The n-Category Café --- 1/29/2025
Warming up to a dual concept to magnitude, with an aside on a new connection between magnitude and entropy.
The n-Category Café --- 1/29/2025
Thinking about the cardinality of limits leads to a new numerical invariant of set-valued functors.
Computational Complexity --- 1/29/2025
In writing the drunken theorem post, I realized I never wrote a post on Lautemann's amazing proof that BPP is contained in (\Sigma^p_2), the second level of the polynomial-time hierarchy.Clemens Lautemann, who passed away in 2005 at the too young...
The Aperiodical --- 1/29/2025
Double Maths First Thing is powered by genuine, artisanal intelligence Hello! My name is Colin and I am a mathematician on a mission to spread joy and delight in maths, and to encourage people to take pride in the maths they do. Straight into it...
Crooked Timber --- 1/29/2025
At some indeterminate point in the fairly recent past, citizens and leaders of most liberal democracies probably looked forward to a condition to be realized in the imaginable future that we can, for the sake of a convenient label, call Universal...
On Skolem’s paradox and the transitive submodel theorem, Rust Belt Workshop in the Philosophy of Logic, Language, and Mathematics, February 2025
Joel David Hamkins --- 1/28/2025
This will be a talk for the Rust Belt Workshop in the Philosophy of Logic, Language, and Mathematics, held at Ohio State University in Columbus, Ohio, February 8-9, 2025, University Hall (230 N Oval Mall, Columbus, OH) Room 386B. Abstract. …...
Good Fibrations --- 1/28/2025
Computational Complexity --- 1/27/2025
4444=1936.4545=2025. This year!46*46= 2116.Since my fake birthday is Oct 1, 1960 (I do not reveal my real birthday to try to prevent ID theft), which is past 1936, and I won't live to 2116 unless Quantum-AI finds a way to put my brain in a a vat,...
Fractal Kitty --- 1/27/2025
Do you ever look at knitted items and liken them to dragon scales? - I do. So when I decided to make generative textured hats it made sense to call them Wyrm Beans.Just as we can have generative pixel colors on a screen, we can have generative...
Crooked Timber --- 1/27/2025
The Australian parliament has legislated what’s commonly described as as “social media ban” for people under 16. More precisely, it will require selected social media platforms to implement (unspecified) age verification technology for people...
Crooked Timber --- 1/26/2025
Computational Complexity --- 1/22/2025
What does an 1838 painting tell us about technological change?A colleague and I decided to see how well LLMs could teach us a topic we knew nothing about. We picked the Romanticism art movement. I asked ChatGPT to tutor me on the topic for an hour....
The Aperiodical --- 1/22/2025
DMFT probably needs wider margins. Hello! My name is Colin and I am a mathematician on a mission to spread the joy and delight in how everything fits together. Straight into the links this week! A mini-theme: going back in time My link collection...
Joel David Hamkins --- 1/21/2025
This was a talk I gave at the Set Theory Workshop at the Mathematisches Forschungsinstitut in Oberwolfach, Germany, 12-17 January 2025. Abstract. The principle of covering reflection holds of a cardinal κ if for every structure B in a countable …...
Crooked Timber --- 1/21/2025
Recently I learned that at Yale University a “Report of the Committee on Institutional Voice” was published a few months ago. The committee was chaired by professors “Della Rocca & Rodríguez” and so hereafter, I refer to the report as “Della...
Crooked Timber --- 1/21/2025
I’ve already said all I plan to (for now) about what’s happening in the US. But if others want to discuss it, here’s an open thread.
Crooked Timber --- 1/21/2025
Inspired by Chris’s recent photo-blogging post, I thought I’d share a less well known little gem about (the original) Ravenna: not a byzantine church interior full of mosaics, but the submerged crypt of an early medieval Church (the Basilica of San...
Computational Complexity --- 1/21/2025
I made up a quiz about the American Presidents here. It has 40 questions. In the modern electronic age you can probably look up most or even all of the answers. So what to do about that?1) The quiz is not for money or credits or anything, so if...
Crooked Timber --- 1/19/2025
Good Fibrations --- 1/17/2025
Cech Covers (click the link to listen to us). I wrote this song with my beloved old room mate Christian Gorski in my last year of grad school while I was wrapping up my thesis. For weeks, I was doing nothing but computing etale sheaf cohomologies...
Good Fibrations --- 1/17/2025
This drawing is an old drawing I made when I was preparing for my qualifying exam in my second year of grad school at Northwestern. It is the crystalline period map. The tower to the left is the “Lubin-Tate” tower, the deeper it goes the more level...
Crooked Timber --- 1/16/2025
There’s been a recent fuss in various media arising from a tweet from economist Ben Golub regarding astonishment that economists haven’t “worked through” Smith and Marx. English professor Alex Moskowitz chimed in with a claim that economics can’t...
Good Fibrations --- 1/16/2025
I have been working recently to counter the writers block that has formed insidiously from an unhealthy creeping perfectionism. In order to do this, I will post some old art and music which at the time I felt was “not good enough to share” or...
The n-Category Café --- 1/15/2025
There's more than one reasonable answer to the question "what is the dual concept of injection?"
Computational Complexity --- 1/15/2025
Slide in Lev Reyzin's JMM talk "Problems in AI and ML for Mathematicians" Reyzin is paraphrasing Telgarsky. Posted with permission.Last week I attended the Joint Mathematics Meeting in Seattle with a theme ofWe Decide Our Future: Mathematics in the...
The Aperiodical --- 1/15/2025
DMFT is significantly less perplexing than HMRC Hello! My name is Colin and I am a mathematician on a mission to spread mathematical joy. This week, I’ve made another contribution to the OEIS (currently in review) about the excellent puzzle #23...
Computational Complexity --- 1/12/2025
Guest Post from Nick Sovich. -----------------------------------------------------------------------------Bill Gasarch recently blogged on RANDOM THOUGHTS ON AI here . He is in the realm of theory. I am in the realm of applications so I asked if I...
Crooked Timber --- 1/12/2025
The Aperiodical --- 1/11/2025
Here’s a round-up of some news stories from the last two months of 2024, (mostly) not otherwise covered here on the Aperiodical. Maths Research At the start of December, John Carlos Baez shared on Mathstodon that the moving sofa problem may have...
Joel David Hamkins --- 1/10/2025
I had an enjoyable little discussion with Joe Murray of The Human Podcast, part of his new series, called 10 questions in 10 minutes, in which he asks his interview subjects for short answers to ten quick questions on their … Continue reading →
Crooked Timber --- 1/10/2025
Those of us who live in Europe have reason to be very pessimistic about the next four years. The state that Europeans have relied upon as their security guarantee is now in the hands of the nationalist extreme right and the information space is...
Crooked Timber --- 1/9/2025
Commentators in Europe are understandably agog about Trump’s rumblings that the US might somehow, possibly, annex Greenland at some point in the future. One would think asking Greenlanders how they see their future might have been a better idea....
Computational Complexity --- 1/8/2025
BILL: Good news for Jimmy Carter! He won The Betty White Award! (see here).LANCE: That's not good news. He had to die to get it.BILL: Call it a mixed bag. Good news for me, in that I have a famous person for The Betty White award. And I later...
The Aperiodical --- 1/8/2025
Double Maths First Thing is being written in the dark Hello! My name is Colin and I am a mathematician on a mission to spread mathematical joy and delight. However, at de moment, delights aren’t working; we’re in a power cut and I’m hoping my...
Math ∩ Programming --- 1/8/2025
I’ll be at the Joint Mathematics Meeting in Seattle (starting tomorrow). If you see me there, say hi! I will have a very light schedule, plenty of time for coffee chats. I’ll be attending many of the crypto sessions for the homomorphic encryption...
Computational Complexity --- 1/7/2025
In Jan of 2023 I estabalished the Betty White Award, see here which is given to people who died late in the prior year and hence won't be in the those who we lost in year X articles. I also gave out a few from prior years. Here are past winners,...
Crooked Timber --- 1/5/2025
A curious one, this. We were looking through some old postcards (from the 1930s) and came across one with a picture of this church interior. Where’s that? Well, it turns out that it is just off the motorway on our journey between Bristol and...
Math ∩ Programming --- 1/4/2025
The Hyperfixed Podcast had a lovely episode recently about tape measures. It started from “why does my tape measure seem to always be off a little bit” and went all the way to the inherent limitations of physical measurement at small scales. In...
The Aperiodical --- 1/4/2025
The next issue of the Carnival of Mathematics, rounding up blog posts from the month of December 2024, is now online at John D Cook’s Blog. The Carnival rounds up maths blog posts from all over the internet, including some from our own Aperiodical....
Math ∩ Programming --- 1/3/2025
In this living document, I will document reactions to uses of homomorphic encryption by members of the public. By “member of the public,” I mean people who may be technical, but are not directly involved in the development or deployment of...
Computational Complexity --- 1/2/2025
Bill's SIGACT Open Problems Column remembering Luca Trevisan is out. I chose the problem of whether Promise-ZPP in P implies Promise-BPP in P, an extension of an earlier theorem by Luca and his co-authors, which showed that Promise-RP in P implies...
Crooked Timber --- 1/1/2025
Here’s a virtual toast to your flourishing in 2025. But more so than any other year, our wishes should not just be from person to person, but rather wishes for societies – and the society of societies, global humanity. I haven’t felt so gloomy...
The Aperiodical --- 1/1/2025
Double Maths First Thing is like a tall, dark stranger with some coal and some whisky Hello! My name is Colin and I am a mathematician on a mission to spread mathematical joy into 2025 and beyond. I note that 1/1/2025 is the first day since...
The Aperiodical --- 1/1/2025
There seem to be a lot of numerical coincidences bouncing around concerning the new year 2025. For example, it’s a square number: ( 2025 = 45^2 ). The last square year was (44^2 = 1936), and the next will be (46^2=2116). The other one you...
The Aperiodical --- 12/31/2024
The UK Government have announced the latest list of honours, and we’ve taken a look for the particularly mathematical entries. Here is the selection for this year – if you spot any more, let us know in the comments and we’ll add to the list. Get...
Crooked Timber --- 12/30/2024
I’ve avoided post-mortems on the US election disaster for two reasons. First, they are useless as a guide to the future. The next US election, if there is one [1], will be a referendum on the Trump regime. Campaign strategies that might have gained...
Crooked Timber --- 12/30/2024
The Aperiodical --- 12/28/2024
It’s now been a year since I took over the puzzle column at New Scientist and turned it into the BrainTwisters column. By way of celebration, I thought I’d write up an interesting bit of maths behind one of the puzzles, which I made a note of at...
Computational Complexity --- 12/23/2024
(I wrote this post without any AI help. OH- maybe not- I used spellcheck. Does that count? Lance claims he proofread it and found some typos to correct without any AI help.) Random Thought on AII saw a great talk on AI recently by Bill Regli, who...
Computational Complexity --- 12/23/2024
Back in the day (circa 1989) we studied locally random reductions which would lead to all those exciting interactive proof results. Somehow locally random reductions got rebranded as locally correctable codes and this year's result of the year...
The n-Category Café --- 12/21/2024
More detail on the Cycle Length Lemma, a basic result in the theory of random permutations. We prove a categorified version of this result, which is an equivalence of groupoids, and then derive the original version by taking groupoid cardinalities.
Computational Complexity --- 12/18/2024
I've heard a few times recently the phrase "Information only exists in a physical state". It come from the quantum computing world where they claim quantum changes the game when it comes to representing information.As one who has spent his career...
Bartosz Ciechanowski --- 12/17/2024
In the vastness of empty space surrounding Earth, the Moon is our closest celestial neighbor. Its face, periodically filled with light and devoured by darkness, has an ever-changing, but dependable presence in our skies. In this article, we’ll...
Joel David Hamkins --- 12/13/2024
My Oxford student Emma Palmer and I have been thinking about worldly cardinals and Gödel-Bernays GBC set theory, and we recently came to a new realization. Namely, what I realized is that every worldly cardinal
The n-Category Café --- 12/13/2024
About an early theory in which Mercury and Venus orbit the Sun, while the other planets orbit the Earth.
Computational Complexity --- 12/11/2024
We use grades to evaluate students and motivate them to learn. That works as long as grades remain a reasonably good measure of how well the student understands the material in a class. But Goodhart's law, "When a measure becomes a target, it...
Computational Complexity --- 12/8/2024
In Lance's last post (see here) he listed his favorite theorems from 1965 to 2024.There are roughly 60 Theorems. I mostly agree with his choices and omissions. I will point out where I don't. I could make a comment on every single entry; however,...
Computational Complexity --- 12/6/2024
Now in one place all of my sixty favorite theorems from the six decades of computational complexity (1965-2024).2015-2024Graph Isomorphism (Babai)Sensitivity (Huang)Quantum Provers (Ji-Natarajan-Vidick-Wright-Yuen)Dichotomy (Bulatov, Zhuk)Algebraic...
The n-Category Café --- 12/4/2024
The Eighth International Conference on Applied Category Theory (https://easychair.org/cfp/ACT2025) will take place at the University of Florida on June 2-6, 2025. The conference will be preceded by the Adjoint School on May 26-30, 2025. This...
Computational Complexity --- 12/1/2024
(I got this material from a nice article by Arthur Benjamin here.) Conway suggested the following trick to determine if a number is divisible by each of the following: 2,3,5,7,11,17,19,31Note that( 152=2^3\times 19) (153 =3^2 \times...
Joel David Hamkins --- 12/1/2024
This will be a talk for the Notre Dame Logic Seminar, 3 December 2024, 2:00pm, 125 Hayes-Healey. Abstract. I shall give an account of the theory of computable surreal numbers, proving that these form a real-closed field. Which real numbers …...
Computational Complexity --- 11/25/2024
Will our writing all converge to a generic AI style? Let's take a quick detour into LaTeX. Back in the late '80s, before LaTeX was the standard, there was TeX—a system with no default formatting, which meant everyone had their own unique style for...
The n-Category Café --- 11/22/2024
The penultimate week of this axiomatic set theory course, based on Lawvere's Elementary Theory of the Category of Sets.
The n-Category Café --- 11/22/2024
The final chapter of this course on secretly-categorical set theory.
The n-Category Café --- 11/22/2024
Want to work on applied category theory? Apply to the Adjoint School before December 1, 2024!
Fractal Kitty --- 11/21/2024
Over the last week I have been drawing circles along paths and then shading them with 2 colors using random numbers to determine their radii. For example, the one on this card was drawn with 10 evenly spaced circles along a line using the random...
For what d is the following true: For all 2-colorings of (R^d) has a mono unit square (Answering(?) the Question)
Computational Complexity --- 11/20/2024
In my last post (see here) I invited you to work on the following question:Find a (d) such that--There is a 2-coloring of (R^d) with no mono unit square.--For all 2-colorings of (R^{d+1}) there is a mono unit square. Actually I should have...
Abuse of Notation --- 11/18/2024
Teacher (grumpy): “How did everyone on the class came to know this, and you are the only one who is still clueless?” Me (entusiastically): “Interesting question! In fact I myself have been wondering the same thing?”
For what d is the following true: for all 2-colorings of (R^d) there is a mono unit square (Asking the Question)
Computational Complexity --- 11/17/2024
In this post I give a question for you to think about. My next post will have the answer and the proof. 1) The following are known and I have a set of slides about it herea) For all 2-colorings of (R^2) there exists two points an inch apart that...
Abuse of Notation --- 11/17/2024
Society in which you step over homeless folks on your way to pointlessly click and type on the computer all day, is deeply fucked up. And you have to be veeery brainwashed to not realize that. Society where everyone’s main activity is to do a “job”...
Abuse of Notation --- 11/16/2024
There is an ability I call unity of thought, for a lack of a better word, which is essential for acquiring intelligence/wisdom. It is the ability to connect each new idea/view that you have, with all the other ideas/views in your mind, to make it...
Math ∩ Programming --- 11/15/2024
In my little corner of the FHE world, things have been steadily heating up. For those who don’t know, my main work project right now is HEIR (Homomorphic Encryption Intermediate Representation), a compiler toolchain for fully homomorphic encryption...
Math ∩ Programming --- 11/15/2024
Editor’s note: This essay was originally published in 2019. I have made minor edits in this republishing. There was a MathOverflow thread about mathematically interesting games for 5–6 year olds. A lot of the discussion revolved around how young...
The n-Category Café --- 11/15/2024
Our ETCS-based but category-free course now reaches the theory of ordinals, a.k.a. well ordered sets.
Computational Complexity --- 11/14/2024
October EditionI had a tough choice for my final favorite theorem from the decade 2015-2024. Runners up include Pseudodeterministic Primes and Hardness of Partial MCSP. But instead in memory of the recently departed Steven Rudich, this month's...
Computational Complexity --- 11/11/2024
Complexity theorist Steven Rudich passed away on October 29 at the age of 63. His works on Natural Proofs and Program Obfuscation were both highly influential. Russell Impagliazzo had a great result with him on showing that one-way permutations...
The n-Category Café --- 11/8/2024
Defining N, Z, Q and R in the Elementary Theory of the Category of Sets.
Computational Complexity --- 11/7/2024
It feels eerie as pretty much everyone seemingly avoided talking about the election. But Trump back in the White House will likely have a profound effect on US Colleges and Universities. Trump is no fan of universities and his vice-president once...
The n-Category Café --- 11/6/2024
How can you get a regular icosahedron using Thurston's method?
Computational Complexity --- 11/5/2024
(Lance posted on the search for Mersenne primes in 2006 after a new one was discovered. I will comment on his post later. ADDED LATER- You Tube Video by Matt Parker on the new prime, here)A Mersenne Prime is a prime of the form (2^n-1) where n is...
Computational Complexity --- 11/5/2024
Here are my random thoughts on the election:1) Here is a list of things I DONT care about a) Candidates Gender or Race. The people who say its about time we had a female president might not want to vote for a President Marjorie Taylor Green. (A...
The n-Category Café --- 11/2/2024
You can now apply for the 2025 Summer Research Associate program at the Topos Institute! The deadline to apply is January 17, 2025.
The n-Category Café --- 11/1/2024
Quotients, disjoint unions, families of sets, and the Cantor-Bernstein theorem.
Math ∩ Programming --- 11/1/2024
Welcome to the 233rd Carnival of Mathematics! Who can forget 233, the 6th Fibonacci prime? Hey, not all numbers are interesting. Don’t ask me about the smallest positive uninteresting number. You can’t make it interesting with your feeble mind...
Math ∩ Programming --- 10/31/2024
This article will explain how the blog is organized at a technical level, and show how I implemented various IndieWeb features. Table of Contents: Motivation Structure and Deployment Static search index Running scripts via GitHub Actions Social...
Computational Complexity --- 10/30/2024
Junior/Senior lunch in 80°F ChicagoLast summer I attended the Complexity Conference in Ann Arbor for the first time in eight years largely because it was within driving distance. So with FOCS in Chicago this year I didn't have much of an excuse to...
The n-Category Café --- 10/30/2024
What do triangulations of the 2-sphere have to do with complexified 10-dimensional Minkowski spacetime?
DEONTOLOGISTICS --- 10/18/2024
Here’s a recent thread musing about problems with Bayesian conceptions of general intelligence and the more specific variants based on Solmonoff induction, such as AIXI. I’ve been thinking about these issues a lot recently, in tandem with the...
Proses.ID --- 10/16/2024
I’ve been wanting to reflect of the past 2 years and 2 months I spent not working, but it never felt urgent or important (to…
Math ∩ Programming --- 10/15/2024
Kristin Lauter and her colleagues at Facebook research recently announced a project to benchmark attacks against LWE. The announcement was on the post-quanum crypto mailing list. They state: “Our approach is motivated by the need to study more...
Determinateness of truth does not come for free from determinateness of objects, Singapore, November 2024
Joel David Hamkins --- 10/14/2024
This will be a talk for the (In)determinacy in Mathematics conference at the National University of Singapore, 20-22 November 2024 Abstract. I shall discuss the question whether we may regard determinateness of truth as flowing from determinateness...
Joel David Hamkins --- 10/14/2024
This will be a talk at the UW Madison Logic Seminar on 22 October 2024. Abstract. The principle of covering reflection holds of a cardinal κ if for every structure B in a countable first-order language there is a structure … Continue reading →
Fractal Kitty --- 10/1/2024
It’s mathober! I will be updating this page with my sketches (in procreate and code for this year. Procreate sketches: (not all will be done this month – they take longer): P5js sketches: The code is on my codepen collection here.
Math ∩ Programming --- 9/12/2024
This is a story about a failure to apply dynamic programming to a woodworking project. I’ve been building a shed in my backyard, and for one section I decided to build the floor by laying 2x4 planks side by side. I didn’t feel the need to join them...
Fractal Kitty --- 9/10/2024
Mathober is just around the corner, and I can’t wait to see everyone’s creative take on this year’s prompts! If you’ve never participated before, now’s the perfect time to jump in! The goal of Mathober is simple: have fun, learn, grow, and play...
Math ∩ Programming --- 9/7/2024
In my recent overview of homomorphic encryption, I underemphasized the importance of data layout when working with arithmetic (SIMD-style) homomorphic encryption schemes. In the FHE world, the name given to data layout strategies is called...
Math ∩ Programming --- 9/2/2024
In my recent overview of homomorphic encryption, I underemphasized the importance of data layout when working with arithmetic (SIMD-style) homomorphic encryption schemes. In the FHE world, the name given to data layout strategies is called...
Fractal Kitty --- 8/21/2024
I coded a generative zine to bring to XOXO 2024. It is different every time it loads with a sampling of p5.js sketches. Each zine has hundreds of thousands to millions of generated shapes and points using random numbers. The github is here (the...
Fractal Kitty --- 8/20/2024
I made Curve, a coloring book, to bring to XOXO 2024 this week. If you’d like to print/play: See the Pen Curve Zine by Sophia (fractal kitty) (she/her) (@fractalkitty) on CodePen.
Infinite-time computable analogues of the universal algorithm, Generalized Computability Theory Workshop, Spain, August 2024
Joel David Hamkins --- 8/18/2024
This will be a talk at the Generalized Computability Theory workshop in Castro Urdiales, Spain, a beautiful setting on the sea near Bilbao, 19-23 August 2024. Abstract. I shall present infinite-time computable analogues of the universal algorithm,...