A group blog on math, physics and philosophy

- The Stable Homotopy Hypothesis and Categorified Abelian Groups (Jan 29, 2018)
- Guest post by Nick Gurski
- FreeTikZ (Jan 23, 2018)
- Sketch a string diagram by hand, and a new tool by Chris Heunen will automatically convert it into TikZ code.
- Entropy Modulo a Prime (Continued) (Dec 16, 2017)
- The right definition of the entropy of a probability distribution whose “probabilities” are integers mod p.
- Entropy Modulo a Prime (Dec 14, 2017)
- The analogue of Shannon entropy in the field of p elements (after Kontsevich).
- Unconscious Bias in Recruiting (Oct 23, 2017)
- Online tool for testing whether your job ad is likely to sustain gender inequality.
- Magnitude Homology in Sapporo (Aug 11, 2017)
- Magnitude homology of metric spaces: the short version.
- Laws of Mathematics "Commendable" (Jul 14, 2017)
- Malcolm Turnbull wants to be able to read your messages.
- Functional Equations, Entropy and Diversity: A Seminar Course (Apr 19, 2017)
- Notes from a seminar course.
- The Diversity of a Metacommunity (Apr 19, 2017)
- Conditional entropy, joint entropy and mutual information all have something important to say about the biological diversity of a metacommunity.
- Value (Apr 14, 2017)
- What is the value of the whole in terms of the values of its parts? This apparently very general question, when made precise in a reasonable way, has a surprisingly specific answer.
- Functional Equations IX: Entropy on a Metric Space (Apr 5, 2017)
- A family of entropy measures for probability distributions on metric spaces. Or viewed another way: how to measure biological diversity while taking into account the varying similarities between species.
- Functional Equations VIII: Measuring Biodiversity (Mar 29, 2017)
- What should the word “diversity” mean for a biological (or other) community, and how is related to the concepts of entropy, mean and norm?
- Functional Equations VII: The
*p*-Norms (Mar 22, 2017) - A theorem saying what’s so special about the p-norms.
- Functional Equations VI: Using Probability Theory to Solve Functional Equations (Mar 15, 2017)
- How the probabilistic theory of large deviations and the analytic tool of convex conjugacy combine to provide a tool for solving functional equations.
- Postdocs in Sydney (Mar 10, 2017)
- Two postdoc positions in category theory, geometry and logic at Macquarie University, Sydney.
- Functional Equations V: Expected Surprise (Mar 8, 2017)
- A family of q-deformations of Shannon entropy, interpreted as expected surprise. Plus, a simple characterization theorem for these “surprise entropies”, and another for relative surprise entropy.
- Functional Equations IV: A Simple Characterization of Relative Entropy (Feb 28, 2017)
- Relative entropy is almost uniquely characterized by the chain rule.
- Variance in a Metric Space (Feb 26, 2017)
- “Variance” makes sense for probability distributions on a metric space. How should we interpret the variance of the maximum entropy distribution?
- Functional Equations III: Explaining Relative Entropy (Feb 22, 2017)
- An explanation of relative entropy in terms of coding.
- Functional Equations II: Shannon Entropy (Feb 14, 2017)
- Shannon entropy and the functional equation that characterizes it.
- The Heilbronn Institute and the University of Bristol (Feb 10, 2017)
- News report on Bristol University and the surveillance state.
- Functional Equations I: Cauchy's Equation (Feb 7, 2017)
- First instalment of a course on functional equations. This time: which functions f: R –> R satisfy f(x + y) = f(x) + f(y)?
- The Brunn-Minkowski Inequality (Jan 31, 2017)
- A highly general inequality for volume, and how it implies the isoperimetric inequality.
- Papers Written While Drunk (Jan 24, 2017)
- Chemicals in the bloodstream and their influence on mathematics.
- Category Theory in Barcelona (Jan 10, 2017)
- Joachim Kock and I are teaching category theory in Barcelona. We like slogans.
*Basic Category Theory*Free Online (Jan 2, 2017)- My introductory textbook Basic Category Theory is now available free online.
- NSA Axes Math Grants (Dec 31, 2016)
- NSA suspends its Mathematical Sciences Program. Fewer moral dilemmas for mathematicians.
- Field Notes on the Behaviour of a Large Assemblage of Ecologists (Dec 15, 2016)
- What this mathematician made of the British Ecological Society’s annual meeting.
- Quarter-Turns (Nov 29, 2016)
- For which linear operators T is T(x) always orthogonal to x? The answer depends very much on the field you’re working over.
- Jobs at Edinburgh (Oct 10, 2016)
- Advertisement for two jobs in Algebra, Geometry and Topology at the University of Edinburgh.
- Disaster at Leicester (Sep 15, 2016)
- Leicester University is imposing very severe cuts on the Mathematics department. You can add your voice to those objecting.
- Magnitude Homology (Sep 6, 2016)
- A new homology theory for enriched categories, created by Mike Shulman, which looks very much as if it categorifies magnitude.
- Economy of Style (Sep 3, 2016)
- An extreme example of mathematical writing style.
- Jobs at Heriot-Watt (Aug 27, 2016)
- Permanent positions are available at Heriot-Watt University, Edinburgh.
- Monoidal Categories with Projections (Aug 25, 2016)
- A defence of the technical-sounding concept of “semicartesian monoidal category”.
- A Survey of Magnitude (Aug 11, 2016)
- Magnitude is a highly informative invariant of metric spaces. Here are some of the highlights of the theory, now collected together in a survey paper by Mark Meckes and me.
- In Praise of the Gershgorin Disc Theorem (Aug 9, 2016)
- An elementary and very useful lemma in basic linear algebra. (Why wasn’t I taught it?)
- How the Simplex is a Vector Space (Jun 11, 2016)
- The interior the n-simplex can be given the structure of a real vector space in a natural way. How, and what use is it?
- Man Ejected from Flight for Solving Differential Equation (May 8, 2016)
- Yes, really.
- Weil, Venting (Nov 10, 2015)
- Passive-aggressive Weil.
- Where Does The Spectrum Come From? (Sep 14, 2015)
- A categorical account of the most important invariant of a linear operator.
- Rainer Vogt (Sep 3, 2015)
- An appreciation of Rainer Vogt, who died last month.
- How Do You Handle Your Email? (Sep 2, 2015)
- What’s the best way for academics to manage the day-to-day deluge of email?
- Mary Shelley on Invention (Jul 8, 2015)
- A thought on invention from the introduction to Frankenstein.
- How to Acknowledge Your Funder (May 22, 2015)
- Acknowledging the NSA’s support when you object to universal surveillance.
- Five Quickies (Apr 6, 2015)
- 1. Reflexive completion. 2. Mathematicians accepting and declining overtures from GCHQ. 3. The Euler characteristic of an algebra. 4. Review of Nick Gurski’s higher categories book. 5. Are lectures the best way to teach?
- More on the AMS and NSA (Feb 4, 2015)
- Letter from cryptographers and mathematicians to the AMS.
- The AMS Must Justify Its Support of the NSA (Jan 9, 2015)
- Letter in the Notices of the AMS.
- Effective Sample Size (Dec 18, 2014)
- Yet another place where the concept of magnitude turns up: the statistical notion of effective sample size.
- Welcome, Qiaochu! (Dec 18, 2014)
- Qiaochu Yuan joins us as a host of the Café.
- Turing's Legacy (Dec 16, 2014)
- The legacy of Alan Turing’s work at Bletchley Park (later GCHQ).
- Graph Colouring and Cartesian Closed Categories (Dec 5, 2014)
- A categorical reformulation of Hedetniemi’s conjecture on colouring a product of graphs.
- Grothendieck's Activism And What It Says About The World (Nov 26, 2014)
- Ivar Ekeland’s reflections on the activist phase of Grothendieck’s life.
- Barbie on Monads (Nov 22, 2014)
- The definition of monad in a Barbie cartoon.
- General Topology Notes (Nov 19, 2014)
- Undergraduate notes from a course on General Topology.
- The Kan Extension Seminar in the
*Notices*(Nov 18, 2014) - Emily Riehl on running an online graduate reading course.
- Jaynes on Mathematical Courtesy (Nov 16, 2014)
- Edwin Jaynes rails against modern mathematical writing.
- Where Do Probability Measures Come From? (Oct 22, 2014)
- Tom Avery explains a categorical construction of the concept of probability measure.
- New Evidence of the NSA Deliberately Weakening Encryption (Oct 17, 2014)
- Yet more evidence that the NSA wants to make it impossible to communicate privately.
- The Atoms of the Module World (Oct 5, 2014)
- The one-to-one correspondence between simple and projective indecomposable modules.
- Why It Matters (Aug 25, 2014)
- Many mathematicians are uncomfortable about the NSA/GCHQ’s surveillance of entire populations, but fundamentally aren’t that bothered. Does it really matter?
- Holy Crap, Do You Know What A Compact Ring Is? (Aug 20, 2014)
- Every compact ring is totally disconnected.
- Wrestling with Tight Spans (Aug 2, 2014)
- A geometric construction, used in combinatorial optimization and phylogenetic analysis, interpreted categorically.
*Basic Category Theory*(Jul 27, 2014)- My introduction to category theory has just been published.
- The Place of Diversity in Pure Mathematics (Jul 20, 2014)
- Diversity is a canonical concept of pure mathematics.
- Math and Mass Surveillance: A Roundup (Jul 15, 2014)
- List of all the posts so far on mathematics and mass surveillance.
- The Categorical Origins of Lebesgue Integration (Jul 8, 2014)
- A simple universal property characterizing the space L^1 of Lebesgue-integrable functions.
- Should Mathematicians Cooperate with GCHQ? Part 3 (Jun 5, 2014)
- Mathematicians: let’s have a properly evidence-based discussion of the ethics of working for the NSA, GCHQ and other intelligence agencies.
- Categories vs. Algebras (May 14, 2014)
- Understanding the category algebra construction from representation theory.
- Should Mathematicians Cooperate with GCHQ? Part 2 (Apr 30, 2014)
- Reply to Richard Pinch’s article on GCHQ in the London Mathematical Society newsletter.
*New Scientist*Article (Apr 27, 2014)- Mathematicians should not cooperate with agencies of mass surveillance.
- Big Data Power (Apr 1, 2014)
- Guest post by Nils Carqueville and Daniel Murfet: the big data generated by whole-population surveillance.
- Fourier Series and Flipped Classrooms (Mar 30, 2014)
- Moving beyond traditional lecturing.
- Operads of Finite Groups (Mar 5, 2014)
- Guest post by Nick Gurski.
- Should Mathematicians Cooperate with GCHQ? (Mar 2, 2014)
- Article for the LMS Newsletter on an ethical decision facing mathematicians.
- The Deteriorating Relationship Between Academics and the NSA (Feb 11, 2014)
- NSA rejects academics, academics reject NSA.
- Weightings for Compact Metric Spaces (Jan 30, 2014)
- Guest post by Mark Meckes, relating new geometric invariants to old.
- The Magnitude of a Graph (Jan 22, 2014)
- An analogue of cardinality for graphs, associating to each graph a rational function.
- Hilarious Takedown of Bonkers Maths in Top Psychology Journal (Jan 21, 2014)
- Make sure you have at least 2.9013 positive emotions for each negative one.
- The Electronic Frontier Foundation at the Joint Meetings (Jan 13, 2014)
- An antidote to NSA recruitment at the Joint Math Meetings in Baltimore this week.
- Academics Against Mass Surveillance (Jan 8, 2014)
- Add your name to a public declaration against the systematic recording of our daily activities.
- The Long Grind of Writing a Book (Dec 25, 2013)
- Why it so painful?
- Commuting Limits and Colimits over Groups (Dec 21, 2013)
- An apparently new limit-colimit commutation theorem.
- A Technical Innovation (Dec 10, 2013)
- How to view the source TeX of mathematical expressions on this site.
- The 8th Scottish Category Theory Seminar (Nov 21, 2013)
- Announcement of the 8th Scottish Category Theory Seminar
- Four New Talks (Nov 12, 2013)
- The talks I gave in October.
- Severing Ties with the NSA (Nov 10, 2013)
- Should mathematical societies refuse to cooperate with the NSA and GCHQ?
- Unexpected Connections (Oct 6, 2013)
- What are your favourite examples of unexpected connections in mathematics?
- Who Ordered That? (Oct 2, 2013)
- A very odd and elementary conjecture of Kontsevich has now been proved.
- The Definition of Graph is Classified (Sep 24, 2013)
- Top Secret: a graph is a set of objects linked by edges.
- Linear Operators Done Right (Aug 22, 2013)
- On the very useful analogy between complex numbers and linear operators.
- The Shannon Capacity of a Graph, 2 (Aug 18, 2013)
- Connections between Shannon capacity, maximum diversity, and magnitude.
- The Shannon Capacity of a Graph, 1 (Aug 10, 2013)
- When you spell a word down the phone, some letters are more easily confused than others. What’s a good measure of your communication power that takes these confusions into account?
- Keep Calm and Carry One (Jul 30, 2013)
- How New Zealand schoolchildren aren’t learning to add.
- The First Commutative Diagram? (Jul 8, 2013)
- A commutative square from 1919, in a book by Bertrand Russell.
- Whitney Twists (Jul 7, 2013)
- Q. Is the magnitude of a graph invariant under Whitney twists? A. Sometimes.
- Welcome, Emily (Jun 1, 2013)
- Emily Riehl joins us as a host on this blog.
- Bounded Gaps Between Primes (May 14, 2013)
- Report on recent work of Yi Tang Zhang, taking us a step closer to a proof of the Twin Primes Conjecture.
- Tutte Polynomials and Magnitude Functions (Apr 16, 2013)
- The magnitude function is a mysterious invariant of graphs. It’s closely related to invariants that have proven meaningful and useful in other branches of mathematics. But uncovering its meaning for graphs poses a challenge.
- Colouring a Graph (Apr 12, 2013)
- A categorical perspective on graph colourings, and a fundamental gap in our knowledge.
- The Convex Magnitude Conjecture (Mar 30, 2013)
- An unsolved problem about measure of convex sets, arising from category theory.
- Five Positions at Edinburgh (Mar 22, 2013)
- Jobs at Edinburgh.
- Mark on Magnitude (Mar 21, 2013)
- Mark Meckes mouths magnificently on metric magnitude.
- 7th Scottish Category Theory Seminar (Jan 29, 2013)
- Conference announcement.
- This Week's Finds at 20 (Jan 26, 2013)
- The 20th birthday of John’s column.
- Carleson's Theorem (Jan 17, 2013)
- The twisty turny history of beliefs about pointwise convergence of Fourier series.
- Category Theory 2013 (Dec 24, 2012)
- Announcement of Category Theory 2013, Sydney.
- Rethinking Set Theory (Dec 18, 2012)
- Expository paper on set theory done categorically.
- Bob on TAC (Dec 8, 2012)
- Bob Rosebrugh’s article about Theory and Applications of Categories in the Notices of the AMS.
- Research Fellowships in Edinburgh (Dec 7, 2012)
- Five postdoc positions available.
- Almost All of the First 50 Billion Groups Have Order 1024 (Nov 28, 2012)
- More than 99 per cent of the groups of order at most 2000 have order 1024.
- Parametrized Mates and Multivariable Adjunctions (Nov 1, 2012)
- Guest post by Emily Riehl.
- The Zorn Identity (Oct 26, 2012)
- Zorn’s lemma is trivially equivalent to a theorem provable without the axiom of choice.
- The Curious Dependence of Set Theory on Order Theory (Oct 13, 2012)
- Is set theory as dependent on order theory as it seems to be, and if so, why?
- Analysis Fellowship in Edinburgh (Oct 11, 2012)
- Job ad.
- Symmetric Monoidal Bicategories (Oct 4, 2012)
- Guest post by Nick Gurski.
- Where Do Ultraproducts Come From? (Sep 30, 2012)
- Are ultraproducts categorically inevitable?
- Where Do Linearly Compact Vector Spaces Come From? (Sep 25, 2012)
- What should it mean for a topological vector space to be compact?
- Where Do Ultrafilters Come From? (Sep 23, 2012)
- The notion of ultrafilter is inherent in the notion of finiteness of a set.
- Where Do Monads Come From? (Sep 20, 2012)
- Monads don’t just come from adjunctions.
- Jech (1973), page 118 (Aug 29, 2012)
- An amusing page of Thomas Jech’s book “The Axiom of Choice”.
- Integrating Against the Euler Characteristic (May 17, 2012)
- Euler characteristic is something like a measure. So what does integrating against it do?
- The Mathematics of Biodiversity (Apr 26, 2012)
- Conference in Barcelona.
- 10,000 (Apr 23, 2012)
- News on Elsevier boycott.
- Wellcome Joins In (Apr 9, 2012)
- I just looked at the front page of The Guardian, a quality British newspaper that’s especially widely read online… and I was amazed to see that their second-leading story was on The Cost of Knowledge: The news is that…
- Reader Survey: log|
*x*| +*C*(Mar 19, 2012) - On a worrying tradition in the teaching of calculus.
- 6th Scottish Category Theory Seminar (Mar 19, 2012)
- Conference announcement.
- Types, Homotopy and Univalent Foundations: Special Issue (Feb 26, 2012)
- Announcement of a special journal issue on homotopy type theory
- Vorsicht! Funktor! (Jan 15, 2012)
- Sign in Germany.
- A Semigroup Approach to Finite Markov Chains (Jan 3, 2012)
- Guest post by Benjamin Steinberg, proving the basic results on finite Markov chains using semigroup theory.
- The Eventual Image, Part 2 (Dec 25, 2011)
- In which categories do endomorphisms have eventual images? A solution to the problem posed in Part 1.
- On the Law of Large Numbers (Such As 60) (Dec 20, 2011)
- Alex Simpson’s work on randomness and locales.
- What Do You Think of EPSRC Policy? (Dec 19, 2011)
- Problems with EPSRC funding policy, and a call for comments by Burt Totaro and Michael Singer.
- The Eventual Image (Dec 8, 2011)
- Three different categories exhibit very similar dynamical behaviour. Why?
- Measuring Diversity (Oct 23, 2011)
- A way of quantifying diversity, especially biological diversity, that takes similarity between species into account.
- Do You Know This Idempotent? (Oct 19, 2011)
- Every endomorphism of a finite set has a unique idempotent power. Can you describe it?
- Spectra of Operators and Rings (Oct 16, 2011)
- Connecting notions of spectra, and a question on characteristic polynomials.
- The 5th Scottish Category Theory Seminar (Oct 10, 2011)
- Announcement of Scotcats 5.
- Universal Measures (Sep 14, 2011)
- Groemer’s integral theorem, recast as a universal property.
- Hadwiger's Theorem, Part 2 (Aug 29, 2011)
- Generalizing Hadwiger’s theorem to arbitrary metric spaces, especially R^n with the 1-norm.
- Mixed Volume (Aug 26, 2011)
- Several explanations of the mixed volume of convex bodies.
- All Job Ads Should Be Like This (Aug 21, 2011)
- Poetic ad for a job at the University of Strathclyde.
- Definitions of Ultrafilter (Jul 2, 2011)
- Several equivalent definitions of ultrafilter, including a particularly simple one that I’d like to find a reference for.
- Hadwiger's Theorem, Part 1 (Jun 29, 2011)
- Expository account of Hadwiger’s theorem characterizing invariant valuations on convex sets.
- The Magnitude of an Enriched Category (Jun 13, 2011)
- An analogue of cardinality for enriched categories.
- Möbius Inversion for Categories (May 31, 2011)
- The relationships between Euler characteristic and different kinds of Möbius inversion.
- An Operadic Introduction to Entropy (May 18, 2011)
- Summary of recent results on entropy, and introduction to entropy for the categorically minded.
- Entropies
*vs.*Means (May 10, 2011) - Connections between the two concepts.
*The Colours of Infinity*— A Review (Apr 26, 2011)- Review of a popular mathematics book.
- The 4th Scottish Category Theory Seminar (Apr 7, 2011)
- Seminar announcement.
- Which Graphs Can be Given a Category Structure? (Mar 27, 2011)
- A report on Samer Allouch’s thesis.
- Characterizing the Generalized Means (Mar 2, 2011)
- A new characterization of generalized means?
- Characterizing the
*p*-Norms (Mar 1, 2011) - Exactly what makes the p-norms special?
- Category Theory 2011 (Jan 17, 2011)
- The invited speakers for CT2011 have been announced.
- Another Editorial Board Resigns (Jan 11, 2011)
- Not-so-new news about the Journal of Group Theory.
- Magnitude of Metric Spaces: A Roundup (Jan 8, 2011)
- Resources on magnitude of metric spaces.
- An Informal Introduction to Topos Theory (Dec 25, 2010)
- A short paper on basic topos theory.
- The Boyd Orr Centre, or: What is a Severed Horse Leg? (Dec 17, 2010)
- Interdisciplinarity for mathematicians.
- Poetic Refereeing (Dec 17, 2010)
- Snappy excerpts from referees’ reports.
- Pri la Funkcia Ekvacio
*f(x + y) = f(x) + f(y)*(Dec 10, 2010) - Fréchet, a functional equation, and a curious piece of history.
- What is Integral Geometry? (Nov 29, 2010)
- A quick and dirty explanation.
- A Short Warning (Oct 26, 2010)
- On plagiarism.
- Benoît Mandelbrot (Oct 18, 2010)
- Remarks on the Mandelbrot set and its wider context.
- The Scottish Category Theory Seminar (Oct 13, 2010)
- Announcement of the third meeting.
- An Invitation (Oct 8, 2010)
- Come in, come in!
- Fetishizing
*p*-Values (Sep 23, 2010) - The “cult of statistical significance”.
- Bacard on Segal Enriched Categories (Sep 22, 2010)
- Short description of a new paper on homotopy-enrichment
- Integral Geometry in Barcelona (Sep 6, 2010)
- Short report from a meeting.
- The Difference between Measure Zero and Empty Interior (Aug 28, 2010)
- …is the difference between Lebesgue and Riemann integrability.
*Free High School Science Texts*Needs You! (Aug 25, 2010)- Proof-reading needed for open source textbooks.
- What is the Langlands Programme? (Aug 21, 2010)
- Please?
- Means (Aug 7, 2010)
- Means of numbers and means of structures.
- Pullback-homomorphisms (Aug 1, 2010)
- What does it signify for a map of algebras for a monad to be a pullback square?
- The Dold–Kan Theorem: Two Questions (Jul 12, 2010)
- In Ab, many things become true.
- Algebraic Model Structures (Jun 16, 2010)
- Algebraic model categories and weak factorization systems.
- The Scottish Category Theory Seminar (May 8, 2010)
- The second meeting of the Scottish Category Theory Seminar.
- Normal Service is Resumed! (Mar 22, 2010)
- Temporary software glitch is over.
- A Perspective on Higher Category Theory (Mar 8, 2010)
- Extended ramblings.
- Sheaves Do Not Belong to Algebraic Geometry (Feb 5, 2010)
- They come out of thin categorical air.
*F*and the Shibboleth (Jan 14, 2010)- Thompson’s group F and free structured categories.
- What You're Doing is Good for You (Jan 6, 2010)
- Open access articles get cited more.
- An Adventure in Analysis (Nov 4, 2009)
- How a problem about magnitude of metric spaces was solved.

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