When is a topological branched self-cover of the sphere equivalent to a post-critically finite rational map on $\mathbb {C}\mathbb {P}^1$? William Thurston gave one answer in 1982, giving a negative criterion (an obstruction to a map being rational). We give a complementary, positive criterion: the branched self-cover is equivalent to a rational map if and only if there is an elastic graph spine for

We prove affirmatively the one-dimensional case of a conjecture of Stein regarding the $L^p$-boundedness of the Polynomial Carleson operator for $1\lt p\lt \infty $. \par Our proof relies on two new ideas: (i) we develop a framework for \emph higher-order wave-packet analysis that is consistent with the time-frequency analysis of the (generalized) Carleson operator, and (ii) we introduce a \emph local

We prove an explicit formula for the Poincaré polynomials of parabolic character varieties of Riemann surfaces with semisimple local monodromies, which was conjectured by Hausel, Letellier and Rodriguez-Villegas. Using an approach of Mozgovoy and Schiffmann the problem is reduced to counting pairs of a parabolic vector bundle and a nilpotent endomorphism of prescribed generic type. The generating function

The Glauberman correspondence is a fundamental bijection in the character theory of finite groups. In 1994, Hartley and Turull established a degree-divisibility property for characters related by that correspondence, subject to a congruence condition which should hold for the Green functions of finite groups of Lie type, as defined by Deligne and Lusztig. Here, we present a general argument for completing

Let $\psi :\mathbb {N}\to \mathbb {R}_{\geqslant 0}$ be an arbitrary function from the positive integers to the non-negative reals. Consider the set $\mathcal {A}$ of real numbers $\alpha $ for which there are infinitely many reduced fractions $a/q$ such that $|\alpha -a/q|\leqslant \psi (q)/q$. If $\sum _{q=1}^\infty \psi (q)\varphi (q)/q=\infty $, we show that $\mathcal {A}$ has full Lebesgue measure

In this paper, we reveal the formal algebraic structure underlying the intrinsic reconstitution algorithm, introduced by Singer and Shkolnisky in [9], for determining three dimensional macromolecular structures from images obtained by an electron microscope. Inspecting this algebraic structure, we obtain a conceptual explanation for the admissibility (correctness) of the algorithm and a proof of its