Paper derives new geometric invariants from affine connections.
problem Finding new invariants for geometric mappings.
method Generalized previous invariants of symmetric affine connection space.
result Invariants related to Thomas and Weyl projective parameters.
Geometric symbols help compute heat invariants.
problem Computing heat invariants efficiently.
method Geometric symbol calculus of pseudodifferential operators.
result Efficient computation of heat invariants.
Study geometric properties of cuspidal edges with boundary.
problem Differential geometric properties of cuspidal edges with boundary.
method Analysis of differential geometric invariants and their relations.
result Relation between boundary behavior and other invariants.
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.
Study dualities of geometric invariants on cuspidal edges in hyperbolic and de Sitter spaces.
problem Computing and understanding dualities of geometric invariants on cuspidal edges.
method Analyzing differential geometric invariants of cuspidal edges in hyperbolic and de Sitter spaces.
result Identified dualities of invariants on cuspidal edges.
Generators for invariants of geometric mappings are derived.
problem Finding invariants of geometric mappings.
method Generalizing Thomas projective parameter and Weyl projective tensor, obtaining generators for vector spaces of invariants.
result Generators for vector spaces of invariants of geometric mappings are obtained.
Introduces geometric quantization and Witten's quantum invariants.
problem None explicitly stated; focuses on introduction.
method Expository introduction to geometric quantization and Witten's quantum invariants.
result Introduction to geometric quantization and Witten's quantum invariants.
New geometric approach to slow invariant manifolds in dynamical systems.
problem Characterizing slow invariant manifolds in a coordinate-independent manner.
method Exploiting curvature concepts and variational approach in Hamiltonian mechanics.
result Differential geometric definition of slow invariant manifolds proposed.
New geometric system from Hessian operators offers solutions to geometric problems.
problem Solving geometric problems using Hessian operators.
method Introducing a new differential-geometric system based on m-Hessian operators. result Deduced an a priori C1-estimate for solutions to the Dirichlet problem for m-Hessian equations. This work proposes a geometric approach to identify slow invariant manifolds in dynamical systems.
problem Identifying slow invariant manifolds in multiple time-scale dynamical systems.
method Differential geometric concepts for submanifolds, sectional curvature, flow invariance.
result Necessary condition for slow invariant manifold invariance stated in terms of differential geometry.
The paper examines geometric invariants near a specific type of singular point.
problem The behavior of geometric invariants near a singular point of a surface or curve.
method Analysis of geometric invariants for surfaces and curves that are suspensions of singular curves.
result Evaluation of the orders of Gaussian and mean curvatures for the studied surfaces and curves.
Study relates Gaussian curvature signs to cuspidal edge types and geometric invariants.
problem Understanding the relationship between Gaussian curvature and singularities of Gauss maps of cuspidal edges.
method Analyzes geometric invariants and types of singularities of Gauss maps to define and characterize positivity/negativity of cusps.
result Defines and characterizes positivity/negativity of cusps of Gauss maps by geometric invariants of cuspidal edges, and shows relation between sign of cusps and Gaussian curvature.
New invariant for hyperbolic surfaces, geometric criterion for domains.
problem Geometric criterion for bounded domains in complex plane.
method Renormalized volume type invariant on hyperbolic surfaces.
result New geometric criterion for bounded domains in complex plane.
New invariants found for mappings between non-symmetric affine spaces.
problem Finding new invariants for mappings between non-symmetric affine spaces.
method Obtained invariants using factored deformation tensor and novel Weyl type invariants.
result Novel Weyl type invariants for mappings between non-symmetric affine spaces.
In this paper we construct the jet geometrical extensions of the KCC-invariants, which characterize a given second-order system of differential equations on the 1-jet space J1(R,M). A generalized theorem of characterization of our jet geometrical KCC-invariants is also presented.
Geometric deep learning predicts knot invariants.
problem Predicting knot invariants from knot data.
method Constructing a functor from knots to graphs and using graph neural networks.
result High generalization capabilities demonstrated.
Study curves in 3-sphere using invariant geometric flows.
problem Understand geometric invariants of curves in S3. method Investigate invariant evolutions of Legendrian and transverse curves.
result Induce well-known integrable systems and hierarchies.
Study geometric inequalities for CR-submanifolds using curvature invariants.
problem Geometric inequalities for CR-submanifolds in almost Hermitian spaces.
method Comparing mutual curvature invariants with Chen-type invariants and proving geometric inequalities.
result Proved geometric inequalities with intermediate mean curvature squared for CR-submanifolds.
Defines geometric invariant and index for projective umbilics.
problem Characterizing projective umbilics on smooth surfaces.
method Introduces a geometric invariant and index, proving their constancy in families of surfaces.
result The sum of indices remains constant in 1-parameter families of surfaces.
Study wave front singularities and their geometric properties.
problem Characterize singularities of focal surfaces of wave fronts.
method Characterization through differential geometric properties.
result Relationships between focal surfaces and initial wave fronts' geometric invariants.
Geometric proof shows topological invariance of handle homology.
problem Topological invariance of handle homology in manifolds.
method Entirely geometric proof using Cerf theory.
result Proof of ∂2=0 in chain complex defined by handle decomposition. Geometric invariant theory for real Lie groups proved.
problem Closed orbits and null cone stratification in real reductive Lie groups.
method Completely self-contained proof focusing on geometric and analytic methods.
result Applies to non-rational linear actions.
The paper addresses geometric structure existence and invariants in Lie groups.
problem Existence and invariants of geometric structures in Lie groups.
method Analyzes EX(S>M), EXF(S>M), and DL(S>M) for various geometric structures in Lie groups.
result Addresses major geometric structures in Lie groups, providing insights into their existence and invariants.
Positive braids linked to knot invariants and geometric monodromy groups.
problem Understanding knot invariants and geometric monodromy groups for positive braids.
method Associate braid monodromy groups to positive braids, identify these groups with framed mapping class groups for knots, and use these to determine knot invariants.
result Geometric monodromy groups of irreducible singularities are determined by genus and Arf invariant of associated knots.
Researchers address the generation of differential invariants for geometric structures.
problem Finite generation of differential algebra of relative differential invariants.
method Investigation of algebraic and differential properties, localization, weight analysis.
result Localization on a finite set of relative invariants makes the differential algebra finitely generated.
Geometric interpretations and localisation theory for Kane-Mele invariant.
problem Understanding the Kane-Mele invariant in three-dimensional fermionic systems.
method Homotopy theory, geometric interpretations, Mayer-Vietoris Theorem, bundle gerbes.
result Unified cohomological explanation for equivalence between discrete Pfaffian and local geometric computations.
Surveying tools for estimating geometric properties of hyperbolic knots.
problem Determining geometric properties of hyperbolic knots.
method Analyzing link diagrams to estimate geometric invariants.
result Estimating volume, cusp shape, and cusp area of hyperbolic knots.
The geometric Hopf invariant of a stable map F is a stable Z_2-equivariant map h(F) such that the stable Z_2-equivariant homotopy class of h(F) is the primary obstruction to F being homotopic to an unstable map. In this paper we express the geometric Hopf invariant of the Umkehr map F of an immersion f:M^m \to N^n in t…
New method extends invariant reduction to rescaled geometric structures.
problem Computing invariant geometric structures under symmetries.
method Extends invariant reduction to rescaled structures using shift rule.
result Emergence and loss of invariance in reductions.
In this paper we construct some multi-time geometrical extensions of the KCC-invariants, which characterize a given second-order system of PDEs on the 1-jet space J1(T,M). A theorem of characterization of these multi-time geometrical KCC-invariants is given.
This research studies affine invariance in continuous-domain convolutional neural networks.
problem Recognizing patterns and features under affine transformations in continuous domains.
method Introduces a new criterion for assessing affine invariance, embeds images into the affine Lie group, and analyzes convolution over this group.
result Extends the scope of geometrical transformations that deep-learning pipelines can handle.
Study shows how to reduce data needed for learning under geometric constraints.
problem Learning high-dimensional data with geometric priors.
method Spherical harmonic decompositions and kernel methods for invariance and geometric stability.
result Improvements in sample complexity by leveraging group invariance, with asymptotic behavior depending on spectral properties.
Study on wave fronts' singularities and parallel surfaces.
problem Understanding singularities of wave fronts and their parallel surfaces.
method Using geometric invariants to analyze principal curvatures and singular points.
result Criteria for bounded principal curvatures at non-degenerate singular points.
We give a normal form of the cuspidal edge which uses only diffeomorphisms on the source and isometries on the target. Using this normal form, we study differential geometric invariants of cuspidal edges which determine them up to order three. We also clarify relations between these invariants.
Geometric invariant theory introduces stability conditions mirroring abelian category theory.
problem Stability conditions in geometric invariant theory.
method Axiomatic notion of central charge and stability condition on schemes and stacks.
result Introduction of stability conditions for polarized schemes and smooth projective varieties.
Geometric compactification for complex structures on Lie groups.
problem Compactifying moduli stack of complex structures on Lie groups.
method Describes a geometric compactification using CR structures transverse to a real foliation.
result Extra points represent CR structures transverse to a real foliation.
Let N be a nilpotent Lie group and let S be an invariant geometric structure on N (cf. symplectic, complex or hypercomplex). We define a left invariant Riemannian metric on N compatible with S to be "minimal", if it minimizes the norm of the invariant part of the Ricci tensor among all compatible metrics with the same …
We present an invariant of a three-dimensional manifold with a framed knot in it based on the Reidemeister torsion of an acyclic complex of Euclidean geometric origin. To show its nontriviality, we calculate the invariant for some framed (un)knots in lens spaces. Our invariant is related to a finite-dimensional fermion…
Geometric wavelet scattering on manifolds improves neural network understanding.
problem Improving neural network understanding on manifold and graph domains.
method Defining a geometric scattering transform based on wavelet filters and nonlinearities.
result Generalizes deformation stability and local translation invariance to manifolds.
Geometric torsions are torsions of acyclic complexes of vector spaces which consist of differentials of geometric quantities assigned to the elements of a manifold triangulation. We use geometric torsions to construct invariants for a manifold with a triangulated boundary. These invariants can be naturally united in a …
Study on geometric properties of h-conformal semi-invariant submersions.
problem Exploring geometric characteristics of h-conformal semi-invariant submersions.
method Investigation of quaternionic Kähler manifolds and Riemannian manifolds, focusing on integrability of distributions and foliations.
result Established necessary and sufficient conditions for total geodesic submersions and twisted product manifolds.
We study the relationship between Bar-Natan's perturbation in Khovanov homology and Szabo's geometric spectral sequence, and construct a link invariant that generalizes both into a common theory. We study a few properties of the new invariant, and introduce a family of s-invariants from the new theory in the same spiri…
Proves geometric invariance of signature and cohomology for Riemannian foliations.
problem Defining and proving invariance of geometric invariants for Riemannian foliations.
method Analyzes basic signature and Lichnerowicz cohomology under homotopy equivalence.
result Foliated homotopy invariance of basic signature and cohomology.
Investigates geometric mean reversion process using Lie symmetry method.
problem Describes dynamics of short-term interest rates.
method Lie symmetry method and optimal system of invariant solutions.
result Constructs an optimal system of invariant solutions.
The paper is devoted to differential geometric invariants determining a Frenet curve in up to a direct similarity These invariants can be presented by the Euclidean curvatures in terms of an arc lengths of the spherical indicatrices. Then, these invariants expressed by focal curvatures of the curve. And then, we give t…
Geometric Invariant Theory applied to Kähler manifolds yields analytic models for vector bundles.
problem Constructing local models for vector bundles on Kähler manifolds.
method Applying Geometric Invariant Theory to Kähler manifolds to construct analytic GIT-quotients.
result Existence of Weil-Petersson forms on parameter spaces for stable vector bundles.
A new method for group invariant machine learning using geometric projections.
problem Supervised group invariant and equivariant machine learning.
method Geometric topology approach involving projection of input data into a geometric space parametrizing symmetry group orbits.
result Improvement in accuracy compared to existing methods.
New rack invariants detect geometric properties of Legendrian knots.
problem Detecting geometric properties of Legendrian knots.
method Introducing Legendrian racks, a generalization of quandle invariants.
result These invariants form a natural generalization of quandle invariants.