arXiv subjects

Filip Samuelsen

Publications and source records attributed to Filip Samuelsen.

3 recordsLinked to original sources

arXiv2025

On the homology of $B\Gamma_n^\mathbb{C}$ and its application to complex structures on open manifolds

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. To prove the result new $\Gamma_n^\mathbb{C}$-structures on $\mathbb{CP}^n$ are constructed. As a consequence we derive a theorem concerning the homology of Haefligers classifying space, $B\Gamma_n^\mathbb{C}$. The result then follows from obstruction theory.

math.GT↗

arXiv2022

A toy model for viscous liquid dynamics

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.

cond-mat.soft↗

arXiv2021

Conformal renormalization of compact sets

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 $\phi$ mapping a neighbourhood of $E$ into $\mathbb{C}$ such that the equilibrium measure on $K$ restricted to $E$ equals the scaled push-forward by $\phi^{-1}$ of the equilibrium measure on $\phi(E)$. 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.

math.CV↗