Two open problems in Sasaki geometry are discussed.
problem Open problems in Sasaki geometry.
method Discussion and description of open problems.
result Two open problems in Sasaki geometry are identified.
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arXiv research
A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
169,341 papers · 148 categories
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48 results for “geometry problems”
Discusses geometry problems for fun and learning.
problem Geometry problems and their solutions.
method Not original, but related to differential geometry course.
result Relation to differential geometry course.
Surveying probabilistic real algebraic geometry.
problem Classical problems in real algebraic geometry.
method Probabilistic perspective on classical topics.
result Modern approach to Hilbert's Sixteenth Problem.
Abstract collects open problems in billiards and symplectic geometry.
problem Open problems in billiards and symplectic geometry.
method Compilation of open problems from discussions.
result Compilation of open problems.
New approach uses isotropic geometry to solve Euclidean problems.
problem Solving systems of constraints in Euclidean geometry.
method Start with analogous problems in isotropic geometry to initialize optimization algorithms.
result Solutions in isotropic geometry provide insight and initialize Euclidean problem solutions.
Teichmüller discusses generalizing extremal problems in conformal geometry.
problem Generalizing extremal problems in conformal geometry.
method Wide generalization across function theory, topology, and algebra.
result Final chapter in the Handbook of Teichmüller theory.
Geometry of geodesicsmath.DG
Lecture notes on geodesics in differential geometry.
problem Understanding geodesics in differential geometry.
method Expository lecture notes with exercises.
result Explains the geometry of geodesics.
Open geometry puzzles keep the author engaged.
problem Open problems in geometry that challenge the author.
method Collection of open problems based on puzzle-charm.
result No spare hands in solving the problems.
Develops resolvent degree theory for algebraic geometry problems.
problem Hilbert's 13th Problem and related conjectures.
method Extends Brauer's resolvent degree theory to algebraic geometry.
result Hilbert's 13th Problem and related conjectures are equivalent to enumerative geometry problems.
Higher prequantum geometrymath-ph
Surveying higher prequantum geometry and its applications.
problem Prequantizing local field theory locally and invariantly.
method Abstract cohesive homotopy theory.
result Existence of a solution in higher differential geometry.
We review some applications of noncommutative geometry to the study of transverse geometry of Riemannian foliations and discuss open problems.
Proves well-posedness for elliptic problems on domains with singular points.
problem Elliptic boundary value problems on domains with singular points.
method Conformal changes of metric, differential geometry of manifolds with boundary and bounded geometry.
result Proves well-posedness and regularity results for elliptic boundary value problems.
Study CR geometry surface area elements and singular Yamabe problem solutions.
problem Solving the singular CR Yamabe problem in 3D CR geometry.
method Expressed CR invariant surface area elements, deduced Euler-Lagrange equations, provided solutions.
result One energy functional coefficient is shown to be proportional to the log term in volume renormalization.
The paper compares spectral geometry in hyperbolic and spherical manifolds.
problem Understanding spectral geometry in spherical manifolds.
method Survey of known results and open problems.
result Analogous results hold in hyperbolic manifolds but not necessarily in spherical manifolds.
Survey of recent metric geometry in Kähler metrics space.
problem Understanding the metric geometry of Kähler metrics space.
method Survey and highlighting of recent results.
result Highlighting of open problems in the field.
Simpler method derived for path geometries on surfaces, characterizing projective path geometries.
problem Characterizing projective path geometries on surfaces.
method Solving the equivalence problem of sub-Riemannian geometry of signature (1,1) on a contact 3-manifold.
result Characterization of projective path geometries in terms of their chains.
Survey of open problems linking integrable systems and Nijenhuis geometry.
problem Interplay between integrable systems and Nijenhuis geometry.
method Discussion of open problems and questions.
result Challenges and simple questions in the field.
PINNs solve differential geometry problems in complex shapes.
problem Solving differential geometry problems in complex shapes.
method Training neural networks with loss functions inspired by differential conditions.
result PINNs are effective for differential geometry problems.
The study extends inscription problems to non-Euclidean geometries.
problem Generalizing inscription problems to non-Euclidean geometries.
method Symplectic and Riemannian geometry techniques.
result Proved generalized inscription theorems for hyperbolic and spherical surfaces.
Solves index problem for curved BGG sequences in parabolic geometry.
problem Index theory of curved Bernstein-Gelfand-Gelfand sequences.
method Utilizes K-homology and noncommutative geometry.
result Solves the index problem for BGG-sequences on flat parabolic geometry.
New method for creating metrics in complex geometries.
problem Metrizability of parabolic geometries and sub-Riemannian metrics.
method General method for linearizability and classification of cases.
result Natural sub-Riemannian metrics on distributions in geometric control theory.
Machine learning applied to algebraic geometry for physics problems.
problem Reformulating algebraic geometry problems as tensor mappings for machine learning.
method Supervised and unsupervised machine learning techniques applied to algebraic geometry problems.
result Machine learning provides insights into the structure of algebraic geometry data.
The paper outlines key geometry issues in Siegel-Jacobi space.
problem Basic problems in the geometry of the Siegel-Jacobi space.
method Proposes basic problems.
result Outlines key geometry issues in Siegel-Jacobi space.
The paper solves the isoperimetric problem in Riemannian optical geometry, proving circles minimize lengths with area constraints.
problem Optical geometry of static spherically symmetric spacetimes.
method Applying isoperimetric problem results to curves in Riemannian optical geometry.
result Length-minimizing curves with area constraints are circles, with implications for photon spheres.
New heat equation method solves intertwining problems in CR geometry.
problem Intertwining problems in conformal CR geometry.
method Heat equation and extension problems approach.
result New intertwining formulas derived.
Smooth even solutions found for a generalized convex geometry problem.
problem Dual Orlicz-Minkowski problem in convex geometry.
method Geometric flow involving Gauss curvature and normal vectors.
result Existence of smooth even solutions for smooth even measures.
The paper discusses fundamental problems in global Finsler geometry.
problem Fundamental problems in global Finsler geometry.
method Optimization and improvement of Lie derivatives, characterization of gradient vector fields, gradient estimate.
result Gradient estimate for smooth functions on Randers manifolds.
Solves equivalence problem for CR geometries with simple models.
problem Equivalence problem for 2--nondegenerate CR geometries with simple models.
method Uses homogeneous spaces as maximally symmetric models for simple Lie groups.
result Constructs local embeddings of these models into complex space.
Analytic proof solves differential geometry problem.
problem Solving the system of equations , in connected domains.
method Analytic approach to solve differential equations.
result Validated Segre's Theorem in differential geometry.
A survey of mass partitionsmath.CO
Survey of mass partition problems in geometry and topology.
problem Inducing partitions on measures or sets by dividing space.
method Recent progress in topology, discrete geometry, and computer science.
result Connections between different fields.
Problems for the graduate students who want to improve problem-solving skills in geometry. Every problem has a short elegant solution -- this gives a hint which was not available when the problem was discovered.
Study on potential functions for tensors of rank four in differential geometry.
problem Applying potential functions to tensors of rank four.
method Analyzing the inverse problem and intrinsic perspective.
result Negative result: Potential functions cannot be applied to tensors of rank four.
New algebraic-geometry method for Ribaucour transformations.
problem Classical differential geometry problems.
method Algebraic-geometry approach to constructing orthogonal nets.
result Obtains smooth orthogonal nets as Ribaucour transformations.
Classifies homogeneous models of C3 Monge geometries in 8D.
problem Classifying homogeneous models of C3 Monge geometries in 8D.
method General classification algorithm applied to specific problem.
result Classification of homogeneous models of C3 Monge geometries in 8D.
Study variational problems in Kähler geometry to construct metrics.
problem Maximizing/minimizing Monge--Ampère energy on Kähler potentials.
method Prove existence and uniqueness of extremals, use them to construct metrics.
result Existence and uniqueness of extremals with simple characterization.
Paper solves open problem in complex Finsler geometry.
problem Existence of non-Kähler weakly Kähler Finsler metrics.
method Constructs a family of weakly Kähler Finsler metrics.
result Proves uniformization theorem for unitary invariant complex Randers metrics.
I survey methods from differential geometry, algebraic geometry and representation theory relevant for the permanent v. determinant problem from computer science, an algebraic analog of the P v. NP problem.
Solved the equivalence problem for quaternionic contact structures.
problem Finding local diffeomorphism between quaternionic contact manifolds.
method Cartan's method of equivalence and parabolic geometry.
result Explicit construction of Cartan geometry and detailed curvature components.
The paper explores projective structures on curves and their applications in conformal geometry.
problem Finding qualitative information about solutions of Hill equations.
method Detailed description of projective structures and their isomorphism classes, correcting previous inaccuracies.
result The Yamabe problem for curves has no general solutions in a conformal/Möbius ambient space.
A Finsler geometry may be understood as a homogeneous variational problem, where the Finsler function is the Lagrangian. The extremals in Finsler geometry are curves, but in more general variational problems we might consider extremal submanifolds of dimension . In this minicourse we discuss these problems from a ge…
SageMath aids in complex and differential geometry problems.
problem Solving classification problems on Lie group quotients.
method Using SageMath to compute cohomological invariants and classify geometric structures.
result Demonstrates SageMath's utility in tackling complex and differential geometry challenges.
Reduces 3-body problem to shape and spherical geometry.
problem Investigates the motion of three bodies in a plane.
method Geometric reduction using equivariant Riemannian geometry.
result Time parametrization of moduli curve determined by shape curve and potential function.
The paper studies self-Bäcklund curves in centroaffine geometry using elliptic functions.
problem Understanding self-Bäcklund curves in centroaffine geometry.
method Description of general properties and detailed analysis using elliptic functions.
result Provides a detailed description of self-Bäcklund centroaffine curves in terms of elliptic functions.
The resolved conifold geometry is linked to a special Kähler manifold and an instanton-corrected hyperkähler manifold.
problem Understanding the geometry of the resolved conifold and its associated structures.
method Explicit description of ASK and instanton-corrected HK manifolds, relating them to twistor coordinates and solving Riemann-Hilbert problems.
result The instanton-corrected hyperkähler manifold realizes a smoothing of the semi-flat HK metric associated with the ASK geometry.
GeoFunFlow: Geometric Function Flow Matching for Inverse Operator Learning over Complex Geometriescs.LG
GeoFunFlow tackles inverse problems on complex geometries with efficient learning.
problem Challenges in inverse problems governed by PDEs, especially on irregular geometries.
method Combines geometric function autoencoder and latent diffusion model trained via rectified flow.
result Achieves state-of-the-art reconstruction accuracy and efficient inference.
The paper classifies hypersurfaces in a specific 4D geometry.
problem Classify homogeneous hypersurfaces in the four-dimensional Thurston geometry .
method Used geometric conditions to classify hypersurfaces with constant principal curvatures.
result Complete classification of homogeneous hypersurfaces in .
Study compares weak and homotopy moment maps in multisymplectic geometry.
problem Existence and equivariance of moment maps in multisymplectic geometry.
method Comparison of weak and homotopy moment maps.
result Analysis of existence and equivariance phenomena.
Kepler metrics linked to gravity and string theory.
problem Kepler metrics and their geometric properties.
method Sasakian and Hessian geometry approaches.
result Established connection between gravity and string theory.