We give a simple algorithm that determines whether a given postcritically finite topological polynomial is Thurston equivalent to a polynomial. If it is, then the algorithm produces the Hubbard tree; otherwise, the algorithm produces the canonical obstruction. Our approach is rooted in geometric group theory, using iteration on a simplicial complex of trees, and building on work of Nekrashevych. As one application of our methods, we resolve the polynomial case of Pilgrim’s finite global attractor conjecture. We also give a new solution to Hubbard’s twisted rabbit problem, and we state and solve several generalizations of Hubbard’s problem where the number of postcritical points is arbitrarily large.

The one-particle density matrix $\mathit{\gamma }(x,y)$ for a bound state of an atom or molecule is one of the key objects in the quantum-mechanical approximation schemes. We prove the asymptotic formula ${\mathit{\lambda }}_k\sim {(Ak)}^{-8∕3}$, $A\ge 0$, as $k\to \mathrm{\infty }$, for the eigenvalues ${\mathit{\lambda }}_k$ of the self-adjoint operator $\mathrm{\Gamma }\ge 0$ with kernel $\mathit{\gamma }(x,y)$.

Almost one decade ago, Poonen constructed the first examples of algebraic varieties over global fields for which Skorobogatovs étale BrauerManin obstruction does not explain the failure of the Hasse principle. By now, several constructions are known, but they all share common geometric features such as large fundamental groups.

In this article, we construct simply connected fourfolds over global fields of positive characteristic for which the BrauerManin machinery fails. Contrary to earlier work in this direction, our construction does not rely on major conjectures. Instead, we establish a new Diophantine result of independent interest: a Mordell-type theorem for Campanas geometric orbifolds over function fields of positive characteristic. Along the way, we also construct the first example of a simply connected surface of general type over a global field with a nonempty, but non-Zariski-dense set of rational points.