Hyperbolic geometry and local Dirichlet–Neumann map

Hyperbolic Geometry

Algebraic surfaces and hyperbolic geometry

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...

Holomorphic Dynamics and Hyperbolic Geometry

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: Isometry Groups of Hyperbolic Space

The goal of this paper is twofold. First, it consists of an introduction to the basic features of hyperbolic geometry, and the geometry of an important class of functions of the hyperbolic plane, isometries. Second, it identifies a group structure in the set of isometries, specifically those that preserve orientation, and deals with the topological properties of their discrete subgroups. In the...

Emergent Hyperbolic Network Geometry

A large variety of interacting complex systems are characterized by interactions occurring between more than two nodes. These systems are described by simplicial complexes. Simplicial complexes are formed by simplices (nodes, links, triangles, tetrahedra etc.) that have a natural geometric interpretation. As such simplicial complexes are widely used in quantum gravity approaches that involve a ...