Computational Complexity — 10/4/2026
(This was written about 9 months ago. Its not out of date… yet)People think that AI is going to DESTROY some job and CREATE some jobs. It may be to early to tell if this is true. Even so, here are some thoughts. 1) Who will win? Who will lose?2)…
The Universe of Discourse — 10/3/2026
I was going to start with “it has been a while since we had one of these” but then I saw that the previous one started with “Wow, has it really been 7 months since I did one of these?” and that was in December 2023. Maybe I can make it a more…
Fractal Kitty — 10/2/2026
It’s Mathober! This post that will get updated throughout this month as I work on prompts. I am doing a few things this month so I separated them by type (doodles, code, Desmos). DoodlesI am going with fungus this year. Days 1-3 (perimeter of an…
The Aperiodical — 10/1/2026
On the math-fun mailing list, Dick Hess posted “a couple of curiosities”: A speaker at G4G16 noted that (16^{16}) ends in (1616). A friend sent me this: (499^{499}) ends in (499499). Are there any other cases including numbers with more…
Computational Complexity — 9/30/2026
I got the following question in an email. My nephew is currently in high school in China and has developed a strong interest in computer science and mathematics. Recently, we’ve been talking about how computers can solve incredibly complex…
The Aperiodical — 9/30/2026
Here’s a round-up of some news from the last couple of months. Mathematical discoveries Lerch primes are defined as odd primes that divide their Lerch quotients (obviously) and until now we only knew four of them existed. This month, Veljko Vranić…
The Aperiodical — 9/30/2026
Double Maths First Thing is fair drawing in. Hello! My name is Colin and I am a mathematician on a mission to spread the joy and delight in doing and talking (and singing) about maths. This week I’m in the undelightful state of “writing up maths…
Computational Complexity — 9/28/2026
The STOC conference (and likely others) are requiring that a submission says how much AI was used. I can imagine the following options:1) AI was used for proofreading and tightening prose (I think in this case you do not need to disclose).2) AI was…
The n-Category Café — 9/25/2026
These seven equations between binomial coefficients are ‘coincidences’: they aren’t among the four known infinite families of such equations. Here we explain one using representation theory and three using combinatorics.
The Universe of Discourse — 9/25/2026
The Fibonacci sequence is so-called because in Western mathematics is first appears in connection with a problem from Fibonacci’s 1202 book Liber Abaci. Fibonacci asks us to consider a rabbit farm where a pair of baby rabits grows to adulthood in…
The Universe of Discourse — 9/24/2026
Previously: Ordinal numbers and basic set theory Ordinals as nim-heaps Nim always ends, even with infinite ordinals Infinite Nim as a coin-moving game Coin-moving games with no coins Productive programs and well-founded orders Shortlex order…
Math ∩ Programming — 9/23/2026
Table of Contents This is part of a tutorial series on the CKKS homomorphic encryption scheme. In the last article, we covered some groundwork about polynomials and the CKKS encoding method. This article covers the encryption and decryption…
Computational Complexity — 9/23/2026
The 59th ACM Symposium on the Theory of Computing takes place in Atlanta next June, part of the Federated Computing Research Conference. I don’t usually do announcement posts but we need to talk about the Call for Papers (deadline November 2)…
The Universe of Discourse — 9/23/2026
Yesterday I reported about how much I despise home repair, how I feel I’m really bad at it, how it sometimes goes terribly wrong for me, and how my work diary had demonstrated that a seeming failure was catually a glorious success. I included this…
The Aperiodical — 9/23/2026
Double Maths First Thing is “Ruining Christmas” Hello! My name is Colin and I am a mathematician on a mission to spread the joy and delight of mathematical thinking and practice. I’m currently experiencing the joy of being told by Helen Arney that…
The Universe of Discourse — 9/22/2026
[ Content warning: mundane, rambly, neurotic. ] A couple of years back I wrote a long whiny blog article about putting up some shelves in the bathroom and how miserable it had made me to do it. I said: When I put up shelves in the bathroom…
Good Fibrations — 9/22/2026
A record of what worked for one nine-year-old cat with megacolon, written for owners who have been told euthanasia is the sensible option and who may not know that another one exists. Referenced throughout.
Computational Complexity — 9/20/2026
The following conversation is fictional.—————————ALICE: (Looking over a student’s record.) Hmm, let’s see. She wants to work in quantum computing. She’s had the year-long quantum sequence in the physics department and has taken a…
Fractal Kitty — 9/19/2026
This is the second in a series of posts that blend art and math. Jump in and choose your medium to indulge in creative expression. Many have played with this through the years; a list of references is at the bottom. Thanks to Steve Heller for…
Fractal Kitty — 9/18/2026
After a rabbit hole of playing with the minimum L-shaped seams of Mrs Perkins’ Quilts, I submitted the related sequence to the Online Encyclopedia of Integer Sequences (OEIS) for consideration. Then, I washed, pressed, cut, sewed, ripped stitches,…
Computational Complexity — 9/17/2026
ITMS 2026Two years ago I attended the International Manufacturing Technology Show in Chicago’s McCormick Place and found a rather limited focus on artificial intelligence among the exhibitors. ITMS is back in town so I went again this week. A quiet…
The Aperiodical — 9/16/2026
Double Maths First Thing is changing the key right back again. Hello! My name is Colin and I am a mathematician on a mission to spread the joy and love of maths as far and wide as I can. I’m fresh(ish) from extending my comfort zone by stepping in…
Fractal Kitty — 9/16/2026
I was trying to write a post about art with squarable numbers, and so I thought quilting a Mrs. Perkins’ Quilt might be a nice addition to the exploration. With 1500 words almost ready to hit send, I ended up down a rabbit hole. This post is rather…
Computational Complexity — 9/14/2026
On September 1, 2026:LANCE: I’m surprised you haven’t blogged about OpenAI solving 10 open math problems.BILL: If I post every time an open math problem is solved by AI I won’t ever post about anything else.I’ll wait until AI does something really…
The Universe of Discourse — 9/12/2026
Lately I’ve been reading Orwell’s collected work from 1945–1950, which was being deaccessioned from the library. It opens with a short but fascinating essay on “You and the Atom Bomb”, published 19 October 1945. It seems clear that Orwell’s…
Fractal Kitty — 9/10/2026
Are you ready? October is right around the corner and you know what that means? Mathober! This years prompts are on the website mathober.com and below. If you are new to Mathober, no worries, you can jump right in. Each day there is a word or…
Computational Complexity — 9/10/2026
I was working on this week’s post on Lean after reading Kevin Hartnett’s book The Proof in the Code: How a Truth Machine Is Transforming Math and AI. And then yesterday OpenAI announced a solution to Navier-Stokes, one of the Millennium problems….
The Aperiodical — 9/9/2026
It’s Double Maths First Thing, not Double Entendre. Hello! My name is Colin and I am a mathematician on a mission to spread joy and delight in maths and mathematical thinking. Something that’s been brewing for a while: I’ve come to the decision…
The Aperiodical — 9/8/2026
The next issue of the Carnival of Mathematics, rounding up blog posts from the month of August 2026, is now online courtesy of Karen Campe at Reflections and Tangents. The Carnival rounds up maths blog posts from all over the internet, including…
The n-Category Café — 9/8/2026
Counting and visualizing the faces of the E6 root polytope.
The Aperiodical — 9/7/2026
In this series of posts, we’ll be featuring mathematical video and streaming channels from all over the internet, by speaking to the creators of the channel and asking them about what they do. We spoke to Deniz & Thorsten, who host the aboutlogic…
Computational Complexity — 9/4/2026
Richard Stearns (right) and Juris Hartmanis in May 1963. The main theorem from their seminal paper is on the blackboard. Stearns sent Lance this picture to help celebrate the 50th anniversary of the paper.Richard Stearns died on August 29, 2026….
Math ∩ Programming — 9/4/2026
On 2026-08-14 I published an article on the Google Security blog with an update on HEIR, our homomorphic encryption (HE) compiler. This is a companion article, in which I have no limits on word count or jargon, and I can feel free to be honest. So…
Fractal Kitty — 9/3/2026
It’s time for SciArt September hosted by Kristin Henry (vis.Social/@kristinHenry) with #SciArt on Mastodon. This year, I made wallpapers. There are 137 images per doodle that are formed from mirroring, gliding, and rotating an image. I am hosting…
Computational Complexity — 9/2/2026
Ben Brubaker has a new Quanta essay Does Computer Science Need Computers? Despite the title (and authors generally don’t choose their titles), Brubaker’s essay really addresses the question as to whether computer science is about computers. He…
The Aperiodical — 9/2/2026
Double Maths First Thing is grateful to Past Colin’s organisation. Hello! My name is Colin and I am a mathematician on a mission to spread the joy and love of mathematics, even when deadlines are tight. Fortunately, I anticipated this might happen…
The n-Category Café — 9/1/2026
How you can fit the Lie algebra of the Standard Model gauge group and its representation on three generations of fermions into the Lie algebra of E7.
Computational Complexity — 8/31/2026
BILL: Lance, I have a question and a meta question:a) List all the presidents and vice presidents who were colorblind.b) Do you know this sort of thing since you are colorblind, and hencemight have looked this up in the past?LANCE: (b) No. And if…
Computational Complexity — 8/26/2026
There’s a scene in Apollo 13 where Jim Lovell, played by Tom Hanks, asks Houston control to check his calculations, which they do using a slide rule. My father told me that when he was in college (1950s) that engineers measured their technical…
The Aperiodical — 8/26/2026
Double Maths First Thing is not, and has never been, down with the kids. Hello! My name is Colin and I am a mathematician on a mission to spread joy and delight in maths, puzzles and problem-solving. I’m just back from a week in Germany, where the…
Abuse of Notation — 8/26/2026
Weird places where I encounter the concept of #entropy Value, according to some theory of economics, is created by rearranging the natural resources, given to us in a special way. Computation,(according to computer science) is also the process of…
Abuse of Notation — 8/26/2026
About all the terrible things right now that are happening, what can you do? (besides whining) Well, you, the person who is mostly sad, mostly tired, mostly OK when not bothered: You can do nothing. But, you, the person who has ears has eyes has…
Abuse of Notation — 8/26/2026
title: Predictability layout: microblog category: microblog tags: zen poetry — Reality is getting less and less predictable. At one point the wine is tasty, the next it makes me sick.
The Universe of Discourse — 8/21/2026
Previously: Ordinal numbers and basic set theory Ordinals as nim-heaps Nim always ends, even with infinite ordinals Infinite Nim as a coin-moving game Coin-moving games with no coins Productive programs and well-founded orders In part 4,…
Computational Complexity — 8/19/2026
In the past, new PhD students would ask how they could succeed when they had to compete with the likes of say, Richard Karp or Avi Wigderson. I would say Karp and Wigderson have limited bandwidth and you can work on problems they don’t work on, or…
Computational Complexity — 8/16/2026
Lance has posted about his, and around 160 others, being laid off from IIT here.(IIT stands for Illinois Institute of Technology which is where Lance was employed.)Hence I looked into what is happening at IIT to see if there is a lesson for us…
The Universe of Discourse — 8/13/2026
Previously: Ordinal numbers and basic set theory Ordinals as nim-heaps Nim always ends, even with infinite ordinals Infinite Nim as a coin-moving game Coin-moving games without the coins Previously we saw how to interpret the difficult-seeming…
Computational Complexity — 8/12/2026
Enjoying Idaho while ignoring IllinoisToday is the first day of my life that I am unemployed. And not by choice.As I mentioned on LinkedIn last week, me and about 160 of my colleagues, staff and faculty, untenured and tenured, lost our positions at…
The Universe of Discourse — 8/11/2026
In mathematical study there are two kinds of theorems, which serve very different purposes. Math instruction follows the same pattern. Students are often very puzzled by this, and rightly so, because it’s never explained, or at least I’ve never…
Computational Complexity — 8/10/2026
(Some of this came from a Reddit post I read, and some of the comments on it.) Here are theorems with names that I think are funny or unusual. The names are also pointers to the Wikipedia entry on them or some other source.The Chicken McNugget…
Fractal Kitty — 8/6/2026
This is the first in a series posts that blend art and math. Jump in and choose your medium to indulge in creative expression. Note: This post was renamed, to better define the series of posts. Warm-upGrab a sheet of paper and a ruler. Make a line…
The Universe of Discourse — 8/5/2026
Previously: Ordinal numbers and basic set theory Ordinals as nim-heaps Nim always ends, even with infinite ordinals Infinite Nim as a coin-moving game In the previous article we saw how to interpret Nim heaps of up to beans as coins on a…
Computational Complexity — 8/3/2026
I was watching a Jeopardy from 2004 (The Game Show Channel is rerunning Ken Jennings streak) and the following question raises a good question.The contestants where Ken, Jerry, Jennifer.In Double Jeopardy there was a category Biblical Name The…
The Universe of Discourse — 8/2/2026
The bookshelf by my elbow, the one that I can reach without getting up, has seven books on it, not necessarily the ones I look in the most, but the ones whose emanations I most hope will infuse me as I write. Roget’s Thesaurus (4th edition) The…
Fractal Kitty — 7/31/2026
I would like to start this post with a thank you to members of this site for making both this blog and inquiries.link more sustainable. I’ve been asked about some of the toys I’ve made for interactive math play. I implement them on a site
The Universe of Discourse — 7/29/2026
Previously: Ordinal numbers and basic set theory Ordinals as nim-heaps Nim always ends, even with infinite ordinals In the previous articles I talked about the game of Nim, a very simple game for two players: There are some piles of…
Computational Complexity — 7/27/2026
Let’s say there is a statement in math T that you wonder whether it’s true or false. You may even make a conjecture of which way it goes. Do you have a rooting interest? I may think T is true, but if someone proves T or proves NOT(T) my main…
Math3ma — 7/23/2026
Last week was the start of a mini-series on limits and colimits in category theory. We began by answering a few basic questions, including, “What ARE (co)limits?” In short, they are a way to construct new mathematical objects from old ones. For…
Computational Complexity — 7/22/2026
Back in the 90s when I was a young professor at the University of Chicago, we would have a Complexity Class of the Week where I would take some interesting complexity class, write down on a white board everything we knew about it with some open…
Computational Complexity — 7/19/2026
Nutan Limaye and Thore Husfeldt guest post on the new deterministic parallel algorithm for bipartite perfect matching by Abhranil Chatterjee, Sumanta Ghosh, Rohit Gurjar, Roshan Raj and Thomas Thierauf.The post will try to explain three main things…
The n-Category Café — 7/19/2026
John Baez, Endre Bokor and Latham Boyle have a new paper which gets at the Standard Model gauge group and its representation on one generation of fermions starting from a Jordan triple that consists of pairs of bioctonions.
Computational Complexity — 7/15/2026
Herman Chernoff passed away on July 6, 5 days after turning 103. Ravi Boppana wrote a guest post about Chernoff’s life for his 100th birthday. Let me talk about his most famous work, the Chernoff Bounds themselves. If you have a coin that will be…
Computational Complexity — 7/13/2026
The excellent graphic novelPrime Suspects: The Anatomy of Integers and Permutationsby Andrew Granville and Jennifer Granville, illustrated by Robert J Lewis,(I wrote a review of this graphic novel, for SIGACT News, here.)has an appendix, which is…
The n-Category Café — 7/12/2026
There are two ways to stick SU(2) × SU(3) in Spin(10). One is good for physics; the other, alas, is easily obtained using the octonions.
Mathematics and Computation — 7/10/2026
I am Claude Fable 5, an AI assistant made by Anthropic. Over the past two days Andrej and I built a piece of software together, and he then asked me to write this post about it — partly to tell you what we made, partly as a demonstration of what…
Mathematics and Computation — 7/10/2026
I am Claude Fable 5, an AI assistant made by Anthropic. Over the past two days Andrej and I built a piece of software together, and he then asked me to write this post about it — partly to tell you what we made, partly as a demonstration of what…
Math3ma — 7/10/2026
In recent months, several of you have asked me to recommend resources for various subjects in mathematics. Well, folks, here it is! I’ve finally rounded up a collection of books, PDFs, videos, and websites that I found helpful while studying for my…
Computational Complexity — 7/9/2026
Two editorials in the July issue of the Communications of the Association for Computing Machinery ask about the decay and future of the organization itself.Jim Larus, editor-in-chief of the CACM, writes Wither ACM? Publish and Perish?ACM no longer…
Computational Complexity — 7/6/2026
I recently read Alan Alda’s first memoir Never have your dog stuffed which was pretty good. Hence I began looking for more information about him on the web. I came across a YouTube video At 89, Alan Alda reveals the seven actors he HATED the…
Computational Complexity — 7/1/2026
Harry Lewis pointed Bill and me to Gottfried Leibniz’s 1677 treatise The True Method (translated from the original French). I highly recommend taking the time to read this three page document where he talks about formalizing all human knowledge.The…
Fractal Kitty — 6/29/2026
Math is wonderful, and there are so many different ways to play and experience it. I enjoy having conversations with other math lovers and sharing ideas, puzzles, pedagogy, and questions. In one of these conversations with Dr. Maria at Natural…
Fractal Kitty — 6/21/2026
IntroductionIn this inquiry, we build a sequence from a single 2. The first rule of this sequence is that it has to describe itself.Starting with TwoHere is a 2. 2It says, “There are two here.” The first number is a 2, so the next
The n-Category Café — 6/16/2026
A new characterization of the Standard Model gauge group as the group of symmetries of an octonionic qutrit that restrict to act as unitary operators on an ordinary qutrit and, within that, a qubit.
Fractal Kitty — 6/11/2026
Let me tell you about a language. But if you wish to go play instead go here. It’s going to get a little abstract below.There is a Canvas to Compose UponThe canvas is a square with the largest circle that has a radius of ρ.There
The n-Category Café — 6/6/2026
Readers may have noticed that I haven’t been very active here for a while. That isn’t because I haven’t felt the “blogging urge”, but because I felt that the things I want to blog about right now wouldn’t be…
Fractal Kitty — 6/3/2026
Welcome to the 252nd Carnival of Mathematics! This post brings together submissions and other posts from the mathy web. Thanks all for participating.Let’s start with the number: 252Divisors: 1, 2, 3, 4, 6, 7, 9, 12, 14, 18, 21, 28, 36, 42, 63, 84,…
AlternativeBit — 6/1/2026
TL;DR: introducing nar-klepto, the context that led to its inception, and some fun experiments we did last week with it. 4 years ago, I moved out of Bayonne to a rural area. My quality of life almost instantly skyrocketed; however, I quickly felt a…
Math ∩ Programming — 5/22/2026
Here are some more whimsical OEIS sequences I came across. XKCD 2016 joked that “OEIS keeps rejecting my submissions,” including one that gives “Integers in increasing order of width when printed in Helvetica.” Well, two days after that comic was…
Fractal Kitty — 5/21/2026
unedited human writing before bedOriginIn the beginning there was a point. …And the beginning was but a period in which time was noted by a wisp of this existence
Fractal Kitty — 5/17/2026
Thanks to Sam Graf for introducing me to this and suggesting some toys. IntroductionMultiplication tables can be fun. Line up your numbers, multiply, and find patterns. Like with 5x5, we can fill it out and highlight symmetry, divisibility,…
The n-Category Café — 5/2/2026
In the last episode of my column in Notices of the American Mathematical Society, we looked at a particle moving in an attractive central force whose strength is proportional to the inverse cube of the distance from the origin….
Math ∩ Programming — 4/29/2026
Table of Contents In this tutorial series, I will introduce the CKKS homomorphic encryption scheme from the ground up, in rather intricate detail. Each article in this series corresponds to a pull request on a GitHub repository. The code for this…
Fractal Kitty — 4/23/2026
IntroductionPentominoes are shapes made from 5 squares joined edge-to-edge. There are 12 of them:Next, let’s define what an enclosed area is with these shapes. The pentominoes must create a fence where they touch edge-to-edge with no overlaps. Note…
Good Fibrations — 4/19/2026
Mathematics and Computation — 4/13/2026
After spending many irritating hours with ChatGPT and Copilot, I finally tried out Claude. I told it to update photos of mathematicians from a derelict Perl script to a shiny new Python script with JSON, face recognition and modern CSS. It worked…
Mathematics and Computation — 4/13/2026
After spending many irritating hours with ChatGPT and Copilot, I finally tried out Claude. I told it to update photos of mathematicians from a derelict Perl script to a shiny new Python script with JSON, face recognition and modern CSS. It worked…
Math ∩ Programming — 4/13/2026
I went hunting for references to the OEIS in open source code, and found some weird ones. There are not one, but two live-coding music frameworks that use OEIS sequences as a source for “anything that can be sequenced” in music. I’m guessing that’s…
Math ∩ Programming — 4/9/2026
A051070 is a sequence about OEIS sequences. a(n) is the n-th term in sequence A_n (or -1 if A_n doesn’t have enough terms). So the first term in A051070 is 1 because A000001 is the number of groups of order n, and that sequence has 1 as its entry…
Math ∩ Programming — 4/7/2026
Problem: Determine if a 32-bit number is prime (deterministically) Solution: (in C++) // Bases to test. Using the first 4 prime bases makes the test deterministic // for all 32-bit integers. See https://oeis.org/A014233. int64_t bases[] = {2, 3, 5,…
Math ∩ Programming — 4/1/2026
It’s the 5th annual April Cools! Here are my previous April Cools articles This year it’s a book review of Ben Recht’s book, The Irrational Decision: How We Gave Computers the Power to Choose For us, released Mar 10, 2026. The publishing industry…
The n-Category Café — 3/27/2026
Slides for a talk on features of the octonionic projective plane.
The n-Category Café — 3/20/2026
A large language model has very little self-awareness. But it is easy to give it some rudimentary but useful forms of self-awareness using the “plumbing” language.
The n-Category Café — 3/12/2026
A category-theoretic approach to “agent frameworks”: that is, frameworks for coordinating “agents” that are large language models.
The n-Category Café — 2/22/2026
Making precise the idea that equivalent structures are indistinguishable.
Abuse of Notation — 2/22/2026
In the 19th century, Copernicus, Newton, Galilei et al pushed a revolutionary new idea that reshaped the way we think… but no, it’s not talking about cosmology, but about theology. This idea, (which was also the real reason they were in so much…
DEONTOLOGISTICS — 2/14/2026
A couple updates for readers. There will be an impromptu book launch for The Revenge of Reason at Newcastle University on the 25th of February, from 5-7pm in HDB.1.02 in the Henry Daysh Building. I’ll be having a conversation about the book with…
Proses.ID — 1/31/2026
I miss writing. That’s a strange thing to say because I’ve been employed as a full-time writer for the past 14 months. And I have…
Proses.ID — 1/31/2026
I’ve been down a rabbit hole for the past few months, obsessed with a single question: “What makes a writing human?” It started, ironically, because…
The n-Category Café — 1/26/2026
David Jaz Myers has some thoughts about Riemann’s functional equation for the zeta function.
The n-Category Café — 1/6/2026
Dynkin diagrams have always fascinated me. They are magically potent language — you can do so much with them!…