Computational Complexity — 11/1/2026
(Throughout this post I will assume that the recent results by OpenAI and Anthropic contain correct proofs of what they claim.)I wrote the following about 2 years ago but did not post it. It’s dated, but that’s the point.BEGIN DOCUMENT FROM TWO…
Crooked Timber — 11/1/2026
One of the recurring themes of the 2024 election campaign was the argument that “democracy is on the ballot”. Trump had attempted to overthrow US democracy, and regularly mused about the desirability of dictatorship, but had nevertheless been…
Crooked Timber — 10/31/2026
“Spiders don’t socialise. Spiders don’t work together. Spiders don’t hunt as packs. Spiders don’t use tools. Spiders don’t organise large structures. “Because if they did, we’d all be dead.” — As it turns out, this isn’t quite correct. Some…
The n-Category Café — 10/31/2026
There’s a job in ecological modeling with categories you can apply for at Topos Research UK.
The n-Category Café — 10/30/2026
Bruce Bartlett has won a prize for expository writing!
Computational Complexity — 10/30/2026
In this week’s big news (though it’s only Wednesday), OpenAI announced a proof that the exponential-time hypothesis is false, with a Lean proof. Later Anthropic released their own paper. Ryan O’Donnell and Shiva Kintali posted follow-up proofs,…
Crooked Timber — 10/29/2026
Last week, my partner and I performed a familiar ritual of modern co-parenting: I told him that I felt overburdened by mental labor, and we made a plan to adjust. The logistical work of managing our kids’ lives has recently ramped up steeply, and…
The Aperiodical — 10/28/2026
Double Maths First Thing picked the wrong week to give up Lemsip Hello! My name is Colin and I am mathematician on a mission to spread the joy and delight of doing, thinking about, playing with and suffering from mathematics. I am not on a mission…
The n-Category Café — 10/26/2026
The sign of li(x) - pi(x) changes infinitely many times, according to Littlewood. Can we graph this?
Computational Complexity — 10/26/2026
(Lance and Bill did a vidcast about the OpenAI Document Drop. See here. My spellcheck thinks that vidcast and OpenAI are not words.) The Metric TSP problem is TSP where, for all vertices (x,y,z), (w(x,z) \le…
Crooked Timber — 10/26/2026
I went to see NAZA last week. In case you haven’t heard yet, NAZA is a documentary on the Israeli machinery of mass killing in Gaza. It draws a picture of this machinery by interviewing 24 Israeli who work(ed) for the Israeli Defense Forces (IDF)….
Crooked Timber — 10/25/2026
The n-Category Café — 10/24/2026
What is the quasi-Riemann hypothesis, and what does it imply?
Computational Complexity — 10/21/2026
In fall of 1985 when I started graduate school, my advisor Michael Sipser asked me to read an early handwritten manuscript by Johan Håstad of his switching lemma. I spent many hours decoding the notation and working through the messy conditional…
Math ∩ Programming — 10/21/2026
Table of Contents This is part of a tutorial series on the CKKS homomorphic encryption scheme. In the last article, we described the ciphertext-ciphertext addition and multiplication operations. We ended with two challenges, one to restore the…
The Aperiodical — 10/21/2026
Double Maths First Thing is all a bit chaotic Hello! My name is Colin and I am a mathematician on a mission to find and share the joy in maths, puzzles and working things out. I’m likely to have some availability for new students in the next few…
Joel David Hamkins — 10/21/2026
Joel David Hamkins and Atticus Stonestrom, “The pattern of consistency strength for membership in a computably enumerable set,” preprint. arxiv:2610.24919. Abstract. Every computable preorder relation on the natural numbers is the order of…
Crooked Timber — 10/19/2026
Last week I was in the Public Records of Victoria (PROV) looking at records of the Melbourne mint. My substack about the Sydney mint by rights should come first, but I’m fresh-ish off the plane so I’m doing this one first. It was a bit extravagant…
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…
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
Below are three galleries. One of fungi and growths, one with p5js and one with Desmos. Existential ClosureThirty-one fungi, lichen, slime molds in mathober-land. Created in procreate using camera-roll images and the Trudell Mushrooms of the…
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…
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…
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 →
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 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
Note: this post has been updated. MathString.org is the same site as inquiries.link. The domain has been updated to better work with internet filters.I would like to start this post with a thank you to members of this site for making both this blog…
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…
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.
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…