December | 2021 | Matt Baker's Math Blog
4 posts published by Matt Baker during December 2021| Matt Baker's Math Blog
4 posts published by Matt Baker during December 2021| Matt Baker's Math Blog
Nine years ago, I wrote a July 4th blog post about matroids called A Celebration of Independence. Today, I’d like to talk about independence’s lesser-known sibling. In particular, I want to describe a characterization of matroids due to Paul Vaderlind that I feel ought to be better known. In most books and articles on matroid […]| Matt Baker's Math Blog
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” here, I mean that the algorithm exhibits quadratic convergence: the number of correct digits roughly doubles with each iteration. I […]| Matt Baker's Math Blog
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 -torsor is a set on which acts simply and transitively, i.e., for every , there is a unique such that . Torsors are ubiquitous in […]| Matt Baker's Math Blog
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 algebraic values at algebraic points? The connection with is that Lindemann proved in 1882 that the transcendental function takes transcendental values at every nonzero algebraic […]| Matt Baker's Math Blog
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 linearly independent over . (All rings in this post will be nonzero commutative rings with identity.) And how about […]| Matt Baker's Math Blog
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’d like to explain how the uniqueness portion of that theorem is actually a special case of a more general […]| Matt Baker's Math Blog
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’s a standard argument, based on the existence of the Smith Normal Form for a matrix with entries in a PID, but […]| Matt Baker's Math Blog
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 field of combinatorics more broadly, which at one point was not taken seriously enough by the mathematics community […]| Matt Baker's Math Blog
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 trees on vertices. Apertif: a true story In my early twenties, I appeared on the TV show Jeopardy! That’s not what this […]| Matt Baker's Math Blog
Let’s call a function $latex f : {\mathbb Z} \to {\mathbb Z}$ a near-endomorphism of $latex \mathbb Z$ if there is a constant $latex C>0$ such that $latex |f(a+b)-f(a)-f(b)| \leq C$ for al…| Matt Baker's Math Blog
Thoughts on number theory, graphs, dynamical systems, tropical geometry, pedagogy, puzzles, and the p-adics| Matt Baker's Math Blog
In this post and its sequel, I’d like to explain a new perspective on matroid theory which was introduced in this recent preprint which I wrote with Nathan Bowler. The main observation is th…| Matt Baker's Math Blog
