arXiv subjects

Matthew Bolan

Publications and source records attributed to Matthew Bolan.

4 recordsLinked to original sources

arXiv2026

Universality of Infinite Chess

We prove that chess played on the infinite chessboard $\mathbb{Z}^2$ with infinitely many pieces is as powerful as it could possibly be, by showing that every open Gale-Stewart game with draws is strategically equivalent to some infinite chess position and vice versa. As our construction is computable and open Gale-Stewart games are well understood, this allows us to resolve many open questions about the complexity of infinite chess with infinitely many pieces. In particular, all countable ordinals arise as the game value of some such chess position. We also give an alternate construction that realizes all countable ordinals as game values, with the pleasing property that it consists only of the king pair and pawns.

math.LO↗

arXiv2026

The degrees of irreducible factors of binomials and multiplicativity of the n-Hartley condition

We prove that, over a field of characteristic $0$, the degrees of factors of a binomial $t^n-\alpha$ are divisible by the least such degree. As a consequence, we deduce that for relatively prime natural numbers $m,n$, a polynomial has the $mn$-Hartley condition if and only if it has the $m$-Hartley and $n$-Hartley conditions.

math.NT↗

arXiv2025

A genus-zero surface with bounded curvature enclosing less volume than the unit sphere

We produce a family of bodies in $\mathbb R^3$ parameterized by $\varepsilon > 0$, each bounded by a smooth topological sphere with principal curvatures in $[-1, 1]$, and having volume arbitrarily close to $ 16 - 4\sqrt 3 + \left(10 \sqrt 3 - 14\right) \pi - \left(\frac{10}{3} - \sqrt 3\right) \pi^2 \approx 3.70.$ Thus, in contrast to the two-dimensional case, the unit sphere (which bounds a ball of volume $\frac{4 }{ 3} \pi \approx 4.19$) does not enclose the minimal volume among all smooth spheres in $\mathbb R^3$ with principal curvatures in $[-1,1]$. This answers a folklore question of Dmitri Burago and Anton Petrunin.

math.DG↗

arXiv2025

The Equational Theories Project: Advancing Collaborative Mathematical Research at Scale

We report on the Equational Theories Project (ETP), an online collaborative pilot project to explore new ways to collaborate in mathematics with machine assistance. The project successfully determined all 22 028 942 edges of the implication graph between the 4694 simplest equational laws on magmas, by a combination of human-generated and automated proofs, all validated by the formal proof assistant language Lean. As a result of this project, several new constructions of magmas satisfying specific laws were discovered, and several auxiliary questions were also addressed, such as the effect of restricting attention to finite magmas.

math.RA↗