We show that if a complete, doubling metric space is annularly linearly connected, then its conformal dimension is greater than one, quantitatively. As a consequence, we answer a question of Bonk and Kleiner: if the boundary of a one-ended hyperbolic group has no local cut points, then its conformal dimension is greater than one

We study the properties of Eisenstein-Kronecker numbers, which are related to special values of Hecke $L$-functions of imaginary quadratic fields. We prove that the generating function of these numbers is a reduced (“normalized” or “canonical” in some literature) theta function associated to the Poincaré bundle of an elliptic curve. We introduce general methods to study the algebraic and $p$-adic properties of reduced theta functions for abelian varieties with complex multiplication (CM). As a corollary, when the prime $p$ is ordinary, we give a new construction of the two-variable $p$-adic measure interpolating special values of Hecke $L$-functions of imaginary quadratic fields, originally constructed by Višik-Manin and Katz. Our method via theta functions also gives insight for the case when $p$ is supersingular. The method of this article will be used in subsequent articles to study in two variables the $p$-divisibility of critical values of Hecke $L$-functions associated to imaginary quadratic fields for inert $p$, as well as explicit calculation in two variables of the $p$-adic elliptic polylogarithms for CM elliptic curves

Elements of the tropical vertex group are formal families of symplectomorphisms of the $2$-dimensional algebraic torus. We prove that ordered product factorizations in the tropical vertex group are equivalent to calculations of certain genus zero relative Gromov-Witten invariants of toric surfaces. The relative invariants which arise have full tangency to a toric divisor at a single unspecified point. The method uses scattering diagrams, tropical curve counts, degeneration formulas, and exact multiple cover calculations in orbifold Gromov-Witten theory

We define a new monoidal structure on the category of collections (shuffle composition). Monoids in this category (shuffle operads) turn out to bring a new insight in the theory of symmetric operads. For this category, we develop the machinery of Gröbner bases for operads and present operadic versions of Bergman's diamond lemma and Buchberger's algorithm. This machinery can be applied to study symmetric operads. In particular, we obtain an effective algorithmic version of Hoffbeck's Poincaré-Birkhoff-Witt criterion of Koszulness for (symmetric) quadratic operads

This article concerns Hilbert irreducibility for covers of algebraic groups, with results which appear to be difficult to treat by existing techniques. The present method works by first studying irreducibility above “torsion” specializations (e.g., over cyclotomic extensions) and then descending the field (by Chebotarev theorem). Among the results, we offer an irreducibility theorem for the fibers, above a cyclic dense subgroup, of a cover of ${\mathbb{G}}_m^n$ (Theorem 1) and of a power $E^n$ of an elliptic curve without CM (Theorem 2); this had not been treated before for $n>1$. As a further application, in the function field context, we obtain a kind of Bertini's theorem for algebraic subgroups of ${\mathbb{G}}_m^n$ in place of linear spaces (Theorem 3). Along the way we shall prove other results, as a general lifting theorem above tori (Theorem 3.1)