An introductory course on hyperbolic geometry for advanced students.
problem None explicitly stated; focuses on teaching hyperbolic geometry.
method Textbook format, intended for advanced undergraduate and early graduate students.
result Teaches advanced students about hyperbolic geometry.
Formula found for skinning map contraction in hyperbolic geometry.
problem Finding the contraction constant of skinning maps.
method Elementary hyperbolic geometry and moduli spaces of hyperbolic manifolds.
result Explicit formula for contraction constant.
Study hyperbolic geometry to find Fibonacci numbers.
problem Investigate exponential growth and isoperimetric inequality in hyperbolic geometry.
method Combinatorial approximation of hyperbolic plane, calculations.
result Surprising link to Fibonacci numbers.
Book introduces hyperbolic geometry for knot theory.
problem Understanding knots through hyperbolic geometry.
method Explains hyperbolic geometry, geometric structures, and techniques.
result Develops three knot invariants from hyperbolic geometry.
Study connects hyperbolic geometry to membrane shapes.
problem Understanding the shapes of biological membranes.
method Relates geometry of hyperbolic space to Helfrich model.
result Establishes a connection between membrane shapes and hyperbolic geometry.
New gauge condition fixes L2 metric divergence in hyperbolic monopole spaces.
problem Divergence of L2 metric on hyperbolic monopole moduli spaces. method Alternative gauge-fixing condition inspired by supersymmetry.
result Resulting geometry is hyperbolic hyperkähler, analogous to Euclidean monopole spaces.
We introduce a geometric transition between two homogeneous three-dimensional geometries: hyperbolic geometry and anti de Sitter (AdS) geometry. Given a path of three-dimensional hyperbolic structures that collapse down onto a hyperbolic plane, we describe a method for constructing a natural continuation of this path i…
Invariants of braids found using shear coordinates in hyperbolic geometry.
problem Finding invariants of braids.
method Using shear coordinates in hyperbolic geometry.
result Developed a method for calculating braids invariants.
A new learning method using hyperbolic geometry for class labels.
problem Class label representation and learning in machine learning.
method Hyperbolic Prototype Learning with a new loss function based on hyperbolic geometry.
result Hyperbolic Prototype Learning is equivalent to logistic regression in the one-dimensional case.
The paper studies hyperbolic three-manifolds and their geometric constraints.
problem Understanding the interaction between hyperbolic geometry and homology cobordism.
method Derived explicit bounds on relative grading and invariants in monopole Floer homology.
result Explicit bounds on numerical invariants and subgroup structure of homology cobordism.
Explains how surfaces can have hyperbolic geometries and connects them to Higgs bundles.
problem Understanding hyperbolic structures on surfaces and their relation to Higgs bundles.
method Describes how to obtain and parametrize hyperbolic structures on surfaces, introduces Higgs bundles.
result Established a connection between hyperbolic surfaces and Higgs bundles.
The paper compares spectral geometry in hyperbolic and spherical manifolds.
problem Understanding spectral geometry in spherical manifolds.
method Survey of known results and open problems.
result Analogous results hold in hyperbolic manifolds but not necessarily in spherical manifolds.
Study describes moduli of quaternionic hyperbolic triples of points.
problem Tackles the congruence classes of triples of points in quaternionic hyperbolic space.
method Introduces invariants and defines quaternionic Goldman invariants for mixed configurations.
result Defines quaternionic analogues of Goldman invariants for mixed configurations.
The paper extends circle pattern flows to hyperbolic and Euclidean geometry.
problem Extending circle pattern flows to hyperbolic and Euclidean geometry.
method Proving the existence and exponential convergence of combinatorial Calabi flows for ideal circle patterns.
result The solution to combinatorial Calabi flows converges exponentially fast to a flat cone metric.
Study connects flow dynamics to 3D geometry via surface intersections.
problem Relating flow dynamics to geometric properties of 3-manifolds.
method Relates pseudo-Anosov flow dynamics to hyperbolic geometry via curve graphs.
result Established a link between flow invariants and geometric features of 3-manifolds.
Crochet models of a hyperbolic plane is a popular educational tool as they help to visualize complicated objets in hyperbolic geometry. We present another way how to make crochet models when we view them as a part of a triangulated hyperbolic plane. We also provide a model of a cylinder in a hyperbolic space. This appr…
Paper improves estimates for discrete Laplace in hyperbolic geometry.
problem Establishing compactness for discrete Laplace in hyperbolic geometry.
method Explicit estimates for discrete Laplace based on Glickenstein-Thomas formulation.
result New proofs of long time existence for Calabi flows in hyperbolic geometry.
Survey solves curvature problems with hyperbolic spaces.
problem Singularities in hypersurface geometry.
method Hyperbolic unfolding correspondence linking hypersurfaces to Gromov hyperbolic spaces.
result Eliminates hypersurface singularities in scalar curvature geometry.
Study of pulleys and gears in spherical and hyperbolic geometries.
problem Understanding mechanical systems in non-Euclidean spaces.
method Analysis of pulley and gear systems in spherical and hyperbolic geometries.
result Similar laws governing movement in non-Euclidean geometries.
This thesis was motivated by a desire to understand the natural geometry of hyperbolic monopole moduli spaces. We take two approaches. Firstly we develop the twistor theory of singular hyperbolic monopoles and use it to study the geometry of their charge 1 moduli spaces. After this we introduce a new way to study the m…
Geometric constraints help classify hyperbolic polytopes.
problem Classifying reflective anisotropic Lorentzian lattices and cocompact arithmetic hyperbolic reflection groups.
method Established geometric constraints on compact Coxeter polytopes in hyperbolic spaces.
result Geometric constraints are useful for classifying hyperbolic polytopes.
Hyperbolic geometry reveals financial network structure and systemic importance.
problem Understanding the structure and importance of financial networks.
method Data from European banking stress tests, hyperbolic geometry analysis.
result Latent dimensions of `popularity' and `similarity' are strongly associated with systemic importance.
Study of hyperbolic 3-manifolds via fractional Dehn twists and cusp geometry.
problem Understanding the geometry of fibred hyperbolic 3-manifolds via combinatorial data.
method Relating Euclidean cusp geometry to fractional Dehn twist coefficients of monodromies.
result Uniform bounds on fractional Dehn twist coefficients for certain open book decompositions.
The paper studies a group action on a hyperbolic space derived from a lattice Veech group.
problem Investigating the geometry of a Veech group and its extensions.
method Analyzing the fundamental group of a bundle with singular Euclidean-by-hyperbolic geometry, collapsing regions to produce a hyperbolic action.
result The Veech group's fundamental group acts on a hyperbolic space, retaining most of its geometry.
Simplified proof of mass center system's uniqueness and generalized Pappus' theorem across Euclidean, spherical, and hyperbolic geometries.
problem Proving the uniqueness of the mass center system in non-Euclidean geometries and deriving a generalized Pappus' theorem.
method Revisiting and simplifying G.A. Galperin's proof, extending the mass center system to manifolds, and deriving a generalized Pappus' theorem.
result Unified and simpler proofs for Pappus' theorem in Euclidean, spherical, and hyperbolic geometries.
The study proves analogues of the discrete isoperimetric inequality in hyperbolic geometry.
problem Finding the minimum perimeter for polygons with a fixed area in hyperbolic geometry.
method Proving analogues of the discrete isoperimetric inequality for cyclic and tangential polygons in hyperbolic geometry, considering both single and multiple polygons.
result Established two versions of the isoperimetric inequality for multiple polygons in hyperbolic geometry with certain area or perimeter restrictions.
Ricci flow stability on manifolds with bounded geometry ensures convergence to hyperbolic metrics.
problem Stability and convergence of Ricci flow on manifolds with bounded geometry.
method Continuous dependence on initial conditions, sectoriality of Ricci-DeTurck flow generator, and Hölder norm analysis.
result Ricci flow converges to hyperbolic metrics under certain conditions.
Hyperbolic geometry autoencoder outperforms Euclidean in top-N recommendation tasks.
problem Top-N recommendation performance using hyperbolic geometry.
method Simple autoencoder based on hyperbolic geometry with a single hidden layer.
result Outperforms Euclidean models and state-of-the-art methods.
New surfaces defy traditional pants decompositions.
problem Creating surfaces without bounded pants decompositions.
method Constructing quasiconformally homogeneous hyperbolic Riemann surfaces.
result Found surfaces with infinite type and compact boundary components that defy pants decompositions.
Develops orbibundle theory for complex hyperbolic geometry.
problem Calculating invariants of complex hyperbolic disc orbibundles.
method Using diffeology, calculates Euler number and Toledo invariant.
result New tools for invariants of complex hyperbolic orbibundles.
We study geometry, topology and deformation spaces of noncompact complex hyperbolic manifolds (geometrically finite, with variable negative curvature), whose properties make them surprisingly different from real hyperbolic manifolds with constant negative curvature. This study uses an interaction between Kähler geometr…
The study of isospectral surfaces in Euclidean and hyperbolic geometries.
problem Existence of non-isometric surfaces with identical chord length distributions.
method Construction of isospectral pairs of hyperbolic surfaces without common covers.
result Found isospectral pairs of hyperbolic surfaces with no common cover.
The paper studies hyperbolic geometry of links formed by adding trivial components to knots.
problem Characterizing hyperbolic geometry in links formed by adding trivial components to knots.
method Examining generalized augmented links of knots given by positive braids with at least one full twist.
result Conditions for the hyperbolic geometry of these links are identified.
Study non-Weinstein Liouville geometry via hyperbolic dynamics, proving rigidity results.
problem Characterize non-Weinstein Liouville geometry with persistent transverse skeleton.
method Anosov 3-flows, Liouville Interpolation Systems, non-singular partially hyperbolic flows, hyperbolic dynamics.
result Mitsumatsu's examples characterize 4D non-Weinstein Liouville geometry with 3D persistent transverse skeleton.
Paper studies degenerated circle packings in hyperbolic geometry and finds conditions for their existence.
problem Whether a prescribed total geodesic curvature can be realized by a degenerated circle packing.
method Introduced combinatorial Ricci flow to find the desired degenerated circle packed surface, analogous to Chow-Luo and Takatsu methods.
result Fully characterized sufficient and necessary conditions for the existence of degenerated circle packings and showed their uniqueness.
We present a large scale hyperbolic recommender system. We discuss why hyperbolic geometry is a more suitable underlying geometry for many recommendation systems and cover the fundamental milestones and insights that we have gained from its development. In doing so, we demonstrate the viability of hyperbolic geometry f…
Introduces Coxeter polyhedra in various geometries.
problem None explicitly stated, focuses on introducing concepts.
method Introduction of Coxeter polyhedra in spherical, Euclidean, and hyperbolic geometries.
result Fundamental theorems and descriptions of polyhedra and tessellations.
The paper defines and studies hyperbolicity in calibrated manifolds and derives Schwarz lemmas.
problem Defining and studying hyperbolicity in calibrated manifolds.
method Introducing Rφ-hyperbolicity and φ-hyperbolicity, defining the KR φ-metric, and deriving Schwarz lemmas. result Characterization of φ-hyperbolic domains and extension of Schwarz lemma to calibrated geometries. Hyperbolic spaces have recently gained momentum in the context of machine learning due to their high capacity and tree-likeliness properties. However, the representational power of hyperbolic geometry is not yet on par with Euclidean geometry, mostly because of the absence of corresponding hyperbolic neural network lay…
The study explores dilating set properties across Euclidean and hyperbolic geometries.
problem Distributional properties of dilating sets under projection.
method Covering maps and unit tangent bundles, focusing on manifolds of constant curvature.
result Established a precise asymptotic expansion for averages along expanding translates of homogeneous curves in hyperbolic surfaces.
Study of quaternionic hyperbolic space bisectors and their decompositions.
problem Understanding bisectors in quaternionic hyperbolic geometry.
method Developed theory of quaternionic bisectors, showed various decompositions, derived projection formulas.
result Introduced fan decompositions of quaternionic bisectors by totally geodesic submanifolds isometric to complex hyperbolic space.
Study of hyperbolic directions in convex projective geometry.
problem Understanding properties of quasi-geodesics in convex projective geometry.
method Three perspectives: Hilbert metric, boundary projective geometry, and automorphisms.
result Relationship between different definitions of Morse and regular quasi-geodesics.
Survey of combination theorems in geometry and dynamics.
problem Combination theorems in hyperbolic geometry, group theory, and dynamics.
method Survey and focus on Thurston's contributions.
result Thurston's influence on combination theorems.
We prove a persistence result for noncompact normally hyperbolic invariant manifolds in Riemannian manifolds of bounded geometry. The bounded geometry of the ambient manifold is a crucial assumption in order to control the uniformity of all estimates throughout the proof.
Study geometry of tetrahedra in complex hyperbolic space and Hilbert spaces.
problem Understanding geometric relationships between complex hyperbolic spaces and Hilbert spaces.
method Use a complex analog of the cosine of a vertex angle as a novel technical tool.
result Describe possible triangular faces of tetrahedra in hyperbolic space and three-dimensional subspaces in Hilbert spaces with Pick kernels.
HKConv learns hyperbolic features by aggregating kernel points.
problem Challenges in learning good hyperbolic representations using Euclidean operations.
method Proposes HKConv, a trainable hyperbolic convolution that correlates local features with kernel points and aggregates them.
result HKConv learns expressive local features according to hyperbolic geometry and enjoys equivariance to permutation and invariance to parallel transport.
Conditions for hyperbolic and relatively hyperbolic extensions of free groups using automorphisms with fixed points.
problem Conditions for hyperbolic and relatively hyperbolic extensions of free groups.
method Using dynamics of outer automorphisms on the complex of free factors and investigating the geometry of the extension group.
result Conditions for hyperbolic and relatively hyperbolic extensions of free groups using automorphisms with fixed points.
We use hyperbolic geometry to construct simply-connected symplectic or complex manifolds with trivial canonical bundle and with no compatible Kahler structure. We start with the desingularisations of the quadric cone in C^4: the smoothing is a natural S^3-bundle over H^3, its holomorphic geometry is determined by the h…