On the crossing number of arithmetic curve systems
We show that the family of systoles of hyperbolic surfaces associated with congruence lattices in $\mathrm{SL}_2(\mathbb{Z})$ have asymptotically minimal crossing number.
math.GT↗
arXiv subjects
Luca Studer
Publications and source records attributed to Luca Studer.
An asymptotic version of the union-closed sets conjecture
We show that the biggest possible average set size in the complement $2^{\{1,2,\ldots, n\}} \setminus A$ of a union-closed family $A \subset 2^{\{1,2, \ldots, n\}}$ is $\tfrac{n+1}{2}$. With the same proof we get a sharp upper bound for the average frequency in complements of union-closed families. This implies an asymptotic version of the union-closed sets conjecture, formulated in terms of complements of union-closed families.
math.CO↗
Exponential factorizations of holomorphic maps
We show that any element of the special linear group $SL_2(R)$ is a product of two exponentials if the ring $R$ is either the ring of holomorphic functions on an open Riemann surface or the disc algebra. This is sharp: one exponential factor is not enough since the exponential map corresponding to $SL_2(\mathbb{C})$ is not surjective. Our result extends to the linear group $GL_2(R)$.
math.CV↗
Distortion of spheres and surfaces in space
It is known that the surface of a cone over the unit disc with large height has smaller distortion than the standard embedding of the 2-sphere in $\mathbb R^3$. In this note we show that distortion minimisers exist among convex embedded 2-spheres and have uniformly bounded eccentricity. Moreover, we prove that $π/2$ is a sharp lower bound on the distortion of embedded closed surfaces of positive genus.
math.MG↗
A splitting lemma for coherent sheaves
The presented splitting lemma extends the techniques of Gromov and Forstneri\v{c} to glue local sections of a given analytic sheaf, a key step in the proof of all Oka principles. The novelty on which the proof depends is a lifting lemma for transition maps of coherent sheaves, which yields a reduction of the proof to the work of Forstneri\v{c}. As applications we get shortcuts in the proofs of Forster and Ramspott's Oka principle for admissible pairs and of the interpolation property of sections of elliptic submersions, an extension of Gromov's results obtained by Forstneri\v{c} and Prezelj.
math.CV↗
A homotopy theorem for Oka theory
We prove a homotopy theorem for sheaves. Its application shortens and simplifies the proof of many Oka principles such as Gromov's Oka principle for elliptic submersions.
math.CV↗
Note on the distortion of (2,q)-torus knots
We show that the distortion of the (2,q)-torus knot is not bounded linearly from below.
math.GT↗