Approximation Schemes for Partitioning: Convex Decomposition and Surface Approximation

We revisit two NP-hard geometric partitioning problems - convex decomposition and surface approximation. Building on recent developments in geometric separators, we present quasi-polynomial time algorithms for these problems with improved approximation guarantees.Comment: 21 pages, 6 figure

Variational principles for circle patterns

A Delaunay cell decomposition of a surface with constant curvature gives rise to a circle pattern, consisting of the circles which are circumscribed to the facets. We treat the problem whether there exists a Delaunay cell decomposition for a given (topological) cell decomposition and given intersection angles of the circles, whether it is unique and how it may be constructed. Somewhat more generally, we allow cone-like singularities in the centers and intersection points of the circles. We prove existence and uniqueness theorems for the solution of the circle pattern problem using a variational principle. The functionals (one for the euclidean, one for the hyperbolic case) are convex functions of the radii of the circles. The analogous functional for the spherical case is not convex, hence this case is treated by stereographic projection to the plane. From the existence and uniqueness of circle patterns in the sphere, we derive a strengthened version of Steinitz' theorem on the geometric realizability of abstract polyhedra. We derive the variational principles of Colin de Verdi\`ere, Br\"agger, and Rivin for circle packings and circle patterns from our variational principles. In the case of Br\"agger's and Rivin's functionals. Leibon's functional for hyperbolic circle patterns cannot be derived directly from our functionals. But we construct yet another functional from which both Leibon's and our functionals can be derived. We present Java software to compute and visualize circle patterns.Comment: PhD thesis, iv+94 pages, many figures (mostly vector graphics

Deformations of bordered Riemann surfaces and associahedral polytopes

We consider the moduli space of bordered Riemann surfaces with boundary and marked points. Such spaces appear in open-closed string theory, particularly with respect to holomorphic curves with Lagrangian submanifolds. We consider a combinatorial framework to view the compactification of this space based on the pair-of-pants decomposition of the surface, relating it to the well-known phenomenon of bubbling. Our main result classifies all such spaces that can be realized as convex polytopes. A new polytope is introduced based on truncations of cubes, and its combinatorial and algebraic structures are related to generalizations of associahedra and multiplihedra.Comment: 25 pages, 31 figure

On Fenchel-Nielsen coordinates on Teichm\"uller spaces of surfaces of infinite type

We introduce Fenchel-Nielsen coordinates on Teicm\"uller spaces of surfaces of infinite type. The definition is relative to a given pair of pants decomposition of the surface. We start by establishing conditions under which any pair of pants decomposition on a hyperbolic surface of infinite type can be turned into a geometric decomposition, that is, a decomposition into hyperbolic pairs of pants. This is expressed in terms of a condition we introduce and which we call Nielsen convexity. This condition is related to Nielsen cores of Fuchsian groups. We use this to define the Fenchel-Nielsen Teichm\"uller space associated to a geometric pair of pants decomposition. We study a metric on such a Teichm\"uller space, and we compare it to the quasiconformal Teichm\"uller space, equipped with the Teichm\"uller metric. We study conditions under which there is an equality between these Teichm\"uller spaces and we study topological and metric properties of the identity map when this map exists