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…
Crooked Timber — 9/20/2026
At La Pointe courte, in Sète, France, where Agnès Varda shot her film of the same name.
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…
Crooked Timber — 9/17/2026
One of the oddities of our age is that there are two apocalyptic sensibilities that have captured two almost completely distinct groups of the scientifically literate not least among the youth: one can be found among those concerned with the…
Crooked Timber — 9/16/2026
About a year ago, I started doing this thing that the people who do it just call “climbing.” Basically, you go to a giant place where there are things strategically stuck to a wall, and you grab them with your hands and feet trying to climb to the…
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 Speaker: Joel David ……
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…
Crooked Timber — 9/13/2026
Last week, on 9 September, Donald Trump promised that, if the Republicans retained their Congressional majorities in the November elections, he would send $5000 to every adult American citizen. The cost has been estimated at around $1.2 trillion….
Crooked Timber — 9/13/2026
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…
Crooked Timber — 9/8/2026
Here’s another of my posts about being a peace activist in the UK in the early eighties. This one is mainly about my time in secondary school. When I linked to the substack version from Facebook a couple of people recognized some of the people…
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…
Crooked Timber — 9/7/2026
This week all the news services carried a story about the Dutch central bank shipping many, many tons of gold from the Mad King-led United States to the custody of cooler heads (well, cooling, we hope, after that very hot summer) in London. On the…
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…
Crooked Timber — 9/4/2026
… but that won’t change any minds It’s now six months since the US-Israel war on Iran began with air attacks which killed the Iranian leader Ayatollah Khamenei as well as around 200 schoolchildren. The war has dragged on a long time. But before…
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…
Crooked Timber — 9/3/2026
We are being flooded with spam coming from fake gmail.com addresses at the moment. So, for the moment, I’ve set our spam filter to delete everything from gmail.com If you want to post a real comment, just give a spurious address like anon.com and…
Crooked Timber — 9/3/2026
I wrote this post a couple of weeks ago, responding to a challenge* to write about the economic aspects of Taylor Swift. Not long after that, the world lost the marvellous Dolly Parton. Parton’s lifetime income almost certainly exceeded a billion…
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.
Crooked Timber — 8/31/2026
A couple of years ago, I edited a volume on pluralizing political philosophy, in particular in relation to economic and ecological inequalities. It was put together under the Covid-pandemic, so we had to swap the authors-conference for online…
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…
Crooked Timber — 8/28/2026
At the end of a glorious day hanging out at galleries, antique shops and the pub with friend and historian Matt Allen we walked to the station at Blackheath in the Blue Mountains, Australia (where I live) for Matt to get the train back to the city….
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…
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.
Crooked Timber — 8/25/2026
Ceuta, the small Spanish enclave on the coast of North Africa, has been in the news lately. And I’ve seen a couple of articles mentioning that Ceuta has been a Spanish possession since 1668. Which triggered my inner history nerd, because on one…
Crooked Timber — 8/23/2026
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,…
Crooked Timber — 8/21/2026
A summer anecdote: when on holidays in France, on a sunny August day, we visited a castle garden in rural Normandy. When buying the tickets, we got the usual: “Your postcode or country of origin?” We said “Germany,” and suddenly the face of the…
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…
The Aperiodical — 8/19/2026
Double Maths First Thing gets its kicks on a.19792ish Hello! My name is Colin and I am a mathematician on a mission to spread the joy and love in doing maths, figuring stuff out, getting stuck and getting unstuck again. You’ll need to forgive a…
The Aperiodical — 8/17/2026
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 Ellie Sleightholm about her YouTube…
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 Aperiodical — 8/12/2026
Double Maths First Thing has an army of clones Hello! My name is Colin and I am a mathematician on a mission to spread the joy and delight of working things out. I had a delightful chat with young Bill about complex numbers and matrices and noticed…
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…
The Aperiodical — 8/5/2026
Double Maths First Thing is square in hex AND decimal! Hello! My name is Colin and I am a mathematician on a mission to share the joy and love in mathematics, to show that it’s more than the mindless memorisation of arbitrary rules. Guess who…
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…
The Universe of Discourse — 7/27/2026
(Previously) Flukes and Breakthroughs In the NBA 2015–16 season, Steph Curry set the all-time single-season record for three-point field goals, 402, completely crushing the old record of 286. Curry’s record still stands. The New York Times was…
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…
The Universe of Discourse — 7/19/2026
Previously: Ordinal numbers and basic set theory Ordinals as nim-heaps [Yesterday][prev-2] I talked about the game of Nim, which involves two players taking beans from several piles, and an extension that includes green tokens that behave a…
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.
The Universe of Discourse — 7/18/2026
Previously: Ordinal numbers and basic set theory We’re going to get to in a long and roundabout way. First I want to talk about the game of Nim. Nim Nim is a very simple game for two players. There are some piles of beans, which are…
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…
The Universe of Discourse — 7/10/2026
This post is going to be about what infinite ordinal numbers are, and about is in particular. I had a brainwave a while back (18 months now, wow, I have definitely not been blogging enough) and suddenly understood much better than I did before. …
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…
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…
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…
Computational Complexity — 6/28/2026
Peter Brass is a prior NSF theory director. He has written an intelligent guest post on the new NSF guidelines that we present here. You have received many mails regarding the proposed OMB Uniform Guidance for federal grant making. It is a very…
The Universe of Discourse — 6/27/2026
A while back I related how I had been mocked by an English person for using the word “burglarize”. I ended by saying: Okay, whatever. Brits have been mocking the American language for centuries now. Let them go ahead. We all know who won that…
Computational Complexity — 6/25/2026
When you start thinking deeply about a mathematics problem you may enter the “zone”, a period of intense focus where you think solely about the problem and potential solutions, and more importantly block out all other thoughts and even lose track…
Computational Complexity — 6/22/2026
(All references in this blog post can be found in the main article the post is about which is here.)Recall that (R(s,k) ) is the least (n) so that, for all 2-colorings of the edges of (K_n), there is either a RED (s)-clique or a BLUE…
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 ……
Computational Complexity — 6/17/2026
Last week I saw a talk by Northwestern professor Nina Wieda on the history of the Silk Road, a network of trading routes across Asia active from the second century BCE until the mid-15th century. I knew of the Silk Road but was surprised by how…
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
Joel David Hamkins — 6/8/2026
This will be a series of graduate lectures at Peking University, two lectures per week beginning mid-June and proceeding into July. Topics. We shall aim to cover the central results in the modal logic of forcing, including an exploration of ……
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…