We show that for a strongly convergent sequence of geometrically finite Kleinian groups with geometrically finite limit, the Cannon-Thurston maps of limit sets converge uniformly. If however the algebraic and geometric limits differ, as in the well known examples due to Kerckhoff and Thurston, then provided the geometr…
Geometrically realizes Khovanov homology for semiadequate links.
problem Computing Khovanov homology for semiadequate links.
method Introducing partial presimplicial sets and their geometric realization.
result Concrete formula for homotopy type of geometric realization.
The paper examines geometric formality on specific surfaces under the Chern-Ricci flow.
problem Understanding geometric formality on class VII surfaces.
method Analysis of Chern-Ricci flow on specific surfaces.
result Evolution of geometric formality under the Chern-Ricci flow.
Paper studies Hausdorff dimension of limit sets for Kleinian groups.
problem Understanding Hausdorff dimension of limit sets for Kleinian groups.
method Constructs geometrically infinite Fuchsian groups and proves properties for finitely generated groups.
result Hausdorff dimension of nonconical limit set equals zero for some groups.
The paper characterizes arithmetic metrics in coarsely geometric settings.
problem Characterizing arithmetic metrics in coarsely geometric settings.
method Using coarse-geometric commensurators and under the Hilbert-Smith conjecture.
result Positive answer in general and unconditional for specific cases.
The paper extends geometric inequalities for nearly spherical sets in various space forms.
problem Investigating weighted inequalities for nearly spherical sets in space forms.
method Generalizing and extending inequalities for nearly spherical sets in C1 and W2,∞ settings, with convex weight functions. result Quantitative stability estimates for weighted inequalities in Rn+1 and Hn+1. The paper establishes pressure gaps for manifolds with flat subtori singularities.
problem Understanding phase transitions in nonpositively curved manifolds with flat subtori.
method Derives a pressure gap criterion for closed rank 1 manifolds with specific singular sets and proves Hölder continuity of geometric potentials.
result Geometric potentials have pressure gaps and no phase transitions under certain curvature constraints.
Geometric inequalities for equipotential curves derived from convex entropy.
problem Geometric relations for equipotential curves defined by harmonic functions.
method Constructing an entropy for each level set and proving convexity.
result Geometric inequalities for curvature and gradient magnitude on equipotential curves.
Study rigidity of geodesic planes in geometrically finite manifolds with cusps.
problem Rigidity of geodesic planes in geometrically finite manifolds with rank 1 cusps.
method Analysis of horocycles recurrence and density criteria for immersions.
result Closed immersions give rise to surfaces with finitely generated fundamental groups.
Geometric argument proves projection theorems in hyperbolic space.
problem Proving projection theorems for hyperbolic space.
method Geometric argument for orthogonal projections.
result Characterization of purely unrectifiable sets in hyperbolic space.
Deep learning models complex multivariate extremes using geometric shapes.
problem Modeling complex extremal dependencies in high-dimensional data.
method Geometric representation and deep learning for flexible semi-parametric models.
result First approach to modeling limit sets using deep learning for high-dimensional data.
Study on geometric Jensen-Shannon divergence for Gaussian measures in Hilbert space.
problem Computing divergence between Gaussian measures in infinite-dimensional Hilbert space.
method Closed form expression and regularization for divergence calculation.
result Closed form expression and regularization for Geometric Jensen-Shannon divergence.
Equivariant networks improve geometric prediction without scalar approximations.
problem Efficiently predicting geometric tensors in real-world scenarios.
method Equivariant networks for geometric prediction.
result Equivariant networks can generalize to unseen systems for geometric prediction.
The paper extends geometric inequalities from Euclidean space to Riemannian manifolds.
problem Proving geometric inequalities on smooth oriented Riemannian manifolds.
method Introducing symmetric decreasing rearrangement inequalities and testing their applicability to Riemannian manifolds.
result Smooth co-area formula and re-formulated geometric inequalities on Riemannian manifolds.
Study shows continuity and geometric regularity of Kähler-Ricci flow blow-up limits.
problem Geometric regularity of blow-up limits of the Kähler-Ricci flow.
method Established geometric regularity for Type I blow-up limits based on sequences of Ricci vertices.
result The limiting flow is continuous in time in Gromov-Hausdorff and Gromov-W1 distance. Extends geometric approach to model non-stationary extremal dependence.
problem Capturing evolving extremal dependence in multivariate data.
method Geometric framework for non-stationary multivariate extreme value modelling.
result Framework can capture various dependence forms and is robust to different model formulations.
New method proves length spectrum rigidity in various geometric settings.
problem Length spectrum rigidity in geometric settings.
method Combination of dynamical systems and geometric group theory.
result Provides concise proofs and extends classical results.
Researchers geometrically define asymptotic coordinates in General Relativity.
problem Understanding the asymptotic behavior of relativistic initial data sets.
method Geometrization of asymptotic flatness and analysis of geometric invariants.
result Geometrically defined asymptotic coordinates for mass, energy, momentum, and angular momentum.
Study continuity of limit sets in symmetric spaces.
problem Continuity of limit sets for geometrically finite subgroups in symmetric spaces.
method Extended geometrically finite representations theory.
result Limit sets vary continuously with respect to Hausdorff distance under strong convergence.
The horocyclic flow on geometrically infinite surfaces shows recurrent irregular orbits or non-minimal closures.
problem Complex dynamics of horocyclic flow on geometrically infinite surfaces.
method Analyzing the recurrence and minimality of irregular orbits.
result Irregular orbits are recurrent or have non-hR minimal closures. Study of 2D Lorentzian anti-de Sitter plane using geometric control theory.
problem Understanding extremal trajectories and reachable set on anti-de Sitter plane.
method Geometric control theory and differential geometry.
result Construction of optimal synthesis and description of Lorentzian distance.
Prove asymptotics of geometric flows using algebro-geometric methods.
problem Prove asymptotics of geometric flows
method Algebro-geometric methods
result Prove a conjecture of Haiden-Katzarkov-Kontsevich-Pandit
Defines unitary setting for quantum mechanics, explaining time evolution.
problem Completing quantum theory by defining unitary time evolution.
method Introduces geometric space with north and south poles, defines unitary time evolution as vector field flow.
result Unitary time evolution is explained as Lie group-Lie group algebra correspondence.
We give a geometric proof of existence of Whitney stratifications of definable sets in o-minimal structures.
The paper extends geometric finiteness to discrete subgroups of negatively pinched Hadamard manifolds.
problem Characterizing geometrically infinite discrete subgroups of negatively pinched Hadamard manifolds.
method Generalizing Bonahon's characterization and proving a theorem of Bishop's extension.
result Every discrete geometrically infinite isometry subgroup has a set of nonconical limit points of cardinality continuum.
Geometric theory developed for ultradifferentiable functions and their wavefront sets.
problem Developing a geometric theory for ultradifferentiable functions.
method Using Dyn'kin's Theorem and Bony's Theorem, the ultradifferentiable wavefront set is defined and its properties are proven.
result Microlocal elliptic regularity theorem for ultradifferentiable vector bundles is proven.
We study geometric realization questions of curvature in the affine, Riemannian, almost Hermitian, almost para Hermitian, almost hyper Hermitian, almost hyper para Hermitian, Hermitian, and para Hermitian settings. We also express questions in Ivanov-Petrova geometry, Osserman geometry, and curvature homogeneity in ter…
This paper develops a new homology theory for biquandles and discusses geometric realizations.
problem Constructing knot invariants using set-theoretic Yang-Baxter equation.
method Developed a normalized (co)homology theory for biquandles and geometrically realized them.
result Geometric realization of biquandles has finitely generated second homotopy group for finite biquandles.
Proves compactness of geometric models for certain homogeneous spaces.
problem Existence and uniqueness of geometric models for locally homogeneous spaces.
method Proves existence and uniqueness of geometric models in the pointed C1,α-topology. result Compact set of geometric models for sectional curvature ≤ 1.
SGLD proves geometric ergodicity via reflection coupling for nonconvex log-concave distributions.
problem Proving geometric ergodicity of SGLD in nonconvex, log-concave settings.
method Reflection coupling technique to handle SGLD's time discretization and minibatch issues.
result SGLD has an invariant distribution and geometric ergodicity in W1 distance. Develops geometric integration for rough differential forms.
problem Integrating rough differential forms with low regularity.
method Uses rough path theory to construct geometric integration.
result Constructs geometric integration for rough differential forms.
In this paper, we prove a functorial aspect of the formal geometric quantization procedure of non-compact spin-c manifolds.
We study the geometry of complete generic Ricci solitons with the aid of some geometric-analytical tools extending techniques of the usual Riemannian setting.
The study extends Dehn filling to Lie groups, ensuring geometric properties.
problem Generalizing Dehn filling to semisimple Lie groups.
method Analyzing deformations of subgroups and their geometric properties.
result Extended geometrically finite subgroups can be deformed while maintaining properties.
First constructed genus 2 Cantor set in 3D space.
problem Constructing a geometrically self-similar Cantor set of genus 2.
method Geometrically self-similar construction in R3. result First uniformly quasiregular mapping with a genus 2 Cantor set Julia set.
We work in both the complex and in the para-complex categories and examine (para)-Kähler Weyl structures in both the geometric and in the algebraic settings. The higher dimensional setting is quite restrictive. We show that any (para)-Kaehler Weyl algebraic curvature tensor is in fact Riemannian in dimension at least 6…
Given a compact n-dimensional immersed Riemannian manifold Mn in some Euclidean space we prove that if the Hausdorff dimension of the singular set of the Gauss map is small, then Mn is homeomorphic to the sphere Sn. Also, we define a concept of finite geometrical type and prove that finite geometrical type h…
New findings on infinite subgroups in negatively curved spaces.
problem Characterizing infinite discrete isometry subgroups in negatively pinched Hadamard manifolds.
method Generalization of Bonahon's characterization to negatively pinched Hadamard manifolds.
result Every geometrically infinite isometry subgroup has a continuum of nonconical limit points.
The paper geometrically characterizes graded manifolds and proves the Frobenius theorem.
problem Understanding and characterizing graded manifolds.
method Geometric characterization and Frobenius theorem proof.
result Frobenius theorem proven for graded distributions.
Study geometric properties of surfaces with specific formulae.
problem Calculate curvature of surfaces with singular points.
method Investigate singular curvature and limiting normal curvature of surfaces.
result Determine geometric properties of surfaces with singular points.
Transforms hyperbolic to flat data, deriving geometric inequalities.
problem Deriving geometric inequalities in asymptotically AdS hyperbolic spacetimes.
method Constructs transformations preserving physical quantities to relate hyperbolic to flat spacetimes.
result Derives geometric inequalities from flat counterparts.
The paper proves a geometric capacitary inequality for sub-static manifolds with harmonic potentials.
problem Proving a geometric capacitary inequality for sub-static manifolds with harmonic potentials.
method Introducing a one-parameter family of functions that are monotone along the level-set flow of the potential, up to the optimal threshold.
result Proves a geometric capacitary inequality where the capacity of the horizon plays the same role as the ADM mass in the celebrated Riemannian Penrose Inequality.
The volume of a credal set correlates with epistemic uncertainty in binary classification but not in multi-class.
problem Representing and quantifying epistemic uncertainty in machine learning.
method Examined the geometric representation of credal sets as d-dimensional polytopes and their volume as a measure of uncertainty. result The volume of a credal set is a meaningful measure of epistemic uncertainty in binary classification but not in multi-class.
The paper constructs a generalized metrical multi-time Lagrange space, which allows a natural development of relativistic geometrical optics theories, in a general setting.
Characterizes level-set families of harmonic functions without critical points.
problem Understanding level-set families of harmonic functions without critical points.
method Characterization via local differential-geometric condition and construction from geometric data.
result Evolution of gradient of harmonic functions determined by mean curvature of level sets.
Introduces three types of partial bihamiltonian structures.
problem None explicitly stated; focuses on definitions and geometrical objects.
method Definition and study of geometrical objects linked with partial bihamiltonian structures.
result Examples of partial bihamiltonian structures in finite and infinite dimensions.
Develops theory of relatively geometric actions on CAT(0) cube complexes.
problem Understand actions of relatively hyperbolic groups on CAT(0) cube complexes.
method Introduces and studies relatively geometric actions, proving key results.
result Proves full relatively quasi-convex subgroups are convex compact.
New math shows many 3D hyperbolic shapes can fit together.
problem Counting specific 3D shapes that fit together.
method Examined both arithmetic and non-arithmetic shapes, focusing on their volume.
result The number of such shapes grows super-exponentially with volume.