Authors:
Filip Samuelsen
Since the 1970s, it has been known that any open connected manifold of dimension 2, 4 or 6 admits a complex analytic structure whenever its tangent bundle admits a complex linear structure. For half a century, this has been conjectured to hold true for manifolds of any dimension. In this paper, we extend the result to manifolds of dimension 8 and 10.
The result is proved by applying Gromov's h-principle in order to adapt a method of Haefliger, originally used to study foliations, to the holomorphic setting. For dimension 12 and greater, the conjecture remains open.
Authors:
Carsten Lunde Petersen,
Filip Samuelsen
This paper develops a conformal renormalization scheme for compact sets $K \subset \mathbb{C}$. As one application of the conformal renormalization scheme we prove that for every isolated non-trivial connected component $E \subset K$ there exists a conformal homeomorphism $φ$ mapping a neighbourhood of $E$ into $\mathbb{C}$ such that the equilibrium measure on $K$ restricted to $E$ equals the scaled push-forward by $φ^{-1}$ of the equilibrium measure on $φ(E)$. Moreover the proof shows that the condition of connectedness of $E$ can be relaxed considerably. We also introduce an inverse to the procedure of conformal renormalization, which allows one to reconstruct $K$ from its conformal renormalizations.
Authors:
Filip Samuelsen,
Lorenzo Costigliola,
Thomas B. Schrøder
A simple model for viscous liquid dynamics is introduced. Consider the surface of the union of hyper-spheres centered at random positions inside a hypercube with periodic boundary conditions. It is argued and demonstrated by numerical simulations that at high dimensions geodetic flows on this surface is a good model for viscous liquid dynamics. It is shown that this simple model exhibits viscous dynamics for densities above the percolation threshold in 8, 12 and 16 dimensions. Thus the slowing down of the dynamics, measured by the mean-squared displacement, extends to several orders of magnitude similarly to what is observed in other models for viscous dynamics. Furthermore, the shape of the mean-squared displacement is to a very good approximation the same as for the standard model in simulations of viscous liquids: the Kob-Andersen binary Lennard Jones mixture.