Matt Baker's Math Blog » Combinatorics
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Matt Baker articulates his thoughts on number theory, graphs, dynamical systems, tropical geometry, pedagogy, puzzles, and the p-adics in this Math Blog. Read his posts categorized in Combinatorics, covering recent news, solved problems, and developments from the field.
Matt Baker's Math Blog » Combinatorics
1y ago
I recently gave three lectures at Yale University for the Hahn Lectures in Mathematics. The unifying theme of my talks was the notion of break divisor, a fascinating combinatorial concept related to the Riemann-Roch theorem for graphs. Some applications of break divisors to algebraic geometry will be discussed in a follow-up post. Break divisors on ..read more
Matt Baker's Math Blog » Combinatorics
1y ago
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 ..read more
Matt Baker's Math Blog » Combinatorics
1y ago
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 ..read more
Matt Baker's Math Blog » Combinatorics
1y ago
In an earlier post, I described the dollar game played on a finite graph , and mentioned (for the “borrowing binge variant”) that the total number of borrowing moves required to win the game is independent of which borrowing moves you do in which order. A similar phenomenon occurs for the pentagon game described in ..read more
Matt Baker's Math Blog » Combinatorics
1y ago
Everyone who studies elementary number theory learns two different versions of Fermat’s Little Theorem: Fermat’s Little Theorem, Version 1: If is prime and is an integer not divisible by , then . Fermat’s Little Theorem, Version 2: If is prime and is any integer, then . as well as the following extension of Version 1 ..read more
Matt Baker's Math Blog » Combinatorics
1y ago
In this previous post, I described the basic theory of Lorentzian polynomials d’après Brändén and Huh. Now I’d like to describe some of the powerful applications of this theory, culling together results from papers by several different sets of authors. Our first application will be Mason’s Ultra-Log-Concavity Conjecture from 1972, established independently by Brändén-Huh and ..read more
Matt Baker's Math Blog » Combinatorics
1y ago
I’m organizing a reading seminar this semester on Lorentzian polynomials, mainly following this paper by Brändén and Huh but also covering some of the work of Anari et. al. In this post, I’d like to give a quick introduction to this active and beautiful subject. I’ll concentrate on the basic theory for now, and in ..read more
Matt Baker's Math Blog » Combinatorics
1y ago
In this post, I’d like to present an amusing and off-the-beaten-path solution to the classical “Drunkard’s Walk” problem which at the same time derives the well-known generating function for the Catalan numbers. This solution evolved from a suggestion by my former undergraduate student Stefan Froehlich during a discussion in my Math 4802 (Mathematical Problem Solving ..read more
Matt Baker's Math Blog » Combinatorics
1y ago
The Jacobian of a finite graph is a finite abelian group whose cardinality is equal to the number of spanning trees of . In this earlier post, I discussed a family of combinatorial bijections between elements of and the set of spanning trees of . I also discussed the fact that for plane graphs, these ..read more
Matt Baker's Math Blog » Combinatorics
1y ago
I’d like to continue this discussion of break divisors on graphs & tropical curves by describing an interesting connection to algebraic geometry. In this post, I’ll explain a beautiful connection to the theory of compactified Jacobians discovered by Tif Shen, a recent Ph.D. student of Sam Payne at Yale. Review of break divisors Let be ..read more