Geometric inequalities for static convex domains in hyperbolic space proved.
problem Proving geometric inequalities for static convex domains in hyperbolic space.
method Using static convexity of flow hypersurfaces, new inequalities are derived.
result New family of geometric inequalities for static convex domains in hyperbolic space.
Study geometric structures on twistor and reflector spaces of paraquaternionic contact manifolds.
problem Geometric structures on paraquaternionic contact manifolds.
method Investigate fiber bundles (twistor and reflector spaces) over paraquaternionic contact manifolds.
result Intrinsic geometric structures on twistor and reflector spaces are always integrable.
Geometrically transforms word embeddings into a common space for better comparison.
problem Comparing embeddings from different sources is challenging.
method Applies orthogonal rotations and Mahalanobis scaling to transform embeddings into a shared latent space.
result The method improves word similarity and analogy tasks.
We provide examples of homogeneous spaces which are neither symmetric spaces nor real cohomology spheres, yet have the property that every invariant metric is geometrically formal. We also extend the known obstructions to geometric formality to some new classes of homogeneous spaces and of biquotients, and to certain s…
We realize Stasheff's multiplihedron geometrically as the moduli space of stable quilted disks. This generalizes the geometric realization of the associahedron as the moduli space of stable disks. We show that this moduli space is the non-negative real part of a complex moduli space of stable scaled marked curves.
The paper proves geometric inequalities for pinched convex hypersurfaces in de Sitter space.
problem Geometric inequalities for convex hypersurfaces in de Sitter space.
method Locally constrained flows with initial compact spacelike hypersurfaces pinched in de Sitter space.
result Established geometric inequalities related to quermassintegrals and weighted curvature integrals.
Tensor approach simplifies Euclidean space descriptions.
problem Simplifying tensor descriptions of Euclidean spaces.
method Emphasizes geometric vectors in tensor description.
result Proved integral identities with vector integrands.
Study geometric properties of generalized vacuum static spaces.
problem Estimating geometric properties of generalized φ-vacuum static spaces. method Proving estimates for φ-scalar curvature and first eigenvalue of the Jacobi operator, and rigidity under various geometric assumptions. result Proved a result related to the Cosmic no-hair conjecture.
The paper examines geometric properties of domains for the p-Laplacian in Euclidean and hyperbolic spaces.
problem Exploring geometric properties of unbounded extremal domains for the p-Laplacian operator.
method Analyzing properties in Euclidean and hyperbolic spaces, proving constraints on domains and their asymptotic boundaries.
result Extremal domains in two dimensions must be balls, and in hyperbolic space, they have specific geometric constraints.
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.
We study integral geometric properties of non-compact harmonic spaces.
Geometric structures over algebras describe geodesics and spaces.
problem Understanding geodesics in hyperbolic and Euclidean spaces over non-standard algebras.
method Study geometric structures from Hermitian forms on real algebras (dual numbers, split-complex, split-quaternions).
result Presented a projective model for the hyperbolic bidisc.
Extends geometric decompositions to arbitrary meshes and forms.
problem Constructing local bases for finite element spaces on arbitrary meshes.
method Generalizes extension operators to arbitrary meshes and forms, showing they yield geometric decompositions.
result Extension operators yield geometric decompositions for arbitrary meshes and forms.
Satake has constructed compactifications of symmetric spaces D=G/K which (under a condition called geometric rationality by Casselman) yield compactifications of the corresponding locally symmetric spaces. The different compactifications depend on the choice of a representation of G. One example is the Baily-Borel-Sata…
Explains mapping properties of elliptic operators in conical spaces.
problem Understanding mapping properties of geometric elliptic operators in conical spaces.
method Develops an approach based on B.-W. Schulze's work.
result Illustrates versatility of results in Geometric Analysis.
We construct a two parameter family of eleven-dimensional indecomposable Cahen-Wallach spaces with irreducible, non-flat, non-restricted geometric supersymmetry of fraction ν=43. Its compactified moduli space can be parametrized by a compact interval with two points corresponding to two non-isometric, decom…
Geometric techniques reveal new insights into Gromov-Witten invariants.
problem Formulating Gromov-Witten invariants for complete intersections in projective space.
method Combining geometric group theory and geometric topology, focusing on geodesic laminations.
result Primitive cohomologies unify mathematical formulations of Gromov-Witten invariants.
The paper studies asymptotic dimensions of manifolds and spaces, proving key results about their geometric decompositions.
problem Understanding the asymptotic dimensions of manifolds and spaces with geometric decompositions.
method Analyzing the fundamental groups and using geometric decompositions to derive asymptotic dimension bounds.
result Asymptotic dimensions of certain manifolds and spaces are bounded and equal to specific values.
Introduces Kundt spaces using geometric language.
problem Clarify properties of Kundt spaces.
method Geometric approach focusing on key concepts.
result Pedagogical introduction to Kundt spaces.
Extends geometric quantization to singular spaces.
problem Handling singular symplectic spaces in geometric quantization.
method Developed stratified pseudobundles to replace auxiliary information.
result Provided results for singular quotients of toric manifolds and cotangent bundles.
Study geometric flows with varying parameters and prove continuous dependence.
problem Continuous dependence of flows on parameters in geometric settings.
method Derived suitable topologies for vector fields and flows, proved new continuous dependence.
result Proved continuous dependence of flows on parameters in a general topological space.
Introduces string structures linking to loop spaces.
problem Understanding geometric string structures.
method Explains connections to loop spaces.
result String structures linked to loop spaces.
Geometric transformations on null curves in AdS induce KdV solutions.
problem Transforming null curves in AdS to solve KdV equations.
method Geometric transformations that induce Bäcklund transformations for KdV.
result Satisfies permutability theorem for null curves with constant bending.
Combines topological and geometric approaches to data analysis.
problem Understanding when and how geometric objects intersect.
method Connects topological and geometric concepts of curvature.
result Reconceptualizes curvature and links it to hyperconvexity.
Geometrically revisits and models homogeneous spaces of compact Lie group G2.
problem Classifying homogeneous reductive spaces of compact Lie group G2. method Geometrical approach to revisit and model the spaces.
result Explicit relations among geometric models of the spaces.
The paper proposes conjectures about moduli space rigidity.
problem Rigidity of moduli spaces in algebraic geometry.
method Group-theoretic, topological, and holomorphic approaches.
result Some conjectures have been proved.
A new comparison theorem for geometric spaces.
problem Geometric space comparison theorems.
method Relative form of Toponogov comparison theorem.
result New geometric space comparison theorem established.
Constructs geometric models for moduli spaces of Higgs bundles over Riemann sphere.
problem Construction of moduli spaces for Higgs bundles with specific properties.
method Elementary geometric and combinatorial techniques, focusing on orbit stability of automorphism groups.
result Explicit geometric models for moduli spaces of parabolic Higgs bundles over Riemann sphere.
High-dimensional ConvNets detect patterns in 32+ dimensions for geometric registration.
problem Detecting geometric patterns in high-dimensional spaces.
method High-dimensional convolutional networks applied to geometric registration problems.
result High-dimensional ConvNets outperform global pooling approaches in 3D registration and image correspondence.
New geometric flow equations describe how space-time dimensions change.
problem Understanding how the number of space-time dimensions affects geometry.
method Developed D-flow equations to model the variation of space-time geometries.
result Solutions for D-flow equations on D-dimensional spheres and Freund-Rubin Compactification.
Geometrically proves majorizing measure theorem on Hadamard manifolds.
problem Volume size relation between random process index space and its convex hull.
method Assumed Hadamard manifold, derived upper bound for volume ratio, applied to prove majorizing measure theorem.
result Upper bound for volume ratio between index space and convex hull.
This article is a survey article on geometric group theory from the point of view of a non-expert who likes geometric group theory and uses it in his own research. The sections are: classical examples, basics about quasiisometry,properties and invariants of groups invariant under quasiisometry, rigidity, hyperbolic spa…
In this paper we construct the differential equations of the stream lines that characterize plasma regarded as a non-isotropic medium geometrized by a jet rheonomic time-invariant Berwald-Moor metric. Section 1 contains historical notes regarding the Plasma Physics and its geometrical description. Section 2 analyzes th…
Machine learning methods struggle with geometric data, but shape space analysis provides a framework for studying and analyzing geometric variability.
problem Machine learning methods struggle with geometric data
method Shape space analysis provides a mathematical and computational framework
result Characterizes shape variability, compares geometric objects, and analyzes structural trajectories
Study wall singularities in spaces with upper curvature bounds.
problem Understanding singularities in spaces with curvature constraints.
method Geometric structure theorem and geometric characterization for codimension one and two.
result Necessary and sufficient conditions for singular sets to be of codimension at least two.
This paper uses sheaf theory to model virtual knots geometrically.
problem Defining and understanding virtual knots in a geometric framework.
method Sheaf theory applied to virtual knot diagrams to model them geometrically.
result A geometric model for virtual knots formalizes the intuitive notion of knots in a variable ambient space.
We compute the differential geometric invariants of cuspidal edges on flat surfaces in hyperbolic 3-space and in de Sitter space. Several dualities of invariants are pointed out.
The space of n-sided polygons embedded in three-space consists of a smooth manifold in which points correspond to piecewise linear or ``geometric'' knots, while paths correspond to isotopies which preserve the geometric structure of these knots. The topology of these spaces for the case n = 6 and n = 7 is described. In…
Study fundamental groups of geometric transformation groups using loop spaces.
problem Understanding fundamental groups of geometric transformation groups.
method Use differential forms on loop spaces to prove infinite fundamental groups.
result Proves infinite fundamental groups for specific geometric transformation groups.
Study geometric singular solutions of generalized Monge-Ampère equations.
problem Solving generalized Monge-Ampère equations on a plane.
method Using exterior differential systems and Cauchy characteristics.
result Criteria for geometric singular solutions to be equivalent to specific types.
In this paper, we use the inverse curvature flow to prove a sharp geometric inequality on star-shaped and two-convex hypersurface in hyperbolic space.
Unified geometric framework for adiabatic quantum mechanics.
problem Understanding geometric phases and exceptional points in quantum mechanics.
method Formal geometric framework for arbitrary non-degenerate Hamiltonians.
result Generalization of geometric phase to non-Hermitian Hamiltonians.
Geometric framework for Milnor classifying spaces in diffeological spaces.
problem Milnor classifying spaces in diffeological spaces.
method Developed spherical and projective models with natural diffeological structures, constructed Riemannian metrics, defined differential forms, and introduced Clifford structures.
result Established a coherent geometric setting combining classifying spaces, diffeology, and higher geometric structures.
Motivated by Felix Klein's notion that geometry is governed by its group of symmetry transformations, Charles Ehresmann initiated the study of geometric structures on topological spaces locally modeled on a homogeneous space of a Lie group. These locally homogeneous spaces later formed the context of Thurston's 3-dimen…
We discuss the question of geometric formality for rationally elliptic manifolds of dimension 6 and 7. We prove that a geometrically formal six-dimensional biquotient with b2=3 has the real cohomology of a symmetric space. We also show that a rationally hyperbolic six-dimensional manifold with b2≤2 and …
Geometric approach to majorizing measures for polyhedra and general compact objects.
problem Understanding the relationship between a space and its convex hull in geometric measure theory.
method Geometric approach using covering number relationships and volume ratios.
result Established a method to evaluate covering number and volume ratios for various spaces.
In this paper, we investigate the geometric structure of activation spaces of fully connected layers in neural networks and then show applications of this study. We propose an efficient approximation algorithm to characterize the convex hull of massive points in high dimensional space. Based on this new algorithm, four…
Confirming a conjecture, new CAT(0) spaces of higher rank are rigid.
problem CAT(0) spaces of higher rank with geometric group actions.
method Proving rigidity for spaces containing periodic flats and geodesics in flats.
result CAT(0) spaces of higher rank n≥2 are rigid if they contain a periodic n-flat.