Perspective, I (Nov 7, 2010)
 Start to see how enriched profunctors can be viewed as
categorifications of integral kernels.
 Enriching
Over a Category of Subsets (Aug 30, 2010)
 Discover how enriched category theory leads to the definition of
some generalized metrics on the space of continuous functions on the
unit interval.
 On
the Magnitude of Spheres, Surfaces and Other Homogeneous Spaces
(Apr 21, 2010)
 See the details of a new paper on the magnitude of metric spaces.
 Modeling
Surface Diagrams (Mar 24, 2010)
 Watch some videos to see how I’m trying to make 3d models of
categorical surface diagrams.
 Intrinsic
Volumes and Weyl's Tube Formula (Mar 12, 2010)
 Read about how the volume of a tube around a surface in 3space
depends only on intrinsic invariants of the surface.
 A
Look at the Mathematical Origins of Western Musical Scales (Feb 26,
2010)
 See how the rational numbers 2 and 3/2 gave birth to the Western
musical scale.
 More
Magnitude of Metric Spaces and Problems with Penguins (Oct 10, 2009)
 Learn about the tenuous link between emperor penguins and the
magnitude of metric spaces.
Visit my coauthors' websites: