Listen pal, you can't just waltz in here, use my toaster and spout universal truths without qualification!
Hal Hartley -
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
The Catsters consists of my colleague Eugenia Cheng and me. We have put up instructional YouTube videos on Category Theory.
Some graphical experiments
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
Preprints, papers, etc.
magnitude of odd balls via Hankel determinants of reverse Bessel
Discrete Analysis 2020:5, 42 pp.
Categorifying the magnitude of a graph
(with Richard Hepworth)
Homotopy, Homology and Applications 19 (2017) 31-60.
The Legendre-Fenchel transform from a category theoretic perspective
Theory and Applications of Categories (to appear). arxiv:1501.03791.
Spread: a measure of
the size of metric spaces
International Journal of Computational Geometry and Applications,
25 (2015) p. 207.
Isbell completions and semi-tropical modules
Theory and Applications of Categories 28 (2013) 696-732.
magnitude of spheres, surfaces and other homogeneous spaces
Geometriae Dedicata (2013).
asymptotic magnitude of subsets of Euclidean space
(with Tom Leinster)
Geometriae Dedicata 164 (2013) 287-310.
The Mukai pairing, I: a
New York Journal of Mathematics 16 (2010) 61-98.
the Rozansky-Witten weight systems
Algebraic & Geometric Topology 10 (2010) 1455-1519.
See also Justin Roberts'talk.
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
twisted Drinfeld double of a finite group via gerbes and finite
Algebraic & Geometric Topology 8 (2008) 1419-1457
and Homotopy Quantum Field Theories (with U.Bunke and
Algebr. Geom. Topol. 4 (2004) 407-437 math/0201116
field theories and related ideas (with M.Brightwell and P.Turner)
Int. J. Modern Phys. A 18 Supplement (2003) 115-122.
almost-integral universal Vassiliev invariant of knots
Algebraic and Geometric Topology 2 (2002) paper no. 29, 649-664
the first two Vassiliev invariants,
Experimental Mathematics (2002).
groups and finite type invariants of pure braids (with
Math. Proc. Camb. Phil. Soc. 132 (2001) 117-130.
Kontsevich integral and algebraic structures on the space of
Knots in Hellas '98, Series on Knots and Everything vol 24, World
Scientific, 2000, 530-546.
invariants as polynomials,
Knot Theory, Banach Centre Publications 42 (1998) 457-463.
half-integration from weight system to Vassiliev knot invariant,
J. Knot Theory Ramifications, 7 no. 4 (1998) 519-526.
and the Hopf algebra of chord diagrams,
Math. Proc. Camb. Phil. Soc., 119 (1996) 55-65.
On the Vassiliev invariants for knots and for pure braids,
Edinburgh University PhD Thesis, July 1997.
Vassiliev invariants for knots,
Essay for Part III of the Cambridge Mathematical Tripos, May 1993.
Selected writings at the n-Category Café
Metric Spaces as Enriched Categories II
(May 3, 2023)
See the definition of enriched category and marvel at how metric spaces can be treated as enriched categories.
Metric Spaces as Enriched Categories I
(Apr 17, 2023)
Read about how we don’t always want our hom-sets in category theory to be sets!
Optimal Transport and Enriched Categories IV: Examples of Kan-type Centres
(Jan 7, 2022)
Optimal Transport and Enriched Categories III: Duality Within Pricing
(Aug 7, 2021)
Learn how conjugation between optimal prices comes from an
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
Optimal Transport and Enriched Categories I
(Jun 6, 2021)
Read about the optimal transport problem and its dual.
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 Leinster-Shulman 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 Scale-Dependent 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 Legendre-Fenchel Transform II
(May 22, 2014)
- Get the second installment of how Legendre-Fenchel duality is an example of the profunctor nucleus construction.
Enrichment and the Legendre--Fenchel 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
(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.
Concept Analysis (Sep 2, 2013)
- Have a peek at the notion of formal concept analysis
Nucleus of a Profunctor: Some Categorified Linear Algebra (Aug 19,
- Watch some linear algebra being categorified.
and enriched categories (Jun 3, 2013)
- Read about a different take on torsors
Scheduling and Copresheaves (Mar 24, 2013)
- Find out what PERT graphs have to do with enriched categories
spans, Isbell completions and semi-tropical modules (Jan 20, 2013)
- See how these three things are related.
Spread of a Metric Space (Sep 5, 2012)
- Read how this notion of size for metric spaces has some
Transforms and the Pull-Push Perspective, I (Nov 7, 2010)
- Start to see how enriched profunctors can be viewed as
categorifications of integral kernels.
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
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.
Surface Diagrams (Mar 24, 2010)
- Watch some videos to see how I’m trying to make 3d models of
categorical surface diagrams.
Volumes and Weyl's Tube Formula (Mar 12, 2010)
- Read about how the volume of a tube around a surface in 3-space
depends only on intrinsic invariants of the surface.
Look at the Mathematical Origins of Western Musical Scales (Feb 26,
- See how the rational numbers 2 and 3/2 gave birth to the Western
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.