Complementary sets of natural numbers and Galois connections

In this post, I’d like to discuss a beautiful result about complementary sets of natural numbers due to Lambek and Moser. I first learned about their theorem as a high school student (from Ross Honsberger’s delightful book “Ingenuity in Mathematics”), but it’s only more recently that I learned about the “Galois” connection.

To motivate the discussion, consider the following question. Let F = \{ 0,1,4,9,16,25,\ldots \} be the sequence of squares, and let G = \{ 2,3,5,6,7,8,10,\ldots \} be its complement in {\mathbb N} = \{ 0,1,2,3,\ldots \}. What is the n^{\rm th} term of the sequence G? In other words, can we give a formula for the n^{\rm th} non-square? One might imagine that no simple formula exists, but in fact Lambek and Moser showed that the n^{\rm th} non-square is equal to n + \{ \sqrt{n} \}, where \{ x \} denotes the closest integer to x. Similarly, if T = \{ 0,1,3,6,10,\ldots \} denotes the set of triangular numbers, the n^{\rm th} element of the complement of T is equal to n + \{ \sqrt{2n} \}.

Figure by Scott Kim
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Lorentzian Polynomials II: Applications

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 Anari-Liu-Oveis Gharan-Vinzant in 2018.

Theorem: Let M be a matroid on n elements, and let I_k(M) denote the number of independent sets of size k in M. Then the sequence I_k(M) is ultra-log-concave.

A special case of this result (which seems to be no easier to prove than the general case) is the following: Let E be a set of n vectors in some finite-dimensional vector space, and let I_k denote the number of k-element linearly independent subsets of E. Then the sequence I_k is ULC.

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Lorentzian polynomials I: Theory

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 a follow-up post I’ll discuss some of the striking applications of this theory.

One major goal of the theory of Lorentzian polynomials is to provide new techniques for proving that various naturally-occurring sequences of non-negative real numbers a_k are log-concave, meaning that a_k^2 \geq a_{k-1} a_{k+1} for all k. More generally, the authors consider homogeneous multivariate polynomials p(x_1,\ldots,x_n) with non-negative coefficients and study certain natural extensions of log-concavity to this setting. (For some background on log-concave sequences, see this survey paper which I wrote for the Bulletin of the AMS.) So let me begin by introducing two “classical” methods for proving (an even sharper version of) log-concavity of the coefficients of certain polynomials.

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Infinitely many primes via generating functions

A few years ago I discovered an amusing proof of Euclid’s theorem that there are infinitely many primes which I thought I’d record here for posterity. (I subsequently learned that a similar argument appears in this paper by Paul Pollack.)

To motivate the proof, and illustrate its working in an (admittedly silly) special case, suppose that there were just two prime numbers, called p and q. Then by the fundamental theorem of arithmetic (i.e., unique factorization into primes) we would have the following identity between generating functions:

\sum_{n=2}^\infty z^n = \sum_{k=1}^\infty z^{kp} + \sum_{k=1}^\infty z^{kq} - \sum_{k=1}^\infty z^{kpq}.

Indeed, there is precisely one term z^n for each integer n \geq 2 on the left-hand side, and the same is true for the right-hand side (consider separately the cases where p \mid n but q \nmid n, q \mid n but p \nmid n, and pq \mid n). Using the formula for the sum of a geometric series, we can rewrite our formula as an identity between complex-analytic functions valid on the open unit disc {\mathbb D} = \{z \in {\mathbb C} \; : \; |z|<1 \}:

\frac{z^2}{1-z} = \frac{z^p}{1-z^p} + \frac{z^q}{1-z^q} - \frac{z^{pq}}{1-z^{pq}}.

This is impossible, however, as we see by letting z approach a primitive pq^{\rm th} root of unity, since each term stays bounded except for \frac{z^{pq}}{1-z^{pq}}, which tends to infinity.

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The Drunkard’s Walk and Catalan Numbers

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) class at Georgia Tech in Fall 2007.

In case you’re not familiar with it, here’s the problem (in a formulation I learned from this book by Frederick Mosteller):

A drunken man standing one step from the edge of a cliff takes a sequence of random steps either away from or toward the cliff. At any step, his probability of taking a step away from the cliff is p (and therefore his probability of taking a step toward the cliff is 1-p). What is the probability that the man will eventually fall off the cliff?

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The Stern-Brocot tree, Hurwitz’s theorem, and the Markoff uniqueness conjecture

My goal in this post is to describe a surprising and beautiful method (the Stern-Brocot tree) for generating all positive reduced fractions. I’ll then discuss how properties of the tree yield a simple, direct proof of a famous result in Diophantine approximation due to Hurwitz.  Finally, I’ll discuss how improvements to Hurwitz’s theorem led Markoff to define another tree with some mysterious (and partly conjectural) similarities to the Stern-Brocot tree.

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The Balanced Centrifuge Problem

Next week I start a new position as Associate Dean for Faculty Development in the Georgia Tech College of Sciences.  One of the things I’m looking forward to is getting to know more faculty outside the School of Mathematics and learning about their research. My knowledge of biology, in particular, is rather woeful, but I love reading about the latest developments in Quanta Magazine and elsewhere.

The other day I took my 14-year-old daughter, who is hoping to study genetics, to visit the lab of a Georgia Tech colleague in the School of Biology.  During the visit we discussed how expensive it can be not just to purchase but also to maintain certain kinds of lab equipment, for example centrifuges.  This reminded me about a blog post I’ve been meaning to write for a long time now…

Back in 2011-12, I spent a year as a faculty member at UC Berkeley and I became friends with some biologists there.  One weekend afternoon I was chatting with a cancer researcher named Iswar Hariharan at a barbecue, and when he heard that I was a number theorist he told me about a problem he had been thinking about on and off for more than 15 years.   The problem concerns balancing centrifuges. Continue reading

Circles, the Basel Problem, and the Apparent Brightness of Stars

On Pi Day 2016, I wrote in this post about the remarkable fact, discovered by Euler, that the probability that two randomly chosen integers have no prime factors in common is \frac{6}{\pi^2}.  The proof makes use of the  famous identity \sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}, often referred to as the “Basel problem”, which is also due to Euler.  In the 2016 post I presented Euler’s original solution to the Basel problem using the Taylor series expansion for \frac{\sin(x)}{x}.

In honor of Pi Day 2018, I’d like to explain a simple and intuitive solution to the Basel problem due to Johan Wästlund.  (Wästlund’s paper is here; see also this YouTube video, which is where I first heard about this approach – thanks to Francis Su for sharing it on Facebook!)  Wästlund’s approach is motivated by physical considerations (the inverse-square law which governs the apparent brightness of a light source) and uses only basic Euclidean geometry and trigonometry.

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GAGS 2018

The 2018 Georgia Algebraic Geometry Symposium is a wrap!  This was the first time that the annual conference was held at Georgia Tech, and I thought it went very well.  Each of the eight talks seems to have been well-received, and some spectacular new results were announced.  Here’s a quick summary of (what I remember about) the talks: Continue reading

The Circuit-Cocircuit Reversal System and Torsor Structures on Spanning Trees

The Jacobian of a finite graph G is a finite abelian group whose cardinality is equal to the number of spanning trees of G.  In this earlier post, I discussed a family of combinatorial bijections between elements of {\rm Jac}(G) and the set {\mathcal T}(G) of spanning trees of G.  I also discussed the fact that for plane graphs, these Bernardi bijections come from a natural simply transitive action of {\rm Jac}(G) on {\mathcal T}(G) (or, more precisely, a natural isomorphism class of such actions).

In the present post, I’d like to discuss a different family of simply transitive actions of {\rm Jac}(G) on {\mathcal T}(G) discovered by myself, Spencer Backman (a former student of mine), and Chi Ho Yuen (a current student of mine).  One virtue of our construction is that it generalizes in a natural way from graphs to regular matroids.  (We will mostly stick to the graphical case in this post, but will insert some comments about extensions to regular and/or oriented matroids.  A regular matroid can be thought of, rather imprecisely, as the smallest natural class of objects which contain graphs and admit a duality theory generalizing duality for planar graphs. Regular matroids are special cases of the more general concept of oriented matroids.)

One of the main new ideas in [BBY] (as we will henceforth refer to our paper) is to use the torsor {\rm Pic}^{g-1}(G) as an intermediate object rather than {\rm Pic}^{g}(G).  The latter is canonically isomorphic (as a {\rm Jac}(G)-torsor) to the set of break divisors on G; the former is isomorphic to the circuit-cocircuit reversal system, which we now introduce.

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The Geometry of Break Divisors

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. Continue reading

The Combinatorics of Break Divisors

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 graphs

Let G be a connected finite graph of genus g = g(G), where g := |E(G)| - |V(G)| + 1.  Our central object of study will be the notion of a break divisor on G.  Recall that a divisor D on a graph G is an assignment of an integer D(v) to each vertex v of G.   The divisor D is called effective if D(v) \geq 0 for all v; in this case, we often visualize D by placing D(v) “chips” at v.  And the degree of D is the sum of D(v) over all vertices v, i.e., the total number of chips.  By analogy with algebraic geometry, we write divisors on G as formal sums D = \sum_{v \in V(G)} D(v) (v), i.e., as elements of the free abelian group on V(G).

A break divisor on G is an effective divisor D of degree g such that for every connected subgraph H of G, the degree of D restricted to H is at least g(H).  In other words, there are g(G) total chips and each connected subgraph H contains at least genus-of-H of these chips.  One surprising fact, proved in this paper (referred to henceforth as [ABKS]), is that the number of break divisors on G is equal to the number of spanning trees of G. Continue reading

Whitney Encounters of the Second Kind

I’m speaking tomorrow in the AMS Current Events Bulletin about the work of Adiprasito, Huh, and Katz on the Rota-Welsh conjecture and Hodge theory for matroids.   See this previous post for an introduction to their work.  [Note added 9/21/17: My write-up for the Current Events Bulletin can be found here.]

Here’s an excerpt from the last section of my slides which I may or may not have time to discuss in tomorrow’s talk.  It concerns this recent paper of June Huh and Botong Wang.  (Note added: As anticipated I did not have time to cover this material!  Here are the slides themselves: ceb_talk)

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p-adic Numbers and Dissections of Squares into Triangles

My thesis advisor Robert Coleman passed away two years ago today (see this remembrance on my blog).  One of the things I learned from Robert is that p-adic numbers have many unexpected applications (see, for example, this blog post).  Today I want to share one of my favorite surprising applications of p-adic numbers, to a simple problem in Euclidean geometry. Continue reading

Matroids over Hyperfields, Part II

In Part I of this post, we defined hyperrings and hyperfields, gave some key examples, and introduced matroids over (doubly distributive) hyperfields in terms of Grassman-Plücker functions.  If E=\{ 1,\ldots,m \} is a finite set and K is a field, we saw that a K-matroid on E is the same thing as a linear subspace of K^m, and if {\mathbb K} is the Krasner hyperfield then a {\mathbb K}-matroid on E is the same thing as a matroid in the usual sense.  Matroids over the hyperfield {\mathbb S} of signs are the same thing as oriented matroids, and matroids over the tropical hyperfield {\mathbb T} are the same thing as valuated matroids.  In this post we will give some “cryptomorphic” axiomatizations of matroids over hyperfields, discuss duality theory, and then discuss the category of hyperrings in a bit more detail. Continue reading