Many properties of a projective algebraic variety can be encoded by convex cones, such as the ample cone and the cone of curves. This is especially useful when these cones have only finitely many edges, as happens for Fano varieties. For a broader class of varieties which includes Calabi-Yau varieties and many rationally connected varieties, the Kawamata-Morrison cone conjecture predicts the st...

We derive explicit inversion formulae for the attenuated geodesic and horocyclic ray transforms of functions and vector fields on two-dimensional manifolds equipped with the hyperbolic metric. The inversion formulae are based on a suitable complexification of the associated vector fields so as to recast the reconstruction as a Riemann Hilbert problem. The inversion formulae have a very similar ...

Rational maps are self-maps of the Riemann sphere of the form z → p(z)/q(z) where p(z) and q(z) are polynomials. Kleinian groups are discrete subgroups of PSL(2,C), acting as isometries of 3-dimensional hyperbolic space and as conformal automorphisms of its boundary, the Riemann sphere. Both theories experienced remarkable advances in the last two decades of the 20th century and are very active...

Hyperbolic geometry is developed in a purely algebraic fashion from first principles, without a prior development of differential geometry. The natural connection with the geometry of Lorentz, Einstein and Minkowski comes from a projective point of view, with trigonometric laws that extend to ‘points at infinity’, here called ‘null points’, and beyond to ‘ideal points’ associated to a hyperbolo...

A complex hyperbolic orbifold M can be written as HC/Γ where Γ is a discrete, faithful representation of π1(M) to Isom(H 2 C). The group SU(2, 1) is a triple cover of the group of holomorphic isometries of HC and (taking subgroups if necessary) we view Γ as a subgroup of SU(2, 1). Our main goal is to discuss the connection between the geometry of M and traces of Γ. We do this in two specific ca...

Constructing smooth freeform surfaces of arbitrary topology with higher order continuity is one of the most fundamental problems in shape and solid modeling. Most real-world surfaces are with negative Euler characteristic number χ = 2 − 2g − b < 0, where g is genus, b is the number of boundaries. This paper articulates a novel method to construct C∞ smooth surfaces with negative Euler numbers b...

Constructing smooth freeform surfaces of arbitrary topology with higher order continuity is one of the most fundamental problems in shape and solid modeling. Most real-world surfaces are with negative Euler characteristic χ < 0. This paper articulates a novel method to construct C∞ smooth surfaces with negative Euler numbers based on hyperbolic geometry and discrete curvature flow. According to...

The geometry of the Cayley graphs of monoids defined by regular confluent monadic rewriting systems is studied. Using geometric and combinatorial arguments, these Cayley graphs are proved to be hyperbolic, and the monoids to be word-hyperbolic in the Duncan–Gilman sense. The hyperbolic boundary of the Cayley graph is described in the case of finite confluent monadic rewriting systems.

We study the set of bounded geodesies of hyperbolic manifolds. For general Riemann surfaces and for hyperbolic manifolds with some finiteness assumption on their geometry we determine its Hausdorff dimension. Some applications to diophantine approximation are included.

A polygon splitting algorithm is a combinatorial recipe. The description and the implementation of polygon splitting should not depend on the embedding geometry. Whether a polygon is being split in Euclidean, in spherical, in oriented projective, or in hyperbolic geometry should not be part of the description of the algorithm. The algorithm should be purely combinatorial, or geometry free. The ...