Computational Complexity — 12/14/2025
ONEThe following headline confused me: Trump, 79, Deletes Weird AI Video Shilling Magic Beds (see here). Was Weird Al selling magic beds? Magic beds?! How does that relate to President Trump? What’s going on?The problem is the…
The Aperiodical — 12/14/2025
We chatted to Chris Dawson – the creator of Jigsy, a shape-based puzzle app which we’ve enjoyed playing – to find out more about where it came from and how it was developed. What is Jigsy? A brand new spatial reasoning visual puzzle concept. At its…
Crooked Timber — 12/14/2025
Computational Complexity — 12/13/2025
The playwright Tom Stoppard passed away at the age of 88 on Nov. 29, 2025.ONE) He wrote many plays and some movies. Below I highlight his works whose themes I think will be of interest to my readers (Or at least to me—your mileage may…
Computational Complexity — 12/11/2025
Watching Mathematicians at Work (AI generated)The Smithsonian Natural History Museum has a FossiLab where visitors can peek through windows watching scientists prepare fossils for conservation. Maybe we should have a similar exhibit at math museums…
The Aperiodical — 12/10/2025
Double Maths First Thing knows where its towel is Hello! My name is Colin and I am a mathematician on a mission to spread the joy and delight of puzzling stuff out. It’s a busy week here, with client work, children at carol services, Christmas…
Crooked Timber — 12/10/2025
I am at the airport in Melbourne (again). I’m sitting in the window eating one of those excellent boxes of kale, broccoli, beans, seeds, peas and a boiled egg that I am grateful are now available at airports. Next to me a father and daughter are…
Crooked Timber — 12/10/2025
My last post described my attempt to generate a report on housework using Deep Research, and the way it came to a crashing halt. Over the fold, I’ve given the summary from the last version before the crash. You can read the whole report here,…
Crooked Timber — 12/9/2025
I’ve long been interested in the topic of housework, as you can see from this CT post, which produced a long and unusually productive discussion thread [fn1]. The issue came up again in relation to the prospects for humanoid robots. It’s also at…
The Aperiodical — 12/8/2025
Maths Week England happened a couple of weeks ago. I had put my name on the speaker directory, and sure enough a maths lead from a primary school in County Durham emailed me to ask if I could go in and do something for them. It’s been a few years…
Crooked Timber — 12/7/2025
Blog - Logic Matters — 12/6/2025
I’m still hoping to get a new version of Introducing Category Theory out in January. I want to rewrite the chapter on power objects which is a bit dense, and also carefully read through Part II once more for reader-friendliness: but changes should…
Crooked Timber — 12/5/2025
I spent a good chunk of the afternoon watching l’Etabli, the film of Robert Linhart’s book (which I own but have never read). It is an arresting depiction of the brutality of the assembly-line and the racalialised hierarchies at work in the…
The Aperiodical — 12/5/2025
A conversation about mathematics inspired by a light pattern in a tea cup. Presented by Katie Steckles and Peter Rowlett, with special guest Dominika Vasilkova. Cardioids in coffee cups – Chalkdust Cardioids in Coffee Cups – Numberphile…
The n-Category Café — 12/4/2025
An introduction to the bioctonionic plane and the mathematics needed to understand it.
Computational Complexity — 12/4/2025
(ADDED LATER- a Sad Note. The older Charles Lin passed away in December of 2025.)In 2001 I supervised Charles Lin’s Master’s Thesis, which was on Private Information Retrieval.In 2025 I supervised Charles Lin’s Master’s Thesis, which was on Ramsey…
Computational Complexity — 12/4/2025
Inspired by Daniel Litt’s X PostStarted asking mathematicians whose career started before the internet if they think Google, email, etc. have sped up the pace of math research. Wide variety of opinions but the broad consensus seems to be “yes,”…
The n-Category Café — 12/3/2025
A complex-analytic perspective on the indefinite integral of 1/x.
The Aperiodical — 12/3/2025
Double Maths First Thing is making a list but not checking it. Hello! My name is Colin and I am a mathematician on a mission to spread mathematical joy and delight, even when I’m not feeling much of either. December is a tough month for me:…
Crooked Timber — 12/3/2025
I heard a rumour that London IT professionals have selected the pub where they will meet when the internet goes down. It is apocalyptic thinking, perhaps, but it also feels plausible. Though the internet feels permanent, stable and sufficiently…
The Aperiodical — 12/2/2025
The next issue of the Carnival of Mathematics, rounding up blog posts from the month of November 2025, is now online at TomMaths. The Carnival rounds up maths blog posts from all over the internet, including some from our own Aperiodical. See…
The Aperiodical — 12/1/2025
Here’s a round-up of maths news from October and November 2025 which we didn’t otherwise cover on the site. Awards and Grants Applications for participation in the 13th Heidelberg Laureate Forum for Outstanding Young Researchers in Mathematics and…
Computational Complexity — 12/1/2025
BILL: I honestly do not know whether ChatGPT will make programmers more productive. (I am not touching question of whether it puts programmers out of work. That’s a problem for Future Bill.) Who can I ask? I found two people who disagree on the…
Abuse of Notation — 12/1/2025
I should stop doing category theory. What’s the point?
Crooked Timber — 11/30/2025
I’ve been seeing more and more alarmism about the idea that, on current demographic trends, the world’s population might shrink to a billion in a century or two. That distant prospect is producing lots of advocacy for policies to increase birth…
The n-Category Café — 11/30/2025
Tymoczko gave a good talk on the math of music theory.
Crooked Timber — 11/30/2025
Crooked Timber — 11/27/2025
“I am especially to speak to you of the character and mission of the United States, with special reference to the question whether we are the better or the worse for being composed of different races of men.” — Frederick Douglass, Composite Nation,…
Crooked Timber — 11/27/2025
I think a fair bit about how generative AI can help our everydays. (I also think a lot about its challenges, but this post is not about that.) Here is a good example for how it can be useful with a complex meal prep situation for which Thanksgiving…
The Aperiodical — 11/26/2025
DMFT reckons it must be downhill from here Hello! My name is Colin and I am a mathematician on a mission to find and disseminate the joy in doing maths, solving puzzles and getting to the root of things in general. I’m back from an excellent…
Computational Complexity — 11/24/2025
Last week we had a talk by Purdue philosophy professor Eamon Duede Tail Novelty, Knowledge Collapse, and Useful Frictions in Science. In part of the talk he argued that if AI makes writing technical papers easier, researchers will write up small…
The Aperiodical — 11/23/2025
Here’s an update on my progress in the Beach Spectres project. I’ve put out two update videos since the last post but failed to write a post here. I promise I’m trying my best to be more organised than usual! Update 2: it’s all coming together!…
Blog - Logic Matters — 11/22/2025
Not the best few weeks. Old people have health issues. Who knew? We were planning to go to Italy in a week or so, in particular to see the wonderful-sounding Fra Angelico exhibition at Palazzo Strozzi. Hopefully we will still be able to catch that…
Crooked Timber — 11/22/2025
The US is one big grift these days: the Trump Administration, traditional and social media, corporations, crypto, financial markets are all selling some kind of spurious promise. It’s hard to pick the most egregious example. But for me, it’s hard…
The Aperiodical — 11/21/2025
A conversation about mathematics, history, games and more, inspired by a Roman dodecahedron. Presented by Katie Steckles and Peter Rowlett.
Crooked Timber — 11/20/2025
I have a piece over at the London Review of Books Blog about the UK government’s appalling changes to the way refugees are treated in the country. “After the home secretary, Shabana Mahmood, announced the government’s new policies for ‘Restoring…
Computational Complexity — 11/20/2025
Carmichael Numbers are the bane of probabilistic primality algorithms. You have to go through extra steps just to handle these relatively rare numbers. But did you know that the Miller-Rabin algorithm not only determines the compositeness of…
Math ∩ Programming — 11/17/2025
In an earlier article, I covered the basic technique for performing matrix-vector multiplication in fully homomorphic encryption (FHE), known as the Halevi-Shoup diagonal method. This article covers a more recent method for matrix-matrix…
Crooked Timber — 11/17/2025
If you want to know more about how the current form of capitalism is undermining (a thick conception of) democracy, and what can be done about this, then you should read Lisa Herzog’s latest book The Democratic Marketplace. The book is written for…
Computational Complexity — 11/16/2025
Since there is now a CCC Test-of-Time Award, see here, (CCC stands for Computational Complexity Conference), I decided to look at other Test-of-Time awards in computer science.Below is a list of various computer science Test-of-Time awards, along…
Crooked Timber — 11/16/2025
Joel David Hamkins — 11/12/2025
This will be a talk for the Logic Seminar at the University of Notre Dame, Tuesday 18 November 20215 2pm 125 Hayes-Healy Building. Abstract. I shall introduce what I call the first-order elementary theory of surreal arithmetic, a theory that is…
Computational Complexity — 11/12/2025
In 2016, in the pre-transformer times, Georgia Tech professor Ashok Goel gave a prescient TEDx Talk on an AI teaching assistant for his large online Artificial Intelligence course. Students would ask questions to an online forum, and fellow…
Computational Complexity — 11/10/2025
A friend of mine told me that in the last six months, the last grandchild of one of our former presidents (who had already passed away) died.I tried to deduce who it was without checking the web directly. For example, I looked up when various…
Computational Complexity — 11/6/2025
We’re seeing an attack on capitalism on both ends of the political spectrum with with the election of Democratic Socialist Zhoran Mamdani as mayor of New York, and Donald Trump trying to direct the economy through tariffs, less independency of the…
The n-Category Café — 11/6/2025
Newton’s Principia is famous for his investigations of the inverse square force law for gravity. But he also studied the inverse cube law. Why, and what is so good about this law?
The n-Category Café — 11/4/2025
You can tweak Heisenberg’s equation so that instead of using a commutator it uses an associator! Then it applies to Jordan algebras other than that of self-adjoint complex matrices.
Computational Complexity — 11/4/2025
The following excerpt from Abrahim Ladha’s comment on Lance’s post aboutAI and intro theory caught my attention:—————————BEGIN EXCERPTNot just with AI, but in theory and math courses, there have always been Bart Simpsonlevel…
Joel David Hamkins — 11/4/2025
I am honored to be giving the 2025-26 Charles R. DePrima Memorial Lecture for the Mathematics Department of the California Institute of Technology. This lecture series aims to bring mathematical researchers to Caltech to give talks for a primarily…
The n-Category Café — 11/3/2025
Exploiting the secret 4-dimensional symmetry of the Kepler problem to think about the periodic table of elements in a new way.
Blog - Logic Matters — 11/1/2025
I’m slowly working on a corrected update of Introducing Category Theory, perhaps for early in the new year. But you can today download a new PDF version, without any of the revised content, but now with the bookmarks for chapters and sections…
DEONTOLOGISTICS — 11/1/2025
It’s been a long time coming, but my second book, The Revenge of Reason, is finally available to buy. There are so many things in here that were written or given as talks long ago but never actually published, and it’s nice to know people will…
Computational Complexity — 10/31/2025
The advances in artificial intelligence have changed the way we think about computing. For today’s post, how nonuniformity plays a much larger role than I previously believed.First, some background. Circuit complexity gives a different, more…
Blog - Logic Matters — 10/29/2025
We were supposed to be in Bergamo for a few days. But when it came near the time, neither of us were feeling up to going — was it a recurrence of Covid? It was a rather good thing that we didn’t set off, as whatever virus it was led to sudden…
The n-Category Café — 10/28/2025
Applied Category Theory 2026 is taking place 6–10 July in Tallinn, Estonia, and it’s preceded by the Adjoint School Research Week, 29 June – 3 July.
Computational Complexity — 10/27/2025
I sometimes give the following advice for research which I label Bill’s Bad Advice. We will later see who it might be good advice for. Spoiler alert: the number of people for whom it is good advice is shrinking but might include Lance especially…
Computational Complexity — 10/23/2025
This fall, for the first time at Illinois Tech, I’m teaching Introduction to Theory of Computation. While I’ve taught a variation of this course a couple dozen times, I last taught this class Spring 2016 at Georgia Tech. Intro Theory is a course…
Computational Complexity — 10/21/2025
Several people emailed me that September 16, 2025—written as 9-16-25 in the US—represents the integer side lengths of a right triangle.9-16-25 is the only such triple that is also a valid date. This kind of mathematical alignment only happens…
Math ∩ Programming — 10/19/2025
Polyhedral optimization is a tool used in compilers for optimizing loop nests. While the major compilers that use this implement polyhedral optimizations from scratch,1 there is a generally-applicable open source C library called the Integer Set…
Blog - Logic Matters — 10/16/2025
Briefly, Thomas Forster tells me that he has acquired funding for two studentships in Wellington NZ (a delightful place!) for students who want to do a Ph.D. in set theory, in particular working with him on NF or perhaps some adjacent topic. He has…
Abuse of Notation — 10/16/2025
In my last post about generality, I tried to show how our ambition to discover ideas that are all-encompassing and eternal makes our worldview crumble, leaving us unable to think clearly even about simple issues with obvious solutions. Today, I…
Computational Complexity — 10/15/2025
Each fall I try to predict the theory computer science faculty job market to come and give suggestions to those going after them. Get set for a rocky ride, with AI’s disruption of computer science, fiscal stress at universities, and new U.S….
Computational Complexity — 10/8/2025
A few comments to last week’s post Computers Don’t Want suggested that human brains are just advanced computers, yet still possess agency and desires. But are we just Turing machines? I wrote about this question before but let’s revisit in the…
The n-Category Café — 10/7/2025
I’m trying to better characterize two maximal subgroups of the group of automorphisms of the exceptional Jordan algebra, whose intersection is the Standard Model gauge group.
Blog - Logic Matters — 10/6/2025
An accessible, very readable, well-motivated, zestful book on ordinal analysis and proof theory would be a very good thing to have. Arai’s Ordinal Analysis with an Introduction to Proof Theory isn’t it. By a country mile. (I was asked for a…
Computational Complexity — 10/5/2025
You used AI in your work. Do you hide it or brag about it? 1) In 2002 there was a movie Simone about an actress who is really an AI. The Wikipedia entry tells the entire plot. I saved time by reading it in two minutes rather than watching it in 2…
Blog - Logic Matters — 10/4/2025
Most of the writing I have actually done this last week or so has been in tinkering with the category theory notes. For, sad to relate, I still find myself occasionally working through them. Until yesterday, however, it has just been a matter of…
Computational Complexity — 10/1/2025
I read through the new book If Anyone Builds It, Everyone Dies by Eliezer Yudkowsky and Nate Soares. “It” refers to Artificial Super Intelligence (ASI). A very short version of the authors’ argument: You can view advanced AI as though it has its…
Computational Complexity — 9/29/2025
Martin Kruskal was born Sept 28, 1925 and passed away on Dec 26, 2006, at the age of 81 (we did two posts for his memorial, here and here). Today, Sept 28, 2025, is his 100th birthday. His son Clyde Kruskal wrote today’s blog post as a tribute to…
Abuse of Notation — 9/27/2025
“My life is like a prison” I wrote this in my personal website when I was 14. I was quite correct in pinpointing the problem, pinpointing how I, and many other people, felt, but I was off at identifying the cause — I thought, that I was kept in…
Computational Complexity — 9/25/2025
A few weeks ago I took an Uber to a regional airport and was picked up by a Tesla. The driver used FSD, so-called Full Self-Driving, never touching the steering wheel during the entire trip. Should you tip a driver who just sits there? In the end I…
Math3ma — 9/25/2025
Over the past couple of years, I’ve been learning a little about the world of quantum machine learning (QML) and the sorts of things people are thinking about there. I recently gave an high-level talk on some of these ideas in connection to a…
Math3ma — 9/25/2025
Next up on Good Reads: The Princeton Companion to Mathematics, edited by Fields medalist Timothy Gowers. This book is an exceptional resource! With over 1,000 pages of mathematics explained by the experts for the layperson, it’s like an…
Computational Complexity — 9/23/2025
STUDENT: What did you do before the web to find papers?BILL: We went to the library and copied papers to read later.STUDENT: Did you read them later?BILL: Well, uh,hmm, …BILL to a professor in his 80’s: What did you do before copy…
Blog - Logic Matters — 9/20/2025
A great couple of days in London. The high point, a quite outstanding evening at Wigmore Hall — the wondrous Lea Desandre and Thomas Dunford (and the Jupiter ensemble) performing Dowland and Purcell. Their new CD is terrific, and the live version…
Computational Complexity — 9/17/2025
When announcing Open-AI’s latest release last month, Sam Altman said “GPT-5 is the first time that it really feels like talking to an expert in any topic, like a PhD-level expert.” Before we discuss whether GPT-5 got there, what does “PhD-Level…
The n-Category Café — 9/16/2025
The octonions have nontrivial inner automorphisms of order 3. Is this related to triality?
Blog - Logic Matters — 9/13/2025
What have I been reading this week? I finished The Voyage Home, the most recent in Pat Barker’s wonderful series re-imaging episodes from the Trojan War. This time, it’s Agamemnon’s voyage home after the war, and his death at the hands of…
The n-Category Café — 9/13/2025
Adjoint School guest post by Khyathi Komalan and Andrew Krenz
Joel David Hamkins — 9/10/2025
Recently I had the pleasure to give a talk at the Conference on the occasion of Jörg Brendle’s 60th birthday at Kobe University in Japan, and I was invited to make remarks at the conference banquet given in his honor. … Continue reading →
Blog - Logic Matters — 9/6/2025
Just before we went off to Zürich, we had our house painted outside (the doors, the windows, and so forth). It took fourteen days, not because we have a mansion but because a lot of preparatory work was needed, cutting out minor rot, repairing,…
The n-Category Café — 8/30/2025
All equivalences are generated by just the strict, literally surjective ones.
Joel David Hamkins — 8/20/2025
This will be a talk for the Conference on the occasion of Jörg Brendle’s 60th birthday at Kobe University in Kobe, Japan, 2-5 September 2025. Many years ago, I was a JSPS Fellow at Kobe University, at the same time … Continue reading →
Joel David Hamkins — 8/18/2025
This will be a talk I shall give for the History and Philosophy of Science (HPS) Colloquium at the University of Notre Dame, 17 October 2025, 12:30-1:30 pm, 201 O’Shaughnessy Hall. Did Turing ever halt? Abstract. Alan Turing’s 1936 paper … Continue…
The n-Category Café — 8/15/2025
This week, 50 category theorists and software engineers working on “safeguarded AI” are meeting in Bristol.
The n-Category Café — 8/5/2025
Comparing two mathematical notions of diversity.
The n-Category Café — 8/2/2025
I hear that Jack Morava died on August 1, 2025.
The n-Category Café — 7/28/2025
Can the Duflo isomorphism explain the extra 1/2 in the Hamiltonian for the quantum harmonic oscillator?
Math ∩ Programming — 7/25/2025
On Monday, July 14th 2025, I hosted a mini-workshop on homomorphic encryption at Google’s Portland, Oregon office. Though Portland is a small city, it’s becoming a hub for homomorphic encryption. Intel and Google both have a presence here, as well…
Math ∩ Programming — 7/18/2025
I work on homomorphic encryption (HE or FHE for “fully” homomorphic encryption) and I have written a lot about it on this blog (see the relevant tag). This article is a collection of short answers to questions I see on various threads and news…
Joel David Hamkins — 7/16/2025
This will be a talk for the Fudan Logic Seminar at Fudan University, to be followed immediately by two talks for the Fudan Logic student seminar, forming a mini-conference for the logic group on 23 July 2025. Abstract. I shall … Continue reading →
Joel David Hamkins — 7/14/2025
This will be a talk for the Seminar on Frontier Issues in Logic and Philosophy The First Forum on Logic and Philosophy 逻辑与哲学前沿问题研究”学术研讨会暨第一届逻辑与哲学论坛 Changchun, China, 18-20 July 2025 Pointwise definable end-extensions of models of arithmetic and set…
Joel David Hamkins — 7/9/2025
This will be a lecture series on the Philosophy of Mathematics at Fudan University in Shanghai, China, 30 June – 25 July 2025, as a part of the International Summer School program at Fudan University. Lectures given by Ruizhi Yang … Continue…
Joel David Hamkins — 7/9/2025
This will be a talk for the International Conference on the Philosophy of Mathematics, held at Lanzhou University, China, 25-27 July 2025. How the continuum hypothesis might have been a fundamental axiom Abstract. I shall describe a…
Joel David Hamkins — 6/23/2025
This will be a talk for the Conference on Infinity, a collaborative meeting of logicians and specialists in Chinese philosophy here at Peking University, 24 June 2025, in the philosophy department. Abstract. I shall lay out a spectrum of…
DEONTOLOGISTICS — 6/22/2025
Here’s a recent thread on philosophy of AI from Twitter/X, in which I address rather popular arguments made by Emily Bender and others to the effect that LLM outputs are strictly speaking meaningless. I think these argument are flawed, as I explain…
Joel David Hamkins — 6/11/2025
This will be a lecture series at Peking University in Beijing in June 2025. Announcement at Peking University Course abstract. This will be a series of advanced lectures on set theory, treating diverse topics and particularly those illustrating how…
Abuse of Notation — 5/27/2025
title: A monad layout: microblog category: microblog tags: programming haskell — A monad is when you know how to convert $M (M a)$ to $M a$, but not $M a$ to $a$.
Abuse of Notation — 5/27/2025
The level of progress in programming language design: Shortly after the first ever programming language was created, it’s author said that the language’s whole paradigm is flawed and we should do functional programming instead*. That was 46 years…
Abuse of Notation — 5/27/2025
I want to forget Gaza. I want to forget the pain. I want to forget all dumb jokes, old song lyrics. I want to forget that most things ever happened, as remembering makes everything tedious. Forces you to become a bureaucrat of your memories, to…