New family of measurable pseudo-Anosov maps on spheres.
problem Generalizing pseudo-Anosov maps to measurable ones.
method Continuous family of homeomorphisms on sphere, semi-conjugate to core tent map.
result Measurable pseudo-Anosov maps have invariant dense streamlines with uniform measures.
Paper defines and studies measures related to earthquakes and best Lipschitz maps.
problem Understanding Thurston's conjecture about maps and measures on hyperbolic surfaces.
method Examining Lie algebra valued transverse measures and their relation to earthquakes.
result Defines and shows correspondence between best Lipschitz maps and earthquakes.
Study invariant measures on measured laminations for subgroups of mapping class group.
problem Classify invariant Radon measures on space of measured laminations for subgroups of mapping class group.
method Geometric approach, focusing on recurrent measured laminations, explicitly constructing ergodic measures.
result Show uniquely ergodic for divergence-type subgroups, generalize results for full mapping class group.
Study on measurable pseudo-Anosov maps on surfaces.
problem Characterize dynamics of pseudo-Anosov maps on surfaces.
method Analyze measurable pseudo-Anosov homeomorphisms with specific properties.
result Prove transitivity, dense periodic points, sensitivity, and ergodicity.
The paper extends a measure preserving property to bi-Lipschitz maps between Moran sets.
problem Extending the measure preserving property to Moran sets.
method Analyzing bi-Lipschitz maps between Moran sets.
result Bi-Lipschitz maps between Moran sets preserve measure locally.
The paper derives Liouville theorems for various generalized maps on Riemannian manifolds.
problem Deriving Liouville theorems for generalized maps on Riemannian manifolds.
method Using conservation laws and monotonicity formulas, the paper derives Liouville theorems for different types of maps under various conditions.
result The paper establishes Liouville theorems for several types of generalized maps, including φ-F harmonic maps, φ-F symphonic maps, and φ-F-V-harmonic maps. Kim-Milman flow map stable under regular target measures
problem Stability of Kim-Milman flow map under target measure variations
method Stability in relative entropy and 2-Wasserstein distance result Lipschitz stability up to logarithmic factor
Proves the bending map is proper for hyperbolic 3-manifolds.
problem Properness of the bending map in hyperbolic 3-manifolds.
method Analyzes geometric properties and isotopy classes of homeomorphisms.
result Proving the bending map is proper for hyperbolic 3-manifolds.
Maps and measures on surfaces link best Lipschitz and least gradient functions.
problem Analyzing maps between surfaces and their geometric properties.
method Duality between best Lipschitz and least gradient maps, geodesic laminations, and transverse measures.
result The infinity harmonic map defines a geodesic lamination and the least gradient map defines a transverse measure.
We study the statistical meaning of the minimization of distortion measure and the relation between the equilibrium points of the SOM algorithm and the minima of distortion measure. If we assume that the observations and the map lie in an compact Euclidean space, we prove the strong consistency of the map which almost …
Transformers can interpolate between arbitrary measures.
problem Understanding the expressive power of Transformers as measure-to-measure maps.
method Provided an explicit choice of parameters for a single Transformer to match N arbitrary input measures to N arbitrary target measures.
result A single Transformer can interpolate between arbitrary measures.
We show that the mapping class group of a compact orientable surface with higher complexity has the following extreme rigidity in the sense of measure equivalence: if the mapping class group is measure equivalent to a discrete group, then they are commensurable up to finite kernel. Moreover, we describe all lattice emb…
The paper studies harmonic map flows and proves rectifiability of singular sets.
problem Understanding the structure of singular sets in harmonic map flows.
method Investigates the stratification theory for suitable solutions using tangent measures.
result Each time slice of the singular set is rectifiable.
The paper proves a Liouville theorem for specific harmonic maps with free boundary.
problem Analyzing harmonic maps with free boundary conditions.
method Developed Liouville theorem for φ-F-symphonic, φ-F-harmonic, and φ-ΦS,p,ε harmonic maps. result Established Liouville theorem for the specified harmonic maps with free boundary.
The Cannon-Thurston map's pushed measures on the circle are singular with respect to sphere measures.
problem Understanding the behavior of geodesics and measures on fibered hyperbolic 3-manifolds.
method Properties of geodesics and measures on the circle and sphere are analyzed to prove singularity.
result Natural measures on the circle become singular with respect to measures on the sphere.
New measure defined on surface strata, invariant under scaling.
problem Defining an invariant measure on moduli spaces of dilation surfaces.
method Novel computation of cohomology with coefficients for mapping class group.
result SL(2,R)-invariant Lebesgue class measure on strata.
We systematically investigate the problem of representing Markov chains by families of random maps, and which regularity of these maps can be achieved depending on the properties of the probability measures. Our key idea is to use techniques from optimal transport to select optimal such maps. Optimal transport theory a…
The Cannon-Thurston map's measures become singular with respect to sphere measures.
problem Characterizing the behavior of measures under the Cannon-Thurston map.
method Analyzing the geometric properties of hyperbolic geodesics and quasi-geodesics.
result Natural measures on the circle become singular with respect to measures on the sphere.
Characterizes closures of mapping class group orbits on non-orientable surfaces.
problem Understanding closures of orbits in Teichmüller spaces for non-orientable surfaces.
method Analyzes closures in ML and PML for measured laminations, projective measured laminations, and points. result Characterizes closures of weighted two-sided curves in ML. Quadratic-time algorithm computes stretch factors and foliations for pseudo-Anosov mapping classes.
problem Computing stretch factors and foliations for pseudo-Anosov mapping classes efficiently.
method Quadratic-time algorithm using input word and length as complexity measure.
result First algorithm to compute stretch factors and foliations in sub-exponential time.
Classifies invariant measures on specific character varieties.
problem Classifying invariant probability measures on character varieties.
method Measure disintegration along transverse Lagrangian tori fibrations.
result Ergodic measures are either counting measures on finite orbits or Liouville measures.
Landmark2Vec maps unknown landmarks without GPS.
problem Estimate positions of unknown landmarks without GPS.
method Unsupervised neural network trained on landmark signals.
result Maps landmarks up to scale, rotation, and shift.
Transformers preserve support and can approximate any continuous map.
problem Understanding the mathematical properties of transformers.
method Characterizing maps between measures that can be represented as transformers and proving their properties.
result Transformers preserve support and have uniformly continuous Fréchet derivatives.
Injectivity proven for measure homology of certain wild spaces.
problem Injectivity of measure homology for mildly wild spaces.
method Proving injectivity of the canonical map from singular to measure homology.
result Injectivity of measure homology for certain mildly wild spaces.
The Gauss Image Measure uniquely identifies dual convex bodies up to dilation.
problem Identifying dual convex bodies based on their Gauss Image Measure.
method Analyzing the Gauss Image Measure and its properties to establish the uniqueness of dual bodies.
result Dual convex bodies are equal up to a dilation on each path-connected component of the support of the measure.
This work establishes properties on diffeological structures for set-valued maps and measures.
problem Establish rigorous properties on diffeological structures for set-valued maps and measures.
method Using diffeologies, the authors link various structures including set-valued maps, relations, gradients, measures, and shape analysis.
result Established rigorous properties on sample diffeologies.
Polynomial Duistermaat-Heckman measure on symplectic groupoid quotients.
problem Calculating measures on symplectic groupoid quotients.
method Using Hamiltonian groupoid actions and proper moment maps.
result Duistermaat-Heckman measure is polynomial.
Automatic continuity of polynomial maps and cocycles proved.
problem Proving continuity of polynomial maps and cocycles.
method The approach involves proving continuity of polynomial maps and cocycles.
result Automatic continuity of polynomial maps and cocycles.
The paper introduces new volume measures and volume comparison inequalities for Lorentzian spaces.
problem Volume comparison in Lorentzian pre-length spaces.
method Introducing modified timelike Hausdorff measures and establishing volume comparison inequalities using timelike Lipschitz maps.
result Coincidence of modified and original volume measures on smooth spacetimes and some pre-length spaces.
The purpose of this paper is to study the action of the mapping class group on the moduli space of representations of the fundamental group of a non-orientable surface into SU(2). The action is shown to be ergodic with respect to a natural measure. This measure is defined using the push-forward measure associated to a …
The paper develops a theory for one-step Wasserstein-guided models for PDE-induced measures.
problem Theoretical understanding of generative models' accuracy in scientific computing.
method Regularity theory for optimal transport between doubling measures, excess-risk bounds.
result One-step Wasserstein-guided generative models can approximate PDE-induced measures with Hölder continuity.
Researchers found counterexamples to conjectures about optimal transport maps on curved spaces.
problem Extending Caffarelli's contraction theorem to curved spaces.
method Constructing counterexamples to precise conjectures.
result Found counterexamples to Milman's conjectures about optimal transport maps on curved spaces.
The paper proves an energy identity for harmonic maps near singularities.
problem Analyzing the behavior of harmonic maps near singular points.
method Analyzes sequences of stationary harmonic maps with bounded energy, proving an energy identity near singularities.
result The energy density of the defect measure is the sum of the energies of the bubbling maps.
Kernel dependence measures yield accurate estimates of nonlinear relations between random variables, and they are also endorsed with solid theoretical properties and convergence rates. Besides, the empirical estimates are easy to compute in closed form just involving linear algebra operations. However, they are hampere…
We establish orbit equivalence rigidity for any ergodic, essentially free and measure-preserving action on a standard Borel space with a finite positive measure of the mapping class group for a compact orientable surface with higher complexity. We prove similar rigidity results for a finite direct product of mapping cl…
New method uses kernel Stein discrepancy for measure transport without strict continuity constraints.
problem Minimizing Kullback-Leibler divergence for posterior approximation.
method Proposes minimizing kernel Stein discrepancy instead of Kullback-Leibler divergence.
result Demonstrates consistency and competitiveness of the new method.
We provide new conditions that ensure that two metric measure spaces are not quasiconformally equivalent. As an application we deduce that there exists no quasiconformal map between the sub-Riemannian Heisenberg and roto-translation groups.
Formula for subgroup growth in mapping class groups.
problem Growth of subgroups in mapping class groups.
method Asymptotic growth formula with Teichmüller metric.
result Asymptotic growth formula for subgroups.
Investigates maps and properties in spaces with negative dimensions and curvature.
problem Existence of transport maps and local-to-global property in spaces with negative dimensions and bounded Ricci curvature.
method Examines metric measure spaces with negative curvature dimensions and applies reduced curvature-dimension conditions.
result Establishes the existence of transport maps and proves the local-to-global property.
The paper explores geometry of probability measures and barycenter maps.
problem Understanding the space of probability measures and their barycenter.
method Information geometry, Fisher metric, dualistic structures, divergences, geodesics.
result Recent developments in the geometry of probability measures and barycenter.
Let G be a complex semisimple Lie group, K a maximal compact subgroup and V an irreducible representation of K. Denote by M the unique closed orbit of G in P(V) and by O its image via the moment map. For any measure on M we construct a map from the Satake compactification of G/K (associated to V) to the Lie algebra of …
Kaimanovich and Masur showed that a random walk on the mapping class group for an initial distribution with finite first moment and whose support generates a non-elementary subgroup, converges almost surely to a point in the space PMF of projective measured foliations on the surface. This defines a harmonic measure on …
Let G a semisimple Lie group of non-compact type and let XG be the Riemannian symmetric space associated to it. Suppose XG has dimension n and it has no factor isometric to either H2 or SL(3,R)/SO(3). Given a closed n-dimensional Riemannian manifold $…
Study of Matsumoto maps on foliated bundles over hyperbolic manifolds.
problem Characterizing ergodic harmonic measures on foliated bundles.
method Analysis of actions of hyperbolic manifold groups on the circle.
result Suspension of actions with non-discrete images cannot admit Matsumoto maps of type I.
Study connects curvature bounds to map existence and flow solutions.
problem Existence of lower scalar curvature bounds and measures.
method Relates curvature bounds to map existence and backward limit of Ricci flow solutions.
result Sufficient condition for existence of limiting scalar curvature measure.
The bending map of a hyperbolic 3-manifold maps a convex cocompact hyperbolic metric on a hyperbolic 3-manifold with boundary to its bending measured geodesic lamination. In the present paper we study the extension of this map to the space of geometrically finite hyperbolic metrics. We introduce a relationship on the s…
Paper translates train track concepts to cluster algebras for pseudo-Anosov mapping classes.
problem Understanding pseudo-Anosov mapping classes on surfaces.
method Using Goncharov--Shen's potential function, the paper translates train track concepts into cluster algebra language.
result Proves sign stability of general pseudo-Anosov mapping classes.
Computing optimal transport (OT) between measures in high dimensions is doomed by the curse of dimensionality. A popular approach to avoid this curse is to project input measures on lower-dimensional subspaces (1D lines in the case of sliced Wasserstein distances), solve the OT problem between these reduced measures, a…