On Deformation Space Analogies between Kleinian Reflection Groups and Antiholomorphic Rational Maps

In a previous paper, we constructed an explicit dynamical correspondence between certain Kleinian reflection groups and certain anti-holomorphic rational maps on the Riemann sphere. In this paper, we show that their deformation spaces share many striking similarities. We establish an analogue of Thurston's compactness theorem for critically fixed anti-rational maps. We also characterize how deformation spaces interact with each other and study the monodromy representations of the union of all deformation spaces.Comment: 56 pages, 13 figures. Final version, to appear in Geom. Funct. Ana

DISTORTION-CONTROLLED ISOTROPIC SWELLING AND SELF-ASSEMBLY OF TRIPLY-PERIODIC MINIMAL SURFACES

In the first part of this thesis, I propose a method that allows us to construct optimal swelling patterns that are compatible with experimental constraints. This is done using a greedy algorithm that systematically increases the perimeter of the target surface with the help of minimum length cuts. This reduces the areal distortion that comes from the changing Gaussian curvature of the sheet. The results of our greedy cutting algorithm are tested on surfaces of constant and varying Gaussian curvature, and are additionally validated with finite thickness simulations using a modified Seung-Nelson model. In the second part of the thesis, we focus on self-assembly methods as an alternate approach to program specific desired structures. More specifically, we develop theoretical design rules for triply-periodic minimal surfaces (TPMS) and show how their symmetry properties can be used to program a minimum number triangular particle-types that successfully coalesce into the TPMS shape. We finally simulate our design rules with Monte Carlo methods and study the robustness of the self-assembled structures upon changing different system parameters like elastic moduli

Discrete Geometry

A number of important recent developments in various branches of discrete geometry were presented at the workshop. The presentations illustrated both the diversity of the area and its strong connections to other fields of mathematics such as topology, combinatorics or algebraic geometry. The open questions abound and many of the results presented were obtained by young researchers, confirming the great vitality of discrete geometry

Discrete Geometry (hybrid meeting)

A number of important recent developments in various branches of discrete geometry were presented at the workshop, which took place in hybrid format due to a pandemic situation. The presentations illustrated both the diversity of the area and its strong connections to other fields of mathematics such as topology, combinatorics, algebraic geometry or functional analysis. The open questions abound and many of the results presented were obtained by young researchers, confirming the great vitality of discrete geometry

LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum

On the Minimum Ropelength of Knots and Links

The ropelength of a knot is the quotient of its length and its thickness, the radius of the largest embedded normal tube around the knot. We prove existence and regularity for ropelength minimizers in any knot or link type; these are $C^{1,1}$ curves, but need not be smoother. We improve the lower bound for the ropelength of a nontrivial knot, and establish new ropelength bounds for small knots and links, including some which are sharp.Comment: 29 pages, 14 figures; New version has minor additions and corrections; new section on asymptotic growth of ropelength; several new reference