A group blog on math, physics and philosophy

- 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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