Tom Leinster

 

Publications

 

Books (all available free online): Entropy and Diversity: The Axiomatic Approach was published by Cambridge University Press in April 2021 Basic Category Theory was published by Cambridge University Press in 2014 Higher Operads, Higher Categories was published by Cambridge University Press in 2004

Here are my papers, grouped by subject. Size: Spaces of extremal magnitude (with Mark Meckes), arXiv:2112.12889, 7 pages, 2021 The maximum entropy of a metric space (with Emily Roff), arXiv:1908.11184; also Quarterly Journal of Mathematics 72 (2021), 1271–1309 Entropy modulo a prime, arXiv:1903.06961; also Communications in Number Theory and Physics 15(2) (2021), 279–314; discussion A short characterization of relative entropy, arXiv:1712.04903; also Journal of Mathematical Physics 60 (2019), Issue 2, 023302; discussion Magnitude homology of enriched categories and metric spaces (with Michael Shulman), arXiv:1711.00802 (with major rewrite in 2020); also Algebraic & Geometric Topology 21 (2021), 2175–2221; discussion The magnitude of a metric space: from category theory to geometric measure theory (with Mark Meckes), arXiv:1606.00095; also in Nicola Gigli (ed.), Measure Theory in Non-Smooth Spaces, de Gruyter Open, 2017 Maximizing diversity in biology and beyond (with Mark Meckes), arXiv:1512.06314; also Entropy 18 (2016), article 88 On the magnitude of a finite dimensional algebra, (with Joseph Chuang and Alastair King), arXiv:1505.04281; also Theory and Applications of Categories 31 (2016), 63–72 The magnitude of a graph, arXiv:1401.4623; also Mathematical Proceedings of the Cambridge Philosophical Society 166 (2019), 247–264; discussions (1, 2, 3) Codensity and the ultrafilter monad, arXiv:1209.3606; also Theory and Applications of Categories 28 (2013), 332–370; discussions (1, 2, 3, 4) Notions of Möbius inversion, arXiv:1201.0413; also Bulletin of the Belgian Mathematical Society—Simon Stevin 19 (2012), 911–935; discussion Measuring diversity: the importance of species similarity (with Christina Cobbold), Ecology 93 (2012), 477–489 A characterization of entropy in terms of information loss (with John Baez and Tobias Fritz), arXiv:1106.1791; also Entropy 13 (2011), no. 11, 1945–1957; discussion A multiplicative characterization of the power means, arXiv:1103.2574; also Bulletin of the London Mathematical Society 44 (2012), 106–112; discussion The magnitude of metric spaces, arXiv:1012.5857; also Documenta Mathematica 18 (2013), 857–905; discussion Integral geometry for the 1-norm, arXiv:1012.5881; also Advances in Applied Mathematics 49 (2012), 81–96; discussion A maximum entropy theorem with applications to the measurement of biodiversity, arXiv:0910.0906, 27 pages, 2009 On the asymptotic magnitude of subsets of Euclidean space (with Simon Willerton), arXiv:0908.1582; Geometriae Dedicata 164 (2013), 287–310; discussion The Euler characteristic of a category as the sum of a divergent series (with Clemens Berger), arXiv:0707.0835; also Homology, Homotopy and Applications 10(1) (2008), 41–51; discussion The Euler characteristic of a category, math.CT/0610260; also Documenta Mathematica 13 (2008), 21–49; nice description here and discussion here and here Self-similarity and recursion: A general theory of self-similarity, arXiv:1010.4474; also Advances in Mathematics 226 (2011), 2935–3017. Supersedes the preprints 'A general theory of self-similarity I' and 'A general theory of self-similarity II' General self-similarity: an overview, math.DS/0411343, 10 pages, 2004; also in Real and Complex Singularities (Proceedings of the Australian-Japanese Workshop, Sydney, 2005), World Scientific (2007) A general theory of self-similarity II: recognition, math.DS/0411345, 28 pages, 2004. An updated version of this, merged with part I, is published as 'A general theory of self-similarity' (above) A general theory of self-similarity I, math.DS/0411344, 49 pages, 2004. An updated version of this, merged with part II, is published as 'A general theory of self-similarity' (above) Objects of categories as complex numbers (with Marcelo Fiore), math.CT/0212377; also Advances in Mathematics 190 (2005), 264-277 An objective representation of the Gaussian integers (with Marcelo Fiore), math.RA/0211454; also Journal of Symbolic Computation 37 (2004), no. 6, 707-716 Category theory in algebra: An abstract characterization of Thompson's group F (with Marcelo Fiore), math.GR/0508617; also Semigroup Forum 80 (2010), 325–340; informal explanation and discussion Are operads algebraic theories?, math.CT/0404016; also Bulletin of the London Mathematical Society 38 (2006), no. 2, 233–238 Higher category theory: Weak ∞-categories via terminal coalgebras (with Eugenia Cheng), arXiv:1212.5853; also Theory and Applications of Categories 34 (2019), no. 34, 1073–1133 A survey of definitions of n-category, math.CT/0107188; also Theory and Applications of Categories 10 (2002), no. 1, 1–70 Topology and higher-dimensional category theory: the rough idea, math.CT/0106240, 15 pages, 2001 Operads in higher-dimensional category theory (PhD thesis), math.CT/0011106; also Theory and Applications of Categories 12 (2004), no. 3, 73–194 Generalized enrichment of categories, math.CT/0204279; also Journal of Pure and Applied Algebra 168 (2002), 391–406 fc-multicategories, math.CT/9903004, 8 pages, 1999 Generalized enrichment for categories and multicategories, math.CT/9901139, 79 pages, 1999 Basic bicategories, math.CT/9810017, 11 pages, 1998 Structures in higher-dimensional category theory, math.CT/0109021, 81 pages, 1998 General operads and multicategories, math.CT/9810053, 35 pages, 1997 (Polished versions of most of the material in these unpublished papers can be found in my book.) Homotopy algebra: Homotopy algebras for operads, math.QA/0002180, 101 pages, 2000 Up-to-homotopy monoids, math.QA/9912084, 8 pages, 1999 Miscellaneous: The eventual image, arXiv:2210.00302, 39 pages, 2022 Isbell conjugacy and the reflexive completion (with Tom Avery), arXiv:2102.08290; also Theory and Applications of Categories 36 (2021), 306–347 A categorical derivation of Lebesgue integration, arXiv:2011.00412, 24 pages, 2020; also Journal of the London Mathematical Society, to appear; discussions (1, 2) The probability that an operator is nilpotent, arXiv:1912.12562; also American Mathematical Monthly 128 (4) (2021), 371–375 The bijection between projective indecomposable and simple modules, arXiv:1410.3671, 10 pages, 2014; also Bulletin of the Belgian Mathematical Society—Simon Stevin 22 (2015), 725–735 Notes on commutation of limits and colimits (with Marie Bjerrum, Peter Johnstone and Will Sawin), arXiv:1409.7860, 5 pages, 2014; also Theory and Applications of Categories 30 (2015), 527–532 Rethinking set theory, arXiv:1212.6543, 8 pages, 2012; also American Mathematical Monthly 121 (2014), no. 5, 403–415 An informal introduction to topos theory, arXiv:1012.5647, 27 pages; also Publications of the nLab 1 (2011), no. 1 Perfect numbers and groups, math.GR/0104012, 12 pages, 1996ish; associated Sloane's integer sequence For other writing (reviews, magazine articles, blog posts, ...), see Notes below.

 

Talks

 

Here are slides and notes from some talks, grouped by subject. Within each subject, the most recent are listed first. Size: Magnitude homology (for category theorists) Measuring diversity (two talks for a mixed scientific audience) The mathematics of diversity (for mathematicians) What is the uniform distribution? (for statisticians and data scientists) New invariants of metric spaces: magnitude and maximum diversity (for geometers) The magnitude of graphs and finite metric spaces (for topologists and neuroscientists) Unexpected connections (for undergraduates) Magnitude (for category theorists) Magnitude and magnitude homology (for pure mathematicians) Magnitude homology (for applied algebraic topologists) Magnitude in geometry: invariants old and new (for a general mathematical audience) Maximizing biological diversity (for information theorists and biologists) The Euler characteristic of an algebra (for algebraists) The Euler characteristic of an algebra (for category theorists) The place of diversity in pure mathematics (for a life sciences audience) The many faces of magnitude (for a general mathematical audience) The Convex Magnitude Conjecture (for a general mathematical audience) Entropy, diversity and magnitude (for a general mathematical audience) Measuring diversity: the importance of species similarity (for ecologists) Integral geometry for the 1-norm (for convex and integral geometers) Notions of Möbius inversion (for category theorists) The magnitude of metric spaces I (for integral geometers) Magnitude and diversity: how an invariant from category theory solves a problem in mathematical ecology (for category theorists) Size (for a pure-mathematical audience, mostly model theorists) Counting, measure and metrics (for a general mathematical audience) How to measure almost anything (for a general scientific audience) The cardinality of a metric space (shorter) The cardinality of a metric space (longer) New perspectives on Euler characteristic (for a general mathematical audience) The Euler characteristic of a category (for category theorists) Another look at Euler characteristic (for a general pure-mathematical audience) Self-similarity and recursion: The categorical origins of Lebesgue integration The eventual image Terminal coalgebras via modules Coalgebraic topology Periodicity of spaces of walks Jónsson-Tarski toposes Self-similarity and recursion A universal Banach space Category theory in algebra: Codensity monads Codensity and the ultrafilter monad The Thompson groups Nerves of algebras (see also this discussion) Higher category theory: Introduction to higher (especially globular) operads A survey of the theory of bicategories Operads (90-minute tutorial) Higher-dimensional algebra (for politicians) Other: Large sets (mostly for category theorists) The functoriality of the reflexive completion (for category theorists) Cayley's theorem and its linearization The Snowden revelations, five years on (for Ethics in Mathematics 1, Cambridge) The reflexive completion (joint work with Tom Avery) The legacy of Turing's work at Bletchley Park (short talk for general public) The mathematics of biodiversity (public lecture) Unexpected connections (for beginning PhD students) Rethinking set theory The power of abstract thinking (for prospective PhD students) The peculiar traits of human mathematics Talk-related, but not actually talks: The categorical late lunch The categorical lunch The Scottish Category Theory Seminar Research programme: The mathematics of biodiversity at the Centre de Recerca Matemàtica, Barcelona, 18 June–20 July 2012. Activities page here. The programme included a five-day exploratory conference (2–6 July) Extremely short introduction to Beamer Conference in celebration of the 60th birthday of my PhD supervisor, Martin Hyland The 2008 Rankin Lectures, given in Glasgow by John Baez The 83rd Peripatetic Seminar on Sheaves and Logic (held in Glasgow in 2006); includes some notes from the talks Category theory seminars in Cambridge, 1996-2002

 

Notes

 

I'm one of the hosts of The n-Category Café, a research blog on mathematics, physics and philosophy. Here are some of my posts (most recent first): The eventual image, eventually Probability monads as codensity monads (on work of Ruben Van Belle) Ergodic set theory is trivial Spaces of extremal magnitude What is the uniform distribution? Diversity and the mysteries of counting Entropy and Diversity is out! Borel determinacy does not require replacement Large sets 13: summary Large sets 12: replacement Large sets 11: measurability, continued Large sets 10: measurability Large sets 9: inaccessibility Large sets 8: cardinal arithmetic Large sets 7: beth fixed points Large sets 6: beths Large sets 5: alephs Large sets 4: the index of a set Large sets 3: well-ordered sets Large sets 2: limits Large sets 1: introduction The Just Mathematics Collective Algebraic closure The lie of "It's Just Math" (after a blog post by Jade Master) The uniform measure Magnitude homology of enriched categories and metric spaces The categorical origins of Lebesgue integration, revisited Quotienting out the degenerate Counting nilpotents: a short paper A Joyal-type proof of the number of nilpotents A visual telling of Joyal's proof of Cayley's formula Counting nilpotents Geodesic spheres and Gram matrices What happened at the magnitude workshop How much work can it be to add some things up? Entropy mod p A well ordering is a consistent choice function The Fisher metric will not be deformed Entropy modulo a prime (continued) Entropy modulo a prime Magnitude homology in Sapporo Functional equations, entropy and diversity: a seminar course Functional equations XI: the diversity of a metacommunity Functional equations X: value Functional equations IX: entropy on a metric space Functional equations VIII: measuring biodiversity Functional equations VII: the p-norms Functional equations VI: using probability theory to solve functional equations Functional equations V: expected surprise Functional equations IV: a simple characterization of relative entropy Variance in a metric space Functional equations III: explaining relative entropy Functional equations II: Shannon entropy The Heilbronn Institute and the University of Bristol Functional equations I: Cauchy's equation The Brunn–Minkowski inequality Category theory in Barcelona Basic Category Theory free online Field notes on the behaviour of a large assemblage of ecologists Quarter-turns Magnitude homology Monoidal categories with projections A survey of magnitude In praise of the Gershgorin disc theorem How the simplex is a vector space Weil, venting Where does the spectrum come from? Rainer Vogt How do you handle your email? Five quickies More on the AMS and NSA The AMS must justify its support of the NSA Effective sample size Turing's legacy Graph colouring and cartesian closed categories Jaynes on mathematical courtesy New evidence of the NSA deliberately weakening encryption The atoms of the module world Why it matters Holy crap, do you know what a compact ring is? Wrestling with tight spans Basic Category Theory The place of diversity in pure mathematics Math and mass surveillance: a roundup The categorical origins of Lebesgue integration Should mathematicians cooperate with GCHQ? Part 3 (response to Richard Pinch and Malcolm MacCallum) Categories vs. algebras Should mathematicians cooperate with GCHQ? Part 2 New Scientist article Fourier series and flipped classrooms Should mathematicians cooperate with GCHQ? The deteriorating relationship between academics and the NSA The magnitude of a graph The Electronic Frontier Foundation at the Joint Meetings Academics Against Mass Surveillance The long grind of writing a book Commuting limits and colimits over groups Severing ties with the NSA Who ordered that? Linear operators done right The Shannon capacity of a graph, 2 The Shannon capacity of a graph, 1 The first commutative diagram? Whitney twists Tutte polynomials and magnitude functions Colouring a graph Carleson's theorem Rethinking set theory Almost all of the first 50 billion groups have order 1024 The Zorn identity The curious dependence of set theory on order theory Where do ultraproducts come from? Where do linearly compact vector spaces come from? Where do ultrafilters come from? Where do monads come from? Jech (1973), page 118 Integrating against the Euler characteristic log|x| + C The eventual image, part 2 On the law of large numbers (such as 60) The eventual image Measuring diversity, plus version for non-mathematicians on John Baez's blog Do you know this idempotent? Spectra of operators and rings Universal measures Hadwiger's theorem, part 2 Mixed volume Definitions of ultrafilter Hadwiger's theorem, part 1 The magnitude of an enriched category Möbius inversion for categories An operadic introduction to entropy Entropies vs. means Which graphs can be given a category structure? Characterizing the generalized means Characterizing the p-norms Magnitude of metric spaces: a roundup An informal introduction to topos theory The Boyd Orr Centre, or: what is a severed horse leg? What is integral geometry? Benoît Mandelbrot Fetishizing p-values The difference between measure zero and empty interior What is the Langlands Programme? Means Pullback-homomorphisms The Dold–Kan Theorem: two questions A perspective on higher category theory Sheaves do not belong to algebraic geometry (with proof) F and the shibboleth What you're doing is good for you An adventure in analysis Asymptotics of the magnitude of metric spaces Entropy, diversity and cardinality (part 2) Entropy, diversity and cardinality (part 1) The cardinality of a metric space How I learned to love the nerve construction On Linear Algebra Done Right Here are a couple of my comments at the n-Category Café: A short explanation of the Central Limit Theorem and how to view it as a result about maximum entropy Informal introduction to classifying toposes And here are some odds and ends: Review by Stefan Forcey of my book Entropy and Diversity, in the European Mathematical Society Magazine Graph of Covid-19 cases at the University of Edinburgh I am a person Working seminar on persistent homology Review of the book Coherence in Three-Dimensional Category Theory by Nick Gurski (published in the Bulletin of the London Mathematical Society) Article for New Scientist magazine: Ethical calculus (also syndicated in Slate), plus associated post and follow-on articles in Boing Boing, Slashdot, Mediapart (free version here), Zeit Online, and Spiegel Online Two articles for the London Mathematical Society Newsletter, both entitled Should mathematicians cooperate with GCHQ?: first (appears in the April 2014 edition, p.34) and second (appears in the July 2014 edition, p.42) A review of the popular mathematics book The Colours of Infinity An interview with me in the December 2008 issue of The Reasoner, largely about higher categories Doing without diagrams: how to take a proof that uses elements, utter some magic words, and conclude that it's valid in any category My favourite proof of the Fundamental Theorem of Algebra: argument learned from Graeme Segal and notes extracted from a course I taught Coproducts of operads, and the W-construction: an observation that plays a part in the story of the operad of phylogenetic trees.

 

Teaching

 

Edinburgh 4th year Galois Theory 2022–23 Edinburgh Linear Algebra 2018–19: accelerated course for undergraduates entering directly into the second year Tips on writing projects, for my undergraduate project students Edinburgh Functional equations seminar course, 2017; see also this post Edinburgh 4th year General Topology 2014–15, plus introductory lecture and problem sheets Edinburgh 4th year Fourier Analysis 2013–14 (mostly Fourier series), plus problem sheets Tips on mathematical writing for undergraduates Glasgow M.Sci. Category Theory 2007–8 Glasgow Category Theory 2004 Cambridge Part III Category Theory 2000 Cambridge Part IA/IB Linear Maths: some notes on the minimal polynomial and Jordan canonical form

 

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