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
n-Category Café.
The Catsters
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
Research
Preprints, papers, etc.
Preprints
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) 31-60.
arxiv:1505.04125
[Blogpost]
-
The Legendre-Fenchel 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 semi-tropical modules
Theory and Applications of Categories 28 (2013) 696-732.
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) 287-310.
arxiv:0908.1582
[Blogpost]
-
The Mukai pairing, I: a
categorical approach
(with A.Caldararu),
New York Journal of Mathematics 16 (2010) 61-98.
arXiv:0707.2052
-
On
the Rozansky-Witten weight systems
(with J.Roberts),
Algebraic & Geometric Topology 10 (2010) 1455-1519.
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) 1419-1457
math.QA/0503266
-
Gerbes
and Homotopy Quantum Field Theories (with U.Bunke and
P.Turner)
Algebr. Geom. Topol. 4 (2004) 407-437 math/0201116
-
Homotopy quantum
field theories and related ideas (with M.Brightwell and P.Turner)
Int. J. Modern Phys. A 18 Supplement (2003) 115-122.
-
An
almost-integral universal Vassiliev invariant of knots
Algebraic and Geometric Topology 2 (2002) paper no. 29, 649-664
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) 117-130.
-
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, 530-546.
- Vassiliev
invariants as polynomials,
Knot Theory, Banach Centre Publications 42 (1998) 457-463.
-
A combinatorial
half-integration from weight system to Vassiliev knot invariant,
J. Knot Theory Ramifications, 7 no. 4 (1998) 519-526.
-
Vassiliev invariants
and the Hopf algebra of chord diagrams,
Math. Proc. Camb. Phil. Soc., 119 (1996) 55-65.
Popularizations
Theses etc.
-
On the Vassiliev invariants for knots and for pure braids,
(official abstract)
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
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 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
- 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 semi-tropical 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 Pull-Push 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 3-space
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 co-authors' websites: