The univalent Bloch-Landau constant, harmonic symmetry and conformal glueing

By modifying a domain first suggested by Ruth Goodman in 1935 and by exploiting the explicit solution by Fedorov of the Polyá-Chebotarev problem in the case of four symmetrically placed points, an improved upper bound for the univalent Bloch-Landau constant is obtained. The domain that leads to this improved bound takes the form of a disk from which some arcs are removed in such a way that the resulting simply connected domain is harmonically symmetric in each arc with respect to the origin. The existence of domains of this type is established, using techniques from conformal welding, and some general properties of harmonically symmetric arcs in this setting are established.

Topological phases in small quantum Hall samples

Topological order has proven a useful concept to describe quantum phase transitions which are not captured by the Ginzburg-Landau type of symmetry-breaking order. However, lacking a local order parameter, topological order is hard to detect. One way to detect it is via direct observation of anyonic properties of excitations which are usually discussed in the thermodynamic limit, but so far has not been realized in macroscopic quantum Hall samples. Here we consider a system of few interacting bosons subjected to the lowest Landau level by a gauge potential, and theoretically investigate vortex excitations in order to identify topological properties of different ground states. Our investigation demonstrates that even in surprisingly small systems anyonic properties are able to characterize the topological order. In addition, focusing on a system in the Laughlin state, we study the robustness of its anyonic behavior in the presence of tunable finite-range interactions acting as a perturbation. A clear signal of a transition to a different state is reflected by the system's anyonic properties.

Beurling-Landau densities of weighted Fekete sets and correlation kernel estimates

Let $Q$ be a suitable real function on $C$. An $n$-Fekete set corresponding to $Q$ is a subset ${Z_{n1}},\dotsb, Z_{nn}}$ of $C$ which maximizes the expression $\Pi^n_i_{