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Christopher Olah's Blog

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### Ars Mathematica

- Nine Chapters on the Semigroup Art 2015-02-28While Googling something or other, I came across Nine Chapters on the Semigroup Art, which is a leisurely introduction to the theory of semigroups. (While the document is labelled “lecture notes”, the typography is quite beautiful.)Walt
- What Did Grothendieck Do? 2015-01-01Happy New Year! The publicity in the wake of Grothendieck’s death has left a certain number of non-mathematicians with the question of what it was exactly that he did. I wrote an answer elsewhere that people seemed to find informative, … Continue reading →Walt
- Learning about Stochastic Processes the Almost Sure Way 2014-11-09George Lowther at Almost Sure has written a terrific series of posts explaining stochastic processes and the stochastic calculus. Stochastic calculus is widely used in physics and finance, so there are many informal introductions that get across the main ideas … Continue reading →Walt
- Arguesian Lattices 2014-09-23As is well-known, the lattice of submodules of a module is modular. What I did not know is that the converse is not true, and that lattices of submodules must satisfy a stronger property, the arguesian law. The Arguesian law … Continue reading →Walt
- K2, not the mountain 2014-03-20Chandan Singh Dalawat has a nice survey article about K2. It just gives the highlights of the theory, without proofs, so it’s closer to a teaser trailer than it is to full-length movie. But sometimes you just want a teaser … Continue reading →Walt
- Cayley Bacharach Theorem through History 2014-02-10I came across this terrific article that describes a sequence of results beginning with Pappas’ theorem through the Cayley-Bacharach theorem to modern formulations in terms of the Gorenstein (!) condition. The connection between classical topics in algebraic geometry and modern … Continue reading →Walt
- Nonassociative Algebras 2013-12-30I periodically feel like I should learn more about nonassociative algebra. (I’ve studied Lie algebras, and technically Lie algebras are non-associative, but they’re pretty atypical of nonassociative algebras.) There’s a mysterious circle of “exceptional” examples that are all related — … Continue reading →Walt
- Determinacy 2013-11-30One of my ambitions in life is to understand projective determinacy. Fortunately, Tim Gowers has written a series of posts to explain Martin’s proof that Borel sets are determined. The main source of interest in determinacy is that results suggest … Continue reading →Walt
- A Generalized Fermat Equation 2013-08-31I came across a number theory paper Twists of X(7) and Primitive Solutions of x2 + y3 = z7 that I find completely fascinating. I find it fascinating because a) the question is so easy, b) the answer is so … Continue reading →Walt
- Linear Bestiary of Francois Pottier 2013-07-09Ugh, I suck at this blogging thing. I periodically get ambitious, and make big plans. That doesn’t actually lead to any completed posts, just many long half-finished posts, and hundreds of open tabs in Firefox. I think I’ll start with … Continue reading →Walt

### Christopher Olah's Blog

- Deep Learning, NLP, and Representations 2014-07-08In the last few years, deep neural networks have dominated pattern recognition. They blew the previous state of the art out of the water for many computer vision tasks. Voice recognition is also moving that way. But despite the results, we have to wonder… why do they work so well? This post reviews some extremely […]colah
- Fanfiction, Graphs, and PageRank 2014-07-07On a website called fanfiction.net, users write millions of stories about their favorite stories. They have diverse opinions about them. They love some stories, and hate others. The opinions are noisy, and it’s hard to see the big picture. With tools from mathematics and some helpful software, however, we can visualize the underlying structure. […]colah
- Neural Networks, Manifolds, and Topology 2014-04-09Recently, there’s been a great deal of excitement and interest in deep neural networks because they’ve achieved breakthrough results in areas such as computer vision. However, there remain a number of concerns about them. One is that it can be quite challenging to understand what a neural network is really doing. If one trains it well, it […]colah
- Visualizing Functions On Groups 2014-01-16Functions of the form or , where is a group, arise in lots of contexts. One very natural way this can happen is to have a probability distribution on a group, . The probability density of group elements is a function . Another way this can happen is if you have some function and has […]colah
- The Death of a Squirrel 2013-08-25(Trigger warning: descriptions of severe animal injury.) Today a squirrel was hit by a car a few feet away from me while I was walking down the side walk. Three of its legs kept twitching. I thought it had a broken leg. I came out of my stupor and went to grab it and pull […]colah
- Order Statistics 2013-08-16What is the distribution of the maximum of random variables? What started out a utilitarian question in my exploration of some generalized versions of the secretary problem turns out to be quite a deep topic. (Note that I have little background in probability and statistics. Please forgive (and inform me of, so I can fix!) […]colah
- Topology Notes 2013-06-14I’ve been talking about writing a topology textbook introductory notes on topology for years. Basically since I wrote my Rethinking Topology (or a Personal Topologodicy) post 2 years ago — it’s hard to believe it’s been that long! In any case, I finally started writing it. I’ve done a mild review of existing introductions to general topology (ie. I […]colah
- How My Neural Net Sees Blackboards (Part 2) 2013-06-09Previously, I discussed training a neural net to clean up images. I’m pleased to say that, using more sophisticated techniques, I’ve since achieved much better results. My latest approach is a four layer convolutional network. Sadly, the convolution throws away the sides of the images, so we get a black margin. In any case, compare […]colah
- I’m Sick and Tired of 3D Printed Guns 2013-05-29For the last few months, every time someone hears that I work with 3D printers they bring up 3D printed guns. I can’t say how many times it has happened in this month alone. And I’m getting really really tired of it. “They’re the killer app of 3D printers.” What a great pun. You don’t know […]colah
- How My Neural Net Sees Blackboards 2013-05-11For the last few weeks, I’ve been taking part in a small weekly neural net study group run by Michael Nielsen. It’s been really awesome! Neural nets are very very cool! They’re so cool, I had to use them somehow. Having been interested in mathematical handwriting recognition for a long time, I decided to train […]colah

### Planet Sage

- Sébastien Labbé: Tiling a polyomino with polyominoes in SageMath 2020-12-03Suppose that you 3D print many copies of the following 3D hexo-mino at home: sage: from sage.combinat.tiling import Polyomino, TilingSolver sage: p = Polyomino([(0,0,0), (0,1,0), (1,0,0), (2,0,0), (2,1,0), (2,1,1)], color='blue') sage: p.show3d() Launched html viewer for Graphics3d Object You would like to know if you can tile a larger polyomino or in particular a rectangular […]
- William Stein: DataDog: Don't make the same mistake I did -- a followup and thoughts about very unhappy customers 2020-04-13This is a followup to my previous blog post about DataDog billing. TL;DR:- I don't recommend DataDog,- dealing with unhappy customers is hard,- monitoring for data science nerds?Hacker News CommentsDataDog at Google Cloud SummitI was recently at the Seattle Google Cloud Summit and DataDog was well represented, with the biggest booth and top vendor billing […]
- Sébastien Labbé: Computer experiments for the Lyapunov exponent for MCF algorithms when dimension is larger than 3 2020-03-27In November 2015, I wanted to share intuitions I developped on the behavior of various distinct Multidimensional Continued Fractions algorithms obtained from various kind of experiments performed with them often involving combinatorics and digitial geometry but also including the computation of their first two Lyapunov exponents. As continued fractions are deeply related to the combinatorics […]

### Yet Another Mathblog

- A simple trace formula for graphs 2023-09-09Let be a simple, connected graph with vertices and adjacency matrix . We start with the geometric series identity where is the identity matrix. Let denote the orthonormal matrix of normalized eigenvectors, so that , where diag(…) denotes the diagonal matrix with the given entries on the diagonal. Let the multi-set denote the spectrum of […]wdjoyner
- Another mathematician visits the ballpark – WHIP 2023-09-08This is the second in the series of blog posts inspired by the 2004 Ken Ross book entitled A Mathematician at the Ballpark. The first one is here. In this post, again, we illustrate all these notions using the Baltimore Orioles’ 2022 season. For an experienced baseball fan, baseball is a game of patterns. We […]wdjoyner
- Another mathematician visits the ballpark – OPS 2023-03-06Yes, I more-or-less stole the above title from the 2004 Ken Ross book entitled A Mathematician at the Ballpark. Like that book, anyone familiar with middle-school (or junior high school) math, should have no problem with most of what we do here. However, I will try to go into baseball in more detail than the […]wdjoyner
- Harmonic quotients of regular graphs – examples 2023-01-06Caroline Melles and I have written a preprint that collects numerous examples of harmonic quotient morphisms , where is a quotient graph obtained from some subgroup . The examples are for graphs having a small number of vertices (no more than 12). For the most part, we also focused on regular graphs with small degree […]wdjoyner
- A graph_id function in SageMath 2022-09-07While GAP has a group_id function which locates a “small” group in a small groups database (see the SageMath page or the GAP page for more info), AFAIK, SageMath doesn’t have something similar. I’ve written one (see below) based on the mountain of hard work done years ago by Emily Kirkman.wdjoyner
- Rankine’s “The Mathematician in Love” 2022-05-26The 1874 poem “The Mathematician in Love” by Scottish mechanical engineer William Rankine (in the book From Songs and Fables) has been published in many places (e.g., poetry.com, New Scientist and the scanned version is available at the internet archive. However, the mathematical equations Rankine presented at the end of his poem are only available […]wdjoyner
- Let’s do the Landau shuffle 2021-10-23Here’s a shuffle I’ve not seen before: Take an ordinary deck of 52 cards and place them, face up, in the following pattern:Going from the top of the deck to the bottom, placing cards down left-to-right, put 13 cards in the top row:11 cards in the next row:then 9 cards in the next row:then 7 […]wdjoyner
- Coding Theory and Cryptography 2021-10-11This was once posted on my USNA webpage. Since I’ve retired, I’m going to repost it here. Coding Theory and Cryptography:From Enigma and Geheimschreiber to Quantum Theory(David Joyner, ed.) Springer-Verlag, 2000. ISBN 3-540-66336-3 Summary: These are the proceedings of the “Cryptoday” Conference on Coding Theory, Cryptography, and Number Theory held at the U. S. Naval […]wdjoyner
- Shanks’ SQUFOF according to McMath 2021-07-30In 2003, a math major named Steven McMath approached Fred Crabbe and I about directing his Trident thesis. (A Trident is like an honors thesis, but the student gets essentially the whole year to focus on writing the project.) After he graduated, I put a lot of his work online at the USNA website. Of […]wdjoyner
- The truncated tetrahedron covers the tetrahedron 2021-04-29At first, you might think this is obvious – just “clip” off each corner of the tetrahedron to create the truncated tetrahedron (by essentially creating a triangle from each of these clipped corners – see below for the associated graph). Then just map each such triangle to the corresponding vertex of the tetrahedron. No, it’s […]wdjoyner