In celebration of Pi Day 2024, I would like to explain how the “Arithmetic-Geometric Mean” of Gauss and Legendre can be used to give a rapid method for computing the digits of . By “rapid�...
A torsor (or principal homogeneous space) is, informally speaking, a mathematical structure quite similar to a group, but without a natural identity element. More formally, if is a group, a -tors...
https://mattbaker.blog/2023/09/18/torsors-as-proportion-spaces/
In honor of Pi Day 2023, I’d like to discuss Hilbert’s 7th Problem, which in an oversimplified (and rather vague) form asks: under what circumstances can a transcendental function take algebr...
https://mattbaker.blog/2023/03/14/algebraic-values-of-transcendental-functions-at-algebraic-points/
Test your intuition: is the following true or false? Assertion 1: If is a square matrix over a commutative ring , the rows of are linearly independent over if and only if the columns of are linea...
https://mattbaker.blog/2022/12/24/linear-algebra-over-rings/
In my previous post, I presented a proof of the existence portion of the structure theorem for finitely generated modules over a PID based on the Smith Normal Form of a matrix. In this post, I’...
https://mattbaker.blog/2022/11/21/fitting-ideals-of-modules/
I’m teaching Graduate Algebra this semester, and I wanted to record here the proof I gave in class of the (existence part of the) structure theorem for finitely generated modules over a PID. It...
https://mattbaker.blog/2022/11/09/finitely-generated-modules-over-a-p-i-d/
Congratulations to all of the winners of the 2022 Fields Medal! The only one I know personally, and whose work I have studied in detail, is June Huh. I’m happy both for June himself and for the...
https://mattbaker.blog/2022/07/05/a-fields-medal-for-june-huh/
In this post I will provide a gentle introduction to the theory of martingales (also called “fair games”) by way of a beautiful proof, due to Johan Wästlund, that there are precisely labeled...
https://mattbaker.blog/2021/12/21/counting-with-martingales/
Let’s call a function a near-endomorphism of if there is a constant such that for all . The set of near-endomorphisms of will be denoted by . We put an equivalence relation on by declaring that...
As readers of this previous post will know, I’m rather fond of mental calendar calculations. My friend Al Stanger, with whom I share a passion for recreational mathematics, came up with a remar...
https://mattbaker.blog/2021/12/07/calendar-calculations-with-cards/