Computational Complexity — 10/19/2026
So what to make of OpenAI and the future of Math? Here are some points I’ve either:thought of myself or heard from a human or read from an AI.Throughout this post I use the term AI to mean an AI system that produces mathematical results. I decline…
Crooked Timber — 10/18/2026
Crooked Timber — 10/18/2026
Colleagues Cyril Hédoin and Pablo Paniagua are running a symposium on “PPE” (philosophy, politics, and economics) over there. They follow James Buchanan’s classic question, from 1964, of “What should economists do?” Given that Crooked Timber is,…
Crooked Timber — 10/15/2026
From the Fall of the Berlin Wall until about a decade ago my animating intellectual question was pretty much, ‘how did a small number of natural sciences become qualitatively more impressive than the social sciences?’ Because I met the late George…
Computational Complexity — 10/15/2026
In the OpenAI Math Drop some papers have Lean proofs, others don’t, including L = RL = BPL. Why is it hard to formalize results on complexity classes? Let’s consider an example, Lautemann’s proof that BPP is in (\Sigma^p_2). It’s a beautiful…
The Aperiodical — 10/14/2026
Double Maths First Thing had to look up where Wells is Hello! My name is Colin and I am a mathematician on a mission to spread the joy and delight in the beauty of maths, and the human effort of figuring things out. This week I’ve been taking joy…
Joel David Hamkins — 10/14/2026
This will be a talk for the NYC Math Education and Communication Seminar, 20 October 2026, 12:15 – 1:15pm, at the CUNY Graduate Center in midtown Manhattan. Epistemic Logic Puzzles for Mathematicians Joel David Hamkins, O’Hara Professor of Logic,…
Computational Complexity — 10/12/2026
Lance blogged on the OpenAI solutions to many problems here and also created a website of the TCS-related problems OpenAI claims to have solved here.Terry Tao seems to have had an infinite number of guest bloggers talking about AI and Math even…
Crooked Timber — 10/12/2026
(This is an updated version of something I posted 20 years ago, on the 20th anniversary, and 10 years ago, on the 30th anniversary, of my friend’s death) About 43 years and 3 weeks ago the soon-to-be late lamented You and Yours had one of those…
Crooked Timber — 10/12/2026
When I posted a piece on the latest failure of the US Navy a while back, I predicted that I wouldn’t be changing any minds. That was one of my safer predictions, confirmed almost immediately by Robert Farley at Lawyers, Guns and Money. Farley…
Math ∩ Programming — 10/11/2026
Table of Contents This is part of a tutorial series on the CKKS homomorphic encryption scheme. In the last article, we introduced symmetric and public-key encryption and decryption routines, and implemented modular polynomial arithmetic (using NTTs…
Crooked Timber — 10/11/2026
Chris just let me know that Alan Thornett has died. International Viewpoint has a long obituary here. I doubt The Graun will carry an obit, and The Times certainly won’t. I got to know members of the Oxford branch of what was then the Socialist…
Crooked Timber — 10/11/2026
Good Fibrations — 10/10/2026
Memantine is a cheap generic dementia drug that might help with chronic pain. Nobody has tested it properly, and that’s mostly about money.
Math ∩ Programming — 10/8/2026
A quick update on our homomorphic encryption compiler HEIR: our partners at Belfort have a hotword detector in FHE that runs in 350ms. More specifically, they launched a live demo at hotword.belfortlabs.com, that has a little game inspired by the…
Computational Complexity — 10/7/2026
I wrote the post below last week. That was a quaint and quiet time. Last night OpenAI released a treasure trove of 722 manuscripts solving 372 major open problems in mathematics including from theoretical computer science:A proof of the unique…
Crooked Timber — 10/7/2026
Having alighted from the bus in an industrial area I’d never been before, I found the right building and descended the stairs. As instructed by email, I rang the doorbell. I was immediately met by the two cutest dachshunds you’ve ever seen. My…
The Aperiodical — 10/7/2026
Double Maths First Thing is tangled up in knots Hello! My name is Colin and I am a mathematician on a mission to spread the joy and delight in doing and discussing maths. This week I have been in error-recovery mode after drawing a wrong conclusion…
The Aperiodical — 10/6/2026
The next issue of the Carnival of Mathematics, rounding up blog posts from the month of September 2026, is now online at Ben Leis’s Math Off The Grid. The Carnival rounds up maths blog posts from all over the internet, including some from our…
The Universe of Discourse — 10/6/2026
release-ref { color: #b34700 } metainfo-commit { color: #007a5a } tree-commit { color: #0072b2 } While I was still at ZipRecruiter, our Git monorepo was developing an annoying problem: it had too many refs. Whenever there was a deployment of any…
Crooked Timber — 10/5/2026
I am sharing this from Benjamin Ferguson and Roberto Veneziani, with thanks (I will be back to post myself soon, apologies for the long radio silence): We are conducting a survey about academic philosophers’ views of (1) the effect of exploitation…
Computational Complexity — 10/5/2026
(This was written about 9 months ago. Its not out of date… yet)People think that AI is going to DESTROY some jobs and CREATE some jobs. It may be too early to tell if this is true. Even so, here are some thoughts. 1) Who will win? Who will…
Crooked Timber — 10/4/2026
Crooked Timber — 10/4/2026
(A piece I ran in The Guardian) The recent panic about a breach of Medicare computer security by an “AI agent” contrasts sharply with other recent cases such as the Telstra and Optus outages that left many Australians unable to reach Triple Zero….
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 1-10 (perimeter of an…
Joel David Hamkins — 10/1/2026
This is a rough outline syllabus for the class I am teaching this semester at the University of Notre Dame on the philosophy and logic of games. University of Notre Dame … Continue reading →
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…
Crooked Timber — 9/30/2026
People here often ask me what part of England I grew up in, which isn’t a particularly easy question – I was born in the north, but then lived in South Wales and then various place in the South all of which were west of London. Oxford is sort of…
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…
Crooked Timber — 9/30/2026
These days, methodological nationalism is finally sparking a heated debate in philosophy. A flurry of new papers has recently been published in open access: Baycan, 2026; Beaton, 2026; Milioni, 2024; Sager, 2021; Ujlaki, 2026. Naturally, some of…
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…
Crooked Timber — 9/29/2026
I’m mostly a descriptivist when it comes to the meaning of words. If everyone uses “literally” to mean “figuratively”, that’s fine by me. Same if there is no likelihood of confusion, as with “aggravate” for “annoy” as well as “exacerbate”. But that…
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…
Joel David Hamkins — 9/28/2026
This will be a talk for the CUNY Student Logic Seminar on 23rd October 2026, 5pm at the CUNY Graduate Center in midtown Manhattan. On the fundamental theorem of finite games Abstract: We will give five different proofs of the … Continue reading →
Crooked Timber — 9/26/2026
Books that complain about the state of universities, written by university members, are probably as old as the institution itself. And if there is one institution that needs – and should be able to provide – self-reflection, it is universities….
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…
Joel David Hamkins — 9/15/2026
This will be a talk for the new Mathematical Philosophy Seminar here at the University of Notre Dame. Monday, September 21, 2026, 4pm, in Malloy Hall. Title: How the continuum hypothesis could have been a fundamental axiom, one necessary for ……
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.
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 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…
Joel David Hamkins — 8/26/2026
This will be a talk for the Logic Seminar at the University of Notre Dame on 15 September 2026, 2pm in the Hayes-Healy building room 125. The contingent HOD dichotomy Joel David Hamkins Abstract: We shall discuss the contingently contingent ……
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,…
Joel David Hamkins — 8/21/2026
This will be a talk for the CUNY Logic Workshop, 23 October 2026, 2pm, at the CUNY Graduate Center in midtown Manhattan. The Contingent HOD Dichotomy Joel David Hamkins, O’Hara Professor of Logic, University of Notre Dame Abstract: We shall ……
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…
Joel David Hamkins — 8/8/2026
I am very pleased to announce that I have been awarded the Alternative Fields Medal in recognition of “Excellence in exposition of mathematics to a popular audience.” The award cites my various books as well as my participation on MathOverflow, ……
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
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…
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…
Joel David Hamkins — 7/7/2026
This will be a talk for the Fudan University logic group on 16 July 4:00pm in Shanghai. Abstract: The axiom V = HOD is contingently contingent—more precisely, it is class-forcing contingent that V = HOD is contingent with respect to set … Continue…
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
Joel David Hamkins — 6/20/2026
This is a talk for the Workshop on Mereology at Shandong University in Jinan, China, a part of the week-long conference Week of Fusion Philosophy, 22-26 June 2026. The mereology talks are on 22 June 2026. Title: Set-theoretic mereology as ……
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.