The study examines Lipschitz normally embedded Hölder triangles in 4D space.
problem Comparing ambient and outer Lipschitz geometry of Hölder triangles.
method Analyzes Lipschitz normally embedded Hölder triangles in \(\mathbb{R}^4\).
result Infinitely many equivalence classes of microknots.
The thesis defines and proves invariants for manifolds of bounded geometry.
problem Lipschitz-homotopy invariants for manifolds of bounded geometry.
method Definition of a controvariant functor and invariance of the Roe index and ρ-class.
result Lipschitz-homotopy invariants are defined and proven for manifolds of bounded geometry.
We discuss the bi-Lipschitz geometry of an isolated singular point of a complex surface which particular emphasis on when it is metrically conical.
The study connects polyhedral manifolds to Riemannian ones with geometric bounds.
problem Connecting polyhedral manifolds to Riemannian manifolds with geometric constraints.
method Using a theorem by C. Lange and B. Bowditch, the study bounds the curvature and injectivity radius of Riemannian manifolds.
result Polyhedral manifolds with bounded geometry are bi-Lipschitz homeomorphic to Riemannian manifolds with controlled curvature and injectivity radius.
New Lipschitz de Rham theorem for Lp-cohomology.
problem Developing a new de Rham theorem for Lp-cohomology. method Regularization procedure in Lipschitz de Rham calculus applied to metric simplicial complexes.
result Established Lipschitz de Rham theorem for Lp-cohomology. We prove that the Hilbert Geometry of a convex set is bi-lipschitz equivalent to a normed vector space if and only if the convex is a polytope.
Study links between surface germs and knot theory in 4D.
problem Understanding the relationship between surface germs and knot theory in R4. method Constructing surface germs XK linked to knots K in S3 and studying their Lipschitz geometry. result Ambient bi-Lipschitz equivalence of surface germs is related to isotopy of knots, and Jones polynomial can recognize non-equivalent germs.
Complex analytic sets' Lipschitz geometry at infinity characterized.
problem Characterize entire complex analytic sets based on their Lipschitz geometry at infinity.
method Proved a complex non-parametric version of Moser's Bernstein Theorem and characterized algebraicity.
result Entire complex analytic sets at infinity are affine linear subspaces if and only if they are bi-Lipschitz homeomorphic to algebraic sets.
Paper studies geometric and combinatorial properties of circular snakes.
problem Exploring geometric and combinatorial properties of circular snakes.
method Definition and investigation of outer Lipschitz geometry, decomposition of Valette link, construction of combinatorial objects, weakly outer Lipschitz classification.
result Existence of canonical decomposition and necessary/sufficient criteria for removing segments or Hölder triangles.
Study conic singular manifolds, proving Lipschitz normal embedding.
problem Understanding metric properties of conic singular manifolds.
method Analyzing interplay between conic and asymptotically conic behavior.
result Proves Lipschitz normal embedding for conic singular sub-manifolds.
We prove a Lipschitz-Volume rigidity theorem for the non-collapsed Gromov-Hausdorff limits of manifolds with Ricci curvature bounded from below. This is a counterpart of the Lipschitz-Volume rigidity in Alexandrov geometry.
Study contact 3-manifolds using sub-Riemannian geometry, proving new properties of their Lipschitz homotopy groups.
problem Characterize the Lipschitz homotopy groups of contact 3-manifolds.
method Sub-Riemannian geometry, geometric measure theory, biLipschitz equivalence, purely unrectifiable sets.
result Contact 3-manifolds are K(π,1) spaces with uncountably generated first homotopy groups. Conic singular sub-manifolds are Lipschitz Normally Embedded in compact non-Euclidean manifolds.
problem Understanding the Lipschitz geometry of conic singular sub-manifolds.
method Analyzing the metric properties of conic singular sub-manifolds in compact non-Euclidean manifolds.
result Connected conic singular sub-manifolds are Lipschitz Normally Embedded.
We show that a weighted homogeneous complex surface singularity is metrically conical (i.e., bi-Lipschitz equivalent to a metric cone) only if its two lowest weights are equal. We also give an example of a pair of weighted homogeneous complex surface singularities that are topologically equivalent but not bi-Lipschitz …
We prove a Lipschitz-Volume rigidity theorem in Alexandrov geometry, that is, if a 1-Lipschitz map f:X=⨿Xℓ→Y between Alexandrov spaces preserves volume, then it is a path isometry and an isometry when restricted to the interior of X. We furthermore characterize the metric structure on Y with re…
Bi-Lipschitz Autoencoder ensures robust manifold preservation.
problem Non-injective autoencoders lead to poor convergence and distorted latent representations.
method Injective regularization and bi-Lipschitz relaxation.
result BLAE consistently outperforms existing methods in manifold preservation.
MLDL preserves manifold geometry in vector transformations.
problem Geometric deterioration in neural network transformations.
method Locally isometric smoothness (LIS) and Markov random field (MRF) encoding.
result Enhanced vector transformations into well-behaved metric homeomorphisms.
Curvature measures uniquely determined by invariance under embeddings.
problem Characterizing curvature measures uniquely.
method Applied Weyl principle and Künneth-type formula.
result Curvature measures uniquely characterized by invariance under isometric embeddings.
Hyperbolic space outperforms Euclidean in learning hierarchical data.
problem Learning hierarchical data in Euclidean space requires exponentially many samples.
method Established geometric obstruction in Euclidean space and showed hyperbolic space's advantage.
result Hyperbolic space enables learning with O(mRlogm) samples, matching information-theoretic optimum. Lipschitz equivalence of self-similar sets is an important area in the study of fractal geometry. It is known that two dust-like self-similar sets with the same contraction ratios are always Lipschitz equivalent. However, when self-similar sets have touching structures the problem of Lipschitz equivalence becomes much …
The paper proves Lipschitz continuity of cut times in spacetimes.
problem Lipschitz continuity of cut times in globally hyperbolic spacetimes.
method Adapted Itoh-Tanaka method to Lorentzian setting.
result Lipschitz continuity of cut times with quantitative estimates.
The paper establishes a uniform Lipschitz bound on the square root of the systole function in Teichmüller space.
problem Uniform Lipschitz bounds on geometric functions in Teichmüller space.
method Injectivity radius analysis and Lipschitz bounds on systole function.
result Uniform Lipschitz constant for the square root of the systole function on Teichmüller space.
The paper proves properties of Lipschitz spacetimes with bounded Ricci curvature.
problem Properties of Lipschitz spacetimes with bounded Ricci curvature.
method Globally hyperbolic spacetimes with locally Lipschitz metrics and timelike Ricci curvature.
result New comparison theorems for Lipschitz spacetimes.
We prove that the Hilbert geometry of a product of convex sets is bi-lipschitz equivalent the direct product of their respective Hilbert geometries. We also prove that the volume entropy is additive with respect to product and that amenability of a product is equivalent to the amenability of each terms.
Local Lipschitz continuity of sub-elliptic harmonic maps into CAT(0) spaces proved.
problem Proving Lipschitz continuity of sub-elliptic harmonic maps between singular spaces.
method Analyzing sub-elliptic harmonic maps from the Heisenberg group into CAT(0) spaces.
result Local Lipschitz continuity established for sub-elliptic harmonic maps.
Recover simple irreversible Finsler geometry from travel time data
problem Stable recovery of a simple irreversible Finsler geometry
method Use a Gromov-Hausdorff distance adapted to irreversible metric spaces
result Unique and Lipschitz-stable recovery
Extends spinor-horosphere correspondence to higher dimensions and new spinor types.
problem Constructing isomorphisms between spinors and horospheres in hyperbolic spaces.
method Generalizes Mathews' results to Lipschitz spinors and null multiflags in higher-dimensional hyperbolic spaces.
result Equivariant correspondence between two-component Lipschitz spinors and higher-dimensional horospheres.
Abstract cone operators prove scalar curvature comparisons on singular manifolds.
problem Proving scalar curvature inequalities on manifolds with cone singularities.
method Using index theory for twisted Dirac operators on Lipschitz bundles.
result Lipschitz rigidity for scalar curvature on odd-dimensional manifolds.
Study bi-Lipschitz equivalence of mixed polynomials under specific conditions.
problem Classify bi-Lipschitz equivalence of mixed polynomials with inner non-degeneracy.
method Defined metric links and introduced new data to determine bi-Lipschitz equivalence.
result Neither Newton boundary nor C-face diagram is an invariant for bi-Lipschitz equivalence.
In this paper we provide an up-to-date survey on the study of Lipschitz equivalence of self-similar sets. Lipschitz equivalence is an important property in fractal geometry because it preserves many key properties of fractal sets. A fundamental result by Falconer and Marsh [On the Lipschitz equivalence of Cantor sets, …
Upper bounds on map degrees for various manifold types.
problem Understanding the maximum degree of maps between different types of manifolds.
method Analyzing Lipschitz maps and dividing manifolds into topological types.
result New upper bounds on map degrees for different manifold types.
Unified high-probability regret bounds for online convex optimisation with randomised gradient estimators.
problem Online convex optimisation with randomised gradient estimators for ℓq-Lipschitz losses. method FTRL with randomised two-point finite-difference gradient estimators based on cone-measure sampling from ℓr-spheres. result Unified high-probability regret bounds for all p,q,r∈[1,∞]. Study de Rham homomorphism for Lipschitz cohomologies on metric simplicial complexes.
problem Triviality of de Rham homomorphism kernel and non-increasing monotonicity of parameters.
method Regularization in Lipschitz de Rham calculus on metric simplicial complexes with bounded geometry.
result Explicit specification of non-trivial cohomology classes for a sequence of parameters.
Smooths metrics on manifolds with curvature bounds and injectivity radius constraints.
problem Smooth metrics on manifolds with curvature and injectivity constraints.
method Bi-Lipschitz smoothing with controlled smoothing and volume lower bounds.
result Proves existence of smooth metrics with curvature bounds and injectivity radius constraints.
Develops new strategy for Hessian estimates in Lagrangian mean curvature equation.
problem Interior Hessian estimates for solutions with prescribed Lipschitz phases.
method Allard-type regularity theorem, geometric measure theory, geometry of Lagrangian graphs, De Giorgi-Nash-Moser iteration.
result Sharp interior Hessian estimates for solutions with critical and supercritical phases.
Injectivity of X-ray transform proven for non-smooth metrics.
problem Injectivity of X-ray transform on non-smooth metrics.
method Microlocal analysis of the normal operator, establishing ellipticity and smoothing properties.
result Injectivity of X-ray transform on L2 for metrics with finitely differentiable tensor. We develop a natural and geometric way to realize the hyperbolic plane as the moduli space of marked genus 1 Riemann surfaces. To do so, a metric is defined on the Teichmüller space of the torus, inspired by Thurston's Lipschitz metric for the case of hyperbolic surfaces. Based on extremal Lipschitz maps, the Teichmüll…
This is one of a series of papers examining the interplay between differentiation theory for Lipschitz maps, X-->V, and bi-Lipschitz nonembeddability, where X is a metric measure space and V is a Banach space. Here, we consider the case V=L^1 where differentiability fails. We establish another kind of differentiability…
The paper sharpens a theorem about surfaces with zero Gaussian curvature.
problem Quantifying the isometric property of surfaces with zero Gaussian curvature.
method Asymptotically sharp quantitative version of a classical theorem using isothermal coordinates.
result An isothermal coordinate map from a Riemannian disc to an Euclidean disc is bi-Lipschitz with a constant of exp(4ε).
New results on geometry of area-preserving diffeomorphisms using braids.
problem Large-scale geometry of area-preserving diffeomorphisms on surfaces.
method Application of Schwarz-Milnor lemma to configuration spaces.
result Quasi-isometric embeddings and Lipschitz properties of quasi-morphisms.
We present a new proof of the bi-Lipschitz model theorem, which occupies the main part of the Ending Lamination Conjecture proved by Minsky and Brock-Canary-Minsky. Our proof is done by using techniques of standard hyperbolic geometry as much as possible.
The paper characterizes the geometry and topology of spin random fields.
problem Understanding the expected geometry and topology of spin random fields.
method Investigating the asymptotic behavior of geometric and topological functionals for spin random fields under scaling assumptions.
result Explicit results for monochromatic fields, showing non-universal asymptotic behavior and new generalized models.
The paper studies maps in the Heisenberg group and their images, called Rickman rugs.
problem Understanding maps and their images in the Heisenberg group.
method Analyzes maps f:WoH, where H is the first Heisenberg group and W is a vertical subgroup. result Rickman rugs in the Heisenberg group admit a corona decomposition by intrinsic bilipschitz graphs.
The paper extends localisation technique to multiple constraints in Euclidean spaces.
problem Proving log-concavity of conditional measures in decomposed convex sets.
method Defining partitions of maximal closed convex sets and proving log-concavity of conditional measures.
result Existence of a partition and log-concavity of conditional measures for almost every set of the partition.
Survey on Thurston metric on Teichmüller space, focusing on extremal maps.
problem Constructing extremal Lipschitz maps between hyperbolic surfaces.
method Review of constructions including Thurston's original work.
result Coarse geometry and isometry rigidity of Thurston metric discussed.
Introduces Grassmann Distance Complexity to measure algebraic set nearest point problems.
problem Measuring complexity of finding nearest points in Grassmannian space.
method Uses Lipschitz critical point theory and o-minimal geometry.
result Establishes fundamental properties of GDC, including bounds and finiteness conditions.
Basic aspects of differential geometry can be extended to various non-classical settings: Lipschitz manifolds, rectifiable sets, sub-Riemannian manifolds, Banach manifolds, Weiner space, etc. Although the constructions differ, in each of these cases one can define a module of measurable 1-forms and a first-order exteri…
Proves rigidity of maps between manifolds with scalar curvature constraints.
problem Lipschitz rigidity problem in scalar curvature geometry.
method Harmonic map heat flow coupled with Ricci flow.
result Continuous maps between manifolds with scalar curvature constraints are either isometries or have scalar curvature strictly less than the sphere.