Thoughts on number theory, graphs, dynamical systems, tropical geometry, pedagogy, puzzles, and the p-adics
My coauthors June Huh, Mario Kummer, Oliver Lorscheid, and I just posted a paper called Lorentzian polynomials and matroids over triangular hyperfields, Part 2: Analytic aspects on the arXiv preprint server. This is the first paper of mine to contain a substantial result proved by AI, and also the first to contain a proof formally [ ]
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 [ ]
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 [ ]
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 [ ]
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 [ ]
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 [ ]
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 [ ]
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 [ ]
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 [ ]
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 [ ]