The paper proves a Basmajian identity for non-Archimedean local fields.
problem Proving Basmajian's identity over non-Archimedean local fields.
method Projective Anosov representations and Berkovich hyperbolic geometry.
result A signed finite sum series identity for Basmajian's identity.
Construct special Lagrangian fibrations on abelian varieties using retraction techniques.
problem Constructing special Lagrangian fibrations on abelian varieties.
method Explicit construction using special techniques in non-Archimedean geometry.
result Solved a conjecture of Kontsevich-Soibelman for finite quotients of abelian varieties.
In this paper, we study the dynamics of degenerating sequences of rational maps on Riemann sphere C^ using R-trees. Given a sequence of degenerating rational maps, we give two constructions for limiting dynamics on R-trees: one geometric and one algebraic. The geometric constructio…
The paper describes a stratification of a compactified Hurwitz space using combinatorial trees.
problem Describing the boundary stratification of a compactified Hurwitz space.
method Using decorated trees to describe the boundary strata and their containment relations.
result The boundary strata of the compactified Hurwitz space are in bijection with decorated trees, and containment is given by edge contraction.
Defines volume and Monge-Ampère energy on polarized affine varieties.
problem Volume and Monge-Ampère energy on polarized affine varieties.
method Definition of volume using asymptotics of jumping numbers, Monge-Ampère energy using forms and currents on Berkovich spaces.
result Monge-Ampère energy agrees with volume of filtrations and recovers known functionals.
We compactify the classical moduli variety of compact Riemann surfaces by attaching moduli of (metrized) graphs as boundary. The compactifications do not admit the structure of varieties and patch together to form a big connected moduli space in which ⊔gMg is open dense. The metrized graphs, which are oft…
The Bergman measure converges to the Zhang measure on a hybrid space.
problem Proving convergence of Bergman measures to Zhang measure.
method Analyzing convergence on a hybrid space and metrized curve complex.
result Bergman measure converges to Zhang measure on a hybrid space.
Lecture notes on using non-Archimedean geometry for complex variety degenerations.
problem Complex algebraic variety degenerations with non-Archimedean Berkovich spaces.
method Hybrid spaces and non-Archimedean pluripotential theory.
result Relation between convergence of psh metrics and Monge-Ampere measures in hybrid spaces.
Synthetic approach to pluripotential theory measures finite energy.
problem Global pluripotential theory on compact Kähler manifolds and projective Berkovich spaces.
method Definition and study of measures of finite energy, introduction of twisted and free energy functionals.
result Coercivity of energy functionals is an open condition with respect to polarization.
The study constructs universal invariants for non-Archimedean metrics on projective varieties.
problem Understanding the singularity of non-Archimedean metrics on projective varieties.
method Constructing partial Okounkov bodies and Duistermaat--Heckman measures for non-Archimedean metrics.
result Generalization of Duistermaat--Heckman measures to finite energy metrics on Berkovich analytifications.
Study shows Calabi-Yau metrics converge to a specific form under certain conditions.
problem Degeneration of Calabi-Yau metrics and their limits.
method Optimal transport problem and minimisation of Kontorovich functional.
result Limit data of Calabi-Yau metrics can be encoded into a unique minimiser.
Solves non-Archimedean Calabi-Yau equation on complex log pairs.
problem Non-Archimedean Monge-Ampère equation on Berkovich analytification.
method Solves complex Monge-Ampère equation, then adapts to non-Archimedean setting.
result Non-Archimedean analog of Ricci-flat metric potentials on complex affine varieties.
This paper constructs a non-Archimedean Teichmüller space using tropical geometry.
problem Constructing a non-Archimedean analogue of Teichmüller space.
method Using techniques from tropical and logarithmic geometry.
result The skeleton of non-Archimedean Teichmüller space is the tropical Teichmüller space.
We study the convergence of volume forms on a degenerating holomorphic family of log-Calabi-Yau varieties to a non-Archimedean measure, extending a result of Boucksom and Jonsson. More precisely, let (X,B) be a holomorphic family of sub log canonical, log-Calabi-Yau complex varieties parameterized by the punctured un…
Mathematically proves SYZ conjecture for conifold transition.
problem Tackles the SYZ conjecture for conifold transition.
method Uses family Floer context and non-archimedean setting.
result Explicitly writes singular T-duality fibers and confirms missing points in mirror cluster variety.
Study Monge-Ampère equations on Calabi-Yau hypersurfaces, proving unique solutions and implications for special Lagrangian fibrations.
problem Existence of special Lagrangian fibrations in Calabi-Yau hypersurfaces.
method Non-Archimedean and tropical Monge-Ampère equations on Berkovich and skeleton spaces, proving uniqueness and deriving solutions.
result Unique solutions to tropical and non-Archimedean Monge-Ampère equations, leading to existence of special Lagrangian fibrations.
Study of non-archimedean μ-entropy and its connection to K-stability.
problem Understanding K-stability in non-archimedean settings.
method Introducing non-archimedean μ-entropy and its properties, connecting it to K-semistability.
result Established a criterion for K-semistability without vector ξ, using the non-archimedean μ-entropy.
Paper generalizes K-stability to non-algebraic spaces for Kähler metrics.
problem Existence of constant scalar curvature Kähler metrics on non-algebraic spaces.
method Developed a non-Archimedean theory of complex spaces and applied it to Kähler manifolds.
result Proved K-stability implies existence of unique constant scalar curvature Kähler metric.
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.
Researchers resolve a SYZ conjecture for A_n singularities using quantum-corrected T-duality.
problem Resolving a mathematically precise SYZ conjecture for A_n singularities.
method Building a quantum-corrected T-duality between two singular torus fibrations.
result Constructing a parameter-dependent SYZ mirror fibration partner with matching singular loci and integral affine structure.
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.
Convexity properties are preserved under radial transformations in hyperbolic and spherical geometries.
problem Preserving convexity in hyperbolic and spherical geometries under radial transformations.
method Used Poincaré disk model for hyperbolic geometry and stereographic projection for spherical geometry to prove preservation of convexity under radial expansion and contraction.
result Radial expansion and contraction preserve hyperbolic and spherical convexity, respectively.
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.