New asymmetric metric on Teichmüller space for surfaces.
problem Defining a new asymmetric weak metric on Teichmüller space.
method Introducing Teichmüller-Randers metric as an asymmetric deformation of the Teichmüller metric.
result Teichmüller geodesics become unique Teichmüller-Randers geodesics under certain conditions.
This paper explores infinite-dimensional Teichmüller spaces and their properties.
problem Teichmüller spaces of infinite-type surfaces are complex and depend on base structures.
method Study various distance functions and Teichmüller spaces associated with infinite-type surfaces.
result Finitely supported Teichmüller space is dense in asymptotically isometric Teichmüller space.
Survey of early moduli and Teichmüller space contributions.
problem Understanding the historical development of moduli and Teichmüller spaces.
method Historical review and explanation of mathematical ideas.
result Confirmation of Teichmüller's results by Ahlfors and Bers.
The paper describes geometric properties of Teichmüller space metrics.
problem Analyzing the geometry of Teichmüller space with weak Finsler metrics.
method Geometric description of unit spheres in weak Finsler metrics.
result Introduced a family of weak Finsler metrics interpolating between Thurston's metric and Teichmüller metric.
Mathematical commentary on Teichmüller's theorem about closed surfaces.
problem Proving the existence of extremal quasiconformal maps on closed Riemann surfaces.
method Defining mappings between Teichmüller space and Fricke space, using Brouwer's invariance of domain theorem.
result Established a homeomorphism between Teichmüller space and Euclidean space of dimension 6g-6.
Constructs deformations of hyperbolic hexagons for new geodesics in Teichmüller spaces.
problem Finding new geodesics in Teichmüller spaces.
method One-parameter family of right-angled hexagons with Lipschitz maps.
result New geodesics for arc and Thurston metrics on Teichmüller spaces.
Teichmuller proved the existence of extremal quasiconformal mappings for pentagons.
problem Existence of extremal quasiconformal mappings for pentagons.
method Proof of existence for quasiconformal mappings in the case of pentagons.
result Existence of extremal quasiconformal mappings for pentagons.
Abstract notes on hyperbolic surfaces and Teichmüller spaces.
problem Understanding the geometry of surfaces and Teichmüller spaces.
method Survey of results on stretch lines and Thurston's metric.
result Analogies between Thurston's metric and Teichmüller's metric.
We construct a new Riemannian metric on Goldman space B(S), the space of the equivalence classes of convex projective structures on the surface S, and then prove the new metric, as well as the metric of Darvishzadeh and Goldman, restricts to be the Weil-Petersson metric on Teichmu¨ller space, embe…
Commentary on Teichmüller's quasiconformal mappings paper, highlighting main results.
problem Exploring extremal quasiconformal mappings and their relation to quadratic differentials.
method Analyzing Teichmüller's original paper and its impact on subsequent research.
result Some results from Teichmüller's paper were rediscovered later without citation.
Teichmuller solved the type problem for Riemann surfaces.
problem Deciding if a Riemann surface is conformally equivalent to the complex plane or unit disc.
method Using line complexes and quasiconformal mappings, Teichmuller proved equivalence of surfaces with the same ramification measure.
result A simply connected Riemann surface is hyperbolic if sufficiently ramified.
Study of Teichmüller space geometry using infinitesimal and global methods.
problem Understanding the geometry of Teichmüller space and its tangent/cotangent spheres.
method Systematic study of Thurston metric's infinitesimal and global properties.
result Rigidity statements for the Thurston metric analogous to Royden theorem.
This paper adapts Thurston's earthquake metric to Riemann surfaces with marked points.
problem Defining a norm and metric on Teichmüller spaces for surfaces of arbitrary genus.
method Adapting Thurston's earthquake norm to Riemann surfaces with marked points and using complex Legendre transforms.
result Establishes a complete analogue of Thurston's earthquake norm in the conformal setting.
Study on Teichmüller space of acute triangles using explicit calculations.
problem Analogue of Thurston's metric on acute triangles.
method Direct calculation of distance function and Finsler structure.
result Explicit expressions and properties of the metric on acute triangles.
Commentary on Teichmüller's 1938 paper on conformal and quasiconformal mappings.
problem Investigations into conformal and quasiconformal mappings and their applications.
method Detailed development of conformal invariants and applications in value distribution theory.
result Insures the almost circularity of certain loci and the circularity near infinity of quasiconformal maps.
Derive derivatives of length functions on Teichmüller spaces using shearing coordinates.
problem Compute derivatives of length functions on Teichmüller spaces.
method Use shearing coordinates and Bonahon's theory of transverse H{ö}lder distributions.
result Hessian of length functions is positive-definite if curves intersect every leaf of a maximal geodesic lamination.
Abstract reviews actions of the absolute Galois group on geometric and topological objects.
problem Understanding the absolute Galois group Γ Q and its actions on geometric/topological structures.
method Exploring Grothendieck's ideas and related works on dessins d'enfant, Teichmüller towers, and nonlinear actions.
result Conjectures and insights into homomorphisms between absolute Galois group and automorphism groups of related objects.
New combinatorial structures for Teichmüller spaces with Thurston's metric are explored.
problem Understanding the combinatorial structures of Teichmüller spaces with Thurston's metric.
method Analyzing the unit tangent and cotangent spheres of Teichmüller space, proving formulas for dimensions and codimensions of faces.
result The combinatorial structure of unit spheres in Teichmüller spaces is independent of the underlying point and is isomorphic to the extended mapping class group.
Explains Teichmüller's 1944 displacement theorem for quasiconformal mappings.
problem Finding the quasiconformal mapping with minimal dilatation.
method Detailed explanation of Teichmüller's solution.
result Solution to extremal problem in quasiconformal mappings.
The paper explores families of Finsler metrics and their properties.
problem Understanding symmetrizations and combinations of Finsler metrics.
method Introducing two families of metrics derived from a general Finsler metric.
result Characterization of geodesics, completeness, and Finsler properties of the introduced metrics.
Grothendieck proposes a new topology for semialgebraic and semianalytic geometry.
problem Topology for semialgebraic and semianalytic geometry.
method New conception of manifold, submanifold, and maps.
result Recasting topology to fit semialgebraic and semianalytic geometry.
Study on earthquake metric on Teichmüller space, proving properties and new completions.
problem Understanding the earthquake metric on Teichmüller space.
method Proofs of properties, new completions, and interpretation of the metric.
result Coincidence of various completions for the earthquake metric.
The arc metric is an asymmetric metric on the Teichm{ü}ller space T(S) of a surface S with nonempty boundary. In this paper we study the relation between Thurston's compactification and the horofunction compactification of T(S) endowed with the arc metric. We prove that there is a natural homeomorphism between the two …
Tissot's indicatrix theory is foundational for quasiconformal mappings.
problem Understanding map distortions in geographical projections.
method Mathematical analysis of map projections and their distortions.
result Tissot's work laid the groundwork for quasiconformal mappings.
Bonahon's method for compactifying Teichmüller space extended to non-compact surfaces.
problem Compactifying Teichmüller space for non-compact finite area surfaces.
method Using geodesic currents, extending Bonahon's construction.
result The method applies to surfaces of finite area.
Analyzes convex structures in Teichmüller space unit tangent spheres.
problem Characterize faces and extreme points of unit tangent spheres in Teichmüller space.
method Analyzes Finsler infinitesimal balls of Thurston metric, characterizes faces, exposed faces, and extreme points.
result Characterizes faces and extreme points of unit tangent spheres in Teichmüller space.
Study on Blaschke locus with covariance metric properties.
problem Riemannian structure of Blaschke locus.
method Analysis of Blaschke locus as a manifold, study of covariance metric properties.
result Geodesics in Blaschke locus have infinite length with respect to the covariance metric.
Study flat spacetimes with BTZ singularities using BTZ-extension.
problem Understanding singularities in flat spacetimes.
method Use BTZ-extension to describe Moduli spaces and construct cauchy-surface.
result Construct convex polyhedral cauchy-surface in Cauchy-compact flat spacetimes with BTZ.
Extends flat spacetimes with BTZ singularities, preserving key properties.
problem Describing globally hyperbolic singular flat spacetimes.
method Lorentzian geometry, Teichmüller space, BTZ singularities.
result BTZ extensions preserve Cauchy-maximality and Cauchy-completeness.
The study bounds the number of closed geodesics in a specific orbit closure of surfaces.
problem Counting closed geodesics in a specific orbit closure of surfaces.
method Analyzes triangulations and Teichmüller geodesics to bound the number of closed geodesics.
result Obtains exponential bounds on the number of closed geodesics of length at most R.
Grötzsch's papers review progress in quasiconformal geometry.
problem Developing quasiconformal mappings theory.
method Analyzing five papers by Grötzsch from 1928-1932.
result Illustrates Grötzsch's motivation and results on quasiconformal mappings.
Continuity of earthquake flow map transfers Teichmüller dynamics results.
problem Transfer results from Teichmüller dynamics to earthquake flow.
method Analyze continuity of earthquake flow map and its inverse.
result Transfer results from Teichmüller dynamics to earthquake flow.
Study on topological properties of Higgs bundles over Riemann surfaces.
problem Topological invariants of parabolic G-Higgs bundles. method Analysis of moduli spaces, orbifold cohomology, and Higgs bundles.
result New topological invariants and connected components counts.
Classifies components of abelian differentials over Teichmüller space.
problem Classifying components of strata of abelian differentials.
method Computing monodromy groups and determining finite generating sets for r-spin stabilizer subgroups. result Complete classification of strata components for g≥5. There are certain families of words and word sequences (words in the generators of a two-generator group) that arise frequently in the Teichm{ü}ller theory of hyperbolic three-manifolds and Kleinian and Fuchsian groups and in the discreteness problem for two generator matrix groups. We survey some of the families of su…
Random walks on hyperbolic spaces show linear growth in translation lengths.
problem Investigate the growth of translation lengths in random walks on hyperbolic spaces.
method Prove linear growth without moment conditions and apply to Teichmüller spaces.
result Linear growth of translation lengths in random walks on hyperbolic spaces.
Maps between acute triangles with minimal stretch found and studied.
problem Finding the minimal stretch between acute triangles.
method Formula for the smallest Lipschitz constant and analysis of the metric space.
result Metric space of pairs of acute triangles with fixed area is Finsler and geodesics determined.
We describe in elementary geometrical terms Teichm\" uller spaces of decorated and holed surfaces. We construct explicit global coordinates on them as well as on the spaces of measured laminations with compact and closed support respectively. We show explicitly that the latter spaces are asymptotically isomorphic to th…
Survey on extending Kleinian group theory to Lorentzian anti-de Sitter space.
problem Applying classical Kleinian group theory to Lorentzian anti-de Sitter space fails due to improper discontinuity and dependence of accumulation points.
method Introducing causality notions to extend limit sets and regularity domains to achronal subgroups.
result The theory of limit sets and regularity domains extends naturally to achronal subgroups in globally hyperbolic spacetimes.
Study harmonic maps and anti-de Sitter 3-manifolds.
problem Existence and uniqueness of harmonic maps in infinite energy settings.
method Generalization of Corlette's result to infinite energy, analysis of asymptotic behavior, use of CAT(-1) Hadamard manifolds.
result Existence of new anti-de Sitter 3-manifolds.
We describe sufficient conditions which guarantee that a finite set of mapping classes generate a right-angled Artin group quasi-isometrically embedded in the mapping class group. Moreover, under these conditions, the orbit map to Teichmuller space is a quasi-isometric embedding for both of the standard metrics. As a …
Finding surface mappings with least distortion arises from many applications in various fields. Extremal Teichmüller maps are surface mappings with least conformality distortion. The existence and uniqueness of the extremal Teichmüller map between Riemann surfaces of finite type are theoretically guaranteed [1]. Rece…
Quantum machine learning models can approximate any continuous function.
problem Theoretical understanding of quantum feature maps in machine learning.
method Proving universal approximation property of quantum machine learning models in quantum-enhanced feature spaces.
result Quantum machine learning models are universal approximators of continuous functions.
Quantum theory of curved tetrahedrons yields quantum group intertwiners.
problem Quantum geometry of curved tetrahedrons and their intertwiners.
method Combinatorial quantization of tetrahedron phase space, relating to SU(2) flat connections.
result Physical Hilbert space coincides with Uq(su(2)) intertwiners, consistent with LQG area spectrum.
Defines coherent manifolds and their quantum applications.
problem Understanding quantum spaces and coherent states.
method Generalizes quantum field theory and Schrödinger equation solutions.
result Solves Schrödinger equation on coherent manifolds.
Riemann moduli spaces are quantum ergodic for certain dimensions.
problem Quantum ergodicity of Riemann moduli spaces.
method Analysis of Weil--Petersson metric and geodesic flow.
result Riemann moduli spaces Mg,n are quantum ergodic for 3g+n≥4. Researchers successfully implemented quantum autoencoders using quantum adders in a cloud quantum computer.
problem Reducing resource usage in quantum computations.
method Experimental implementation of quantum autoencoders using approximate quantum adders in a cloud quantum computer.
result Experimental fidelities are in good agreement with theoretical predictions, proving the feasibility of quantum autoencoders via quantum adders.
Quantum-enhanced feature spaces improve machine learning performance.
problem Large feature spaces and computationally expensive kernel functions in machine learning.
method Two novel quantum methods: quantum variational classifier and quantum kernel estimator.
result Quantum-enhanced classifiers achieve better performance on noisy quantum computers.