We introduce \textcolor{red}{general} new techniques for computing the geometric index of a link in the interior of a solid torus . These techniques simplify and unify previous ad hoc methods used to compute the geometric index in specific examples \textcolor{red}{ and allow the simple computation of geometric i…
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48 results for “geometric computation”
Manifold Markov chain Monte Carlo algorithms have been introduced to sample more effectively from challenging target densities exhibiting multiple modes or strong correlations. Such algorithms exploit the local geometry of the parameter space, thus enabling chains to achieve a faster convergence rate when measured in n…
Unified framework for geometric computation of minimum-area homotopy.
problem Computing the minimum homotopy area of a closed curve.
method Unified combinatorial word approach combining geometric and algebraic methods.
result Unified geometric proof and constructive algorithm for minimum area homotopy.
Efficient Decomposition of Forman-Ricci Curvature on Vietoris-Rips Complexes and Data Applicationsmath.GT
Researchers decompose Forman-Ricci curvature for efficient computation in VR complexes.
problem Efficiently computing Forman-Ricci curvature in higher-dimensional data.
method Decomposition and set-theoretical proof for local computation of FRC in VR complexes.
result Reveals critical geometric insights overlooked by conventional techniques.
Braids can be represented geometrically as curve diagrams. The geometric complexity of a braid is the minimal complexity of a curve diagram representing it. We introduce and study the corresponding notion of geometric generating function. We compute explicitly the geometric generating function for the group of braids o…
Geometric AD framework simplifies derivative computation in JAX.
problem Efficient and accurate automatic differentiation.
method Jet functors and Weil algebras for geometric analysis.
result Unified view of derivative propagation with algebraic exactness.
A new geometric method for clustering SPD data improves upon Euclidean and Riemannian approaches.
problem Skewed interpretations of SPD data in Euclidean analysis and computational inefficiency of Riemannian methods.
method Proposes a geometric method based on the Thompson metric for unsupervised clustering of SPD data.
result Demonstrates improved clustering results using inductive midrange centroid computation.
tf_geometric simplifies graph deep learning in TensorFlow.
problem Efficient graph deep learning in TensorFlow.
method Kernel libraries and infrastructures for GNNs.
result tf_geometric supports various graph tasks and provides efficient GNN models.
Based on Nielsen fixed point theory and Gröbner-Shirshov basis, we obtain a simple method to compute geometric intersection numbers and self-intersection geometric numbers of loops on surfaces.
In this article we develop tools to compute the Geometric Quantization of a symplectic manifold with respect to a regular Lagrangian foliation via sheaf cohomology and obtain important new applications in the case of real polarizations. The starting point is the definition of representation spaces due to Kostant. Besid…
In this paper we prove two results, one semi-historical and the other new. The semi-historical result, which goes back to Thurston and Riley, is that the geometrization theorem implies that there is an algorithm for the homeomorphism problem for closed, oriented, triangulated 3-manifolds. We give a self-contained proof…
New scalable geometric framework for SPD matrices.
problem Costly spectral computations in SPD matrix analysis.
method Efficient computation of extreme generalized eigenvalues through Hilbert and Thompson geometries of the semidefinite cone.
result Existence and uniqueness of a novel iterative mean of SPD matrices.
We show that the computation of the Fredholm index of a fully elliptic pseudodifferential operator on an integrated Lie manifold can be reduced to the computation of the index of a Dirac operator, perturbed by a smoothing operator, canonically associated, via the so-called clutching map. To this end we adapt to our fra…
We use the methods of geometric control theory to study extremal trajectories of vertical rolling disk. We focus on the role of symmetries of the underlying geometric structures. We demonstrate the computations in the CAS Maple package DifferentialGeometry.
Researchers solved the multiple fibration problem for Seifert 3-orbifolds.
problem Determining all inequivalent fibrations of closed orientable Seifert three-orbifolds.
method Geometric and direct arguments for and geometries; computer-assisted for .
result Complete solution for and geometries, recovering previous results.
The problem of evaluating heat invariants can be computerized. Geometric symbol calculus of pseudodifferential operators is the main tool of such computerization.
ICLR 2021 challenge in computational geometry and topology attracted 16 teams.
problem Designing and evaluating computational methods in differential geometry and topology.
method Designing and hosting an open-source competition with repositories Geomstats and Giotto-TDA.
result 16 teams participated in the challenge, showcasing innovative contributions to computational geometry and topology.
Geometric vector perceptrons improve protein structure learning.
problem Learning from protein structure with efficient and natural representations.
method Introducing geometric vector perceptrons to extend dense layers for Euclidean vectors, integrating geometric and relational reasoning.
result Improves model quality assessment and computational protein design over existing methods.
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
GDB bridges geometric states with improved accuracy and generality.
problem Challenges in predicting geometric state evolution in complex systems.
method Geometric Diffusion Bridge (GDB) framework using equivariant diffusion bridges.
result GDB surpasses existing methods in accurately bridging geometric states.
We give a recipe to compute the geometric intersection number of an integral lamination with a particular type of integral lamination on an n-times punctured disk. This provides a way to find the geometric intersection number of two arbitrary integral laminations when combined with an algorithm of Dynnikov and Wiest.
Survey of de Casteljau's algorithm's applications in geometric data analysis.
problem No specific problem stated; focuses on algorithm applications.
method Constructive approach to generalize parametric smooth curves to manifolds.
result Algorithm provides principled way to analyze geometric data.
We describe the cohomology ring of the moduli space of a flexible polygon in geometrically meaningful terms. We propose two presentations, both are computation friendly: there are simple rules for cup product.
Virtual Geometricity is Raremath.GR
We present the results of computer experiments suggesting that the probability that a random multiword in a free group is virtually geometric decays to zero exponentially quickly in the length of the multiword. We then prove this fact.
Classifying spaces for the family of virtually abelian subgroups of orientable -manifold groupsmath.GT
This paper calculates the geometric dimension for 3-manifold groups up to n=2.
problem Computing the geometric dimension for 3-manifold groups.
method Analyzing families of subgroups containing virtually abelian subgroups.
result Computed the geometric dimension for 3-manifold groups for all n ≥ 2.
New invariant simplifies computing geometric invariants of recursive group orbits.
problem Computing geometric invariants of recursive group orbits is hard.
method Introduced new invariant and used it to compute invariants explicitly.
result Explicit formulas for local Euler obstructions and sectional Euler characteristics.
Geometrically computes cohomotopy groups for higher-dimensional manifolds.
problem Computing cohomotopy groups for higher-dimensional manifolds.
method Geometric arguments and introduction of new manifold types.
result Refined Euler class and obstruction to non-vanishing sections.
Generalizes Sobolev IPM for graph-based measures using Orlicz geometric structure.
problem Limitation of Le et al. (2025) framework to geometry.
method Generalizes Sobolev IPM through Orlicz geometric structure, employing convex functions to capture nuanced geometric relationships.
result GSI-M reduces to a simple univariate optimization problem, achieving remarkable computational efficiency.
We compute the differential geometric invariants of cuspidal edges on flat surfaces in hyperbolic -space and in de Sitter space. Several dualities of invariants are pointed out.
We compute the asymptotic expansion of the volume of small sub-Riemannian balls in a contact 3-dimensional manifold, and we express the first meaningful geometric coefficients in terms of geometric invariants of the sub-Riemannian structure
We expose (without proofs) a unified computational approach to integrable structures (including recursion, Hamiltonian, and symplectic operators) based on geometrical theory of partial differential equations. We adopt a coordinate based approach and aim to provide a tutorial to the computations.
We compute the Cheeger constants of a collection of hyperbolic surfaces corresponding to maximal non-compact arithmetic Fuchsian groups, and to subgroups which are the rotation subgroup of maximal reflection groups. The Cheeger constants are geometric quantities, but relate to the smallest eigenvalues of Maass cusp for…
The study examines Eschenburg orbifolds with positive sectional curvature and their geometric/topological properties.
problem Understanding the geometric and topological constraints of positively curved Eschenburg orbifolds.
method Proved restrictions on singular sets and computed orbifold cohomology rings.
result Distinctive behavior in cohomology groups of positively curved Eschenburg orbifolds.
This paper reviews discrete curvature models for geometric data analysis.
problem Capturing intrinsic geometric structure in diverse data representations.
method Comprehensive review of discrete curvature models from Riemannian and metric geometry perspectives.
result Systematic pipeline for curvature-driven data analysis and learning.
The geometric intersection number of a curve on a surface is the minimal number of self-intersections of any homotopic curve, i.e. of any curve obtained by continuous deformation. Given a curve represented by a closed walk of length at most on a combinatorial surface of complexity we describe simple algo…
We develop a geometric version of the inverse problem of the calculus of variations for discrete mechanics and constrained discrete mechanics. The geometric approach consists of using suitable Lagrangian and isotropic submanifolds. We also provide a transition between the discrete and the continuous problems and propos…
Computing PL geometric category in 2D is NP-hard.
problem Determining the PL geometric category of 2D polyhedra.
method Reduction from shellability of 2-complexes, which is known to be NP-hard.
result It is NP-hard to decide whether the PL geometric category of a 2D polyhedron is at most 2.
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.
Study hypoelliptic operators on Carnot manifolds, extending index theory results.
problem Index theory of hypoelliptic operators on Carnot manifolds.
method Operator K-theory and geometric K-homology.
result Compute Fredholm index of hypoelliptic operators on Carnot manifolds.
The discrete sum of geometric Brownian motions plays an important role in modeling stochastic annuities in insurance. It also plays a pivotal role in the pricing of Asian options in mathematical finance. In this paper, we study the probability distributions of the infinite sum of geometric Brownian motions, the sum of …
Quantum algorithms for financial derivatives and credit risk.
problem Estimating credit risk and option pricing in realistic financial models.
method Developed a regime switching volatility model for financial markets, using a Markov chain to determine volatility parameters.
result Quantum algorithms can be applied to realistic financial models, bringing quantum computing closer to practical applications.
Geometric tools study two- and threepeakons in Camassa-Holm equation.
problem Dynamics of peakon solutions in the Camassa-Holm equation.
method Geometric tools applied to prove asymptotic behavior and compute curvature.
result New proofs and curvature computations for two- and threepeakons.
Geometric Algebra Transformer (GATr) handles various geometric data types efficiently.
problem Lack of a single architecture for diverse geometric data types.
method GATr uses projective geometric algebra, equivariant to E(3), and is a Transformer architecture.
result GATr outperforms non-geometric and equivariant baselines in various geometric tasks.
Geometric framework for SPD matrices preserving subspace structures.
problem Processing SPD-valued data with preserved subspace structures.
method Thompson geometry of the semidefinite cone, extreme generalized eigenvalues, geodesic space structure.
result Novel inductive mean of SPD matrices based on Thompson geometry.
Geometrically computes superpotentials for certain 4D N=2 theories.
problem Computing effective twisted superpotentials for 4D N=2 theories.
method Spectral networks and abelianization to compute generating functions of brane opers.
result Geometric recipe for computing effective twisted superpotentials.
Simulating fluid flow in geological formations requires mesh generation, lithology mapping to the cells, and computing geometric properties such as normal vectors and volume of cells. The purpose of this research work is to compute and process the geometrical information required for performing numerical simulations in…
Szabó recently introduced a combinatorially-defined spectral sequence in Khovanov homology. After reviewing its construction and explaining our methodology for computing it, we present results of computations of the spectral sequence. Based on these computations, we make a number of conjectures concerning the structure…
Defines contact structures on Heisenberg groups for geometric interpretation.
problem Finding a geometric interpretation for the Yamabe equation on Heisenberg groups.
method Defines contact structures of Heisenberg type, introduces a natural connection, and computes conformal scalar curvature.
result Establishes equivalence between contact Riemannian manifolds and contact structures of Heisenberg type.