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.
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.
Analytic proof solves differential geometry problem.
problem Solving the system of equations ∣ablau∣=f(u), Δu=g(u) in connected domains. method Analytic approach to solve differential equations.
result Validated Segre's Theorem in differential geometry.
Survey on automating geometry problem solving with large models.
problem Automating geometric problem solving with spatial understanding and logical reasoning.
method Synthesizes GPS advancements through benchmark construction, parsing, and reasoning paradigms.
result Unified analytical paradigm and emerging opportunities identified.
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.
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.
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.
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.
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.
Solves critical LYZ equation in Kähler geometry.
problem Solvability of LYZ equation at critical phase.
method Establishes existence of smooth solutions.
result Solves critical case of LYZ equation.
Researchers solve a Riemannian geometry problem using warped products.
problem Solving a Moser-Bernstein problem in warped Riemannian manifolds.
method Study entire solutions to the minimal hypersurface equation in warped products.
result Solves the Moser-Bernstein problem in a broader class of Riemannian manifolds.
Study domination between non-Fuchsian surface group representations and anti-de Sitter geometry.
problem Domination problem between non-Fuchsian representations of closed surface groups.
method Analysis of branched harmonic immersions and construction of anti-de Sitter 3-manifolds.
result Found that representations admitting branched harmonic immersions dominate other representations, and constructed large families of branched anti-de Sitter 3-manifolds.
Paper solves degenerated circle pattern metric problem in spherical geometry.
problem Existence and rigidity of (degenerated) circle pattern metrics with prescribed total geodesic curvatures.
method Defined prescribed combinatorial Ricci flows and studied their convergence.
result First degenerated result for total geodesic curvatures in spherical background geometry.
We study the geometry of fanning curves in the Grassmann manifold of n-dimensional subspaces of Rkn; we construct a complete system of invariants which solve the congruence problem. The geometry of the invariants themselves and their relation with classical invariants is also studied.
Developed a new formalism to describe Riemannian geometries using geodesic flow bundles.
problem Understanding the consequences of Einstein equations without solving metric equations.
method Using the bundle of arclength parametrized geodesics (geodesic flow bundle GFB) to describe Riemannian geometry.
result Generalized the cosine- and sine-laws for constant curvature to varying curvature fields.
Solves Tian's stabilization problem for toric Fano manifolds.
problem Tian's stabilization problem for equivariant global log canonical thresholds.
method Expressed complex singularity exponents in terms of support and gauge functions from convex geometry.
result First general result on Tian's problem.
We give a short solution to one of the main open problems in subriemannian geometry. Namely, we prove that length minimizers do not have corner-type singularities. With this result we solve Problem II of Agrachev's list, and provide the first general result toward the 30-year-old open problem of regularity of subrieman…
This research solves Plateau's problem for CRPC surfaces.
problem Constructing surfaces with constant ratio of principal curvatures.
method Proposed a family of surfaces containing a given minimal surface without flat points.
result Obtained a partial solution to Plateau's problem for CRPC surfaces.
Solves a specific Calabi conjecture on special nilmanifolds.
problem Solving the quaternionic Monge-Ampère equation on 8D 2-step nilmanifolds.
method Uses HKT geometry and torus fibrations to show solvability for invariant data.
result Shows the quaternionic Monge-Ampère equation can always be solved on these manifolds.
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.
The paper classifies hypersurfaces in a specific 4D geometry.
problem Classify homogeneous hypersurfaces in the four-dimensional Thurston geometry mSol04. method Used geometric conditions to classify hypersurfaces with constant principal curvatures.
result Complete classification of homogeneous hypersurfaces in mSol04. Solves Lie's 3D metric problem for projective vector fields.
problem Describing 3D Levi-Civita metrics with non-trivial projective vector fields.
method Analyzes Riemannian and Levi-Civita metrics of arbitrary signature.
result Solves the analog of Lie's problem in 3D.
Solves multi-class imbalanced data problem with geometry-based sampling and synthetic data.
problem Handling imbalanced multi-class data in classification problems.
method Two novel methods: undersampling and oversampling.
result Efficacy demonstrated through comparison with state-of-the-art methods.
Solves capillary Lp-Christoffel-Minkowski problem in half-space.
problem Capillary surfaces in half-space geometry.
method Non-collapsing estimate for height and capillary support function.
result Extends Christoffel-Minkowski existence result.
Survey solves curvature problems with hyperbolic spaces.
problem Singularities in hypersurface geometry.
method Hyperbolic unfolding correspondence linking hypersurfaces to Gromov hyperbolic spaces.
result Eliminates hypersurface singularities in scalar curvature geometry.
We give an overview of the existence and regularity results for curvature flows and how these flows can be used to solve some problems in geometry and physics.
In general relativity, spatial light rays of static spherically symmetric spacetimes are geodesics of surfaces in Riemannian optical geometry. In this paper, we apply results on the isoperimetric problem to show that length-minimizing curves subject to an area constraint are circles, and discuss implications for the ph…
Reconstructing Finsler manifolds from sphere data.
problem Recovering a Finsler manifold from sphere data.
method Solving the geometrical inverse problem locally along geodesics.
result Local reconstruction of Finsler manifolds.
Deep NURBS improves PINNs for solving PDEs on arbitrary geometries.
problem Solving partial differential equations on complex geometries with physics constraints.
method Combines admissible NURBS parametrizations and PINN solver for arbitrary geometries.
result High convergence rate and accuracy for most PDEs using Deep NURBS.
Geometrization Theorem solves complex geometry problems.
problem Complex geometry problems in differential geometry.
method Based on Hamilton's program, proved by Grigory Perelman.
result Generalized Poincaré's Conjecture.
Complementing the previous paper in the series, this paper classifies ∣2∣-graded parabolic geometries, listing their important properties: the group G0, the graded tangent bundle gr(T) and its algebraïc bracket, the relevant cohomology spaces and the standard Tractor bundle $\mc{T}$. Several of these geometries …
Study reveals CR structure of snake robot's geometry.
problem Understanding the kinematics and geometry of a snake robot.
method Analysis of (2,3,5) distributions and solving Cartan equivalence problem.
result Discovery of a CR structure with CR dimension 1 and real codimension 3.
We study the geometry and topology of immersed surfaces in Euclidean 3-space whose Gauss map satisfies a certain two-piece-property, and solve the ``shadow problem" formulated by H. Wente.
We give an abstract formulation of the formal theory partial differential equations (PDEs) in synthetic differential geometry, one that would seamlessly generalize the traditional theory to a range of enhanced contexts, such as super-geometry, higher (stacky) differential geometry, or even a combination of both. A moti…
Solves Apollonius' problem using oriented circles and inversive geometry.
problem Constructing a circle tangent to three given circles.
method Using oriented circles and inversive invariants, reversing each given circle to find solutions.
result The problem has 0, 1, or 2 solutions, depending on the configuration of given circles.
We solve the problem posed by Boyer and Galicki about the existence of K-contact simply connected manifolds with no Sasakian structure. Although the result lies in the framework of metric contact geometry, our methods come from contact and symplectic geometry and are based on the method of fat bundles developed by Ster…
Stochastic optimization is key to efficient inversion in PDE-constrained optimization. Using 'simultaneous shots', or random superposition of source terms, works very well in simple acquisition geometries where all sources see all receivers, but this rarely occurs in practice. We develop an approach that interpolates d…
In this short note, we solve a Dirichlet problem for a fully nonlinear elliptic equation. The operator is introduced by S. Donaldson and it is relevant to the geometry of the space of volume forms.
In this article, we solve the equivalence problem for 2--nondegenerate CR geometries that have (at every point) a homogeneous space G/H as a maximally symmetric model for G simple real Lie group of CR automorphisms. This completes the classification of real submanifolds in complex space that are maximally symmetric…
Deep learning has achieved remarkable success in diverse applications; however, its use in solving partial differential equations (PDEs) has emerged only recently. Here, we present an overview of physics-informed neural networks (PINNs), which embed a PDE into the loss of the neural network using automatic differentiat…
Space mapping speeds up shape optimization for PDEs.
problem Efficiently solving shape optimization problems constrained by PDEs.
method Combines fine and coarse model optimizations using Riemannian metrics.
result Space mapping methods are highly efficient for complex shape optimization problems.
In this paper, we study Zermelo navigation on Riemannian manifolds and use that to solve a long standing problem in Finsler geometry. Namely, the complete classification of strongly convex Randers metrics of constant flag curvature.
Researchers solve a formally determined inverse problem in Lorentzian geometry.
problem Determining a Lorentzian metric from boundary measurements of the Dirichlet-to-Neumann map.
method New method using distorted plane wave solutions and geometric, topological, and unique continuation arguments.
result A globally hyperbolic metric agreeing with the Minkowski metric outside a compact set and having the same Dirichlet-to-Neumann map must be the Minkowski metric up to diffeomorphism.
The optimal transport problem is studied in the context of Lorentz-Finsler geometry. For globally hyperbolic Lorentz-Finsler spacetimes the first Kantorovich problem and the Monge problem are solved. Further the intermediate regularity of the transport paths is studied. These results generalize parts of Bertrand & Puel…
Potential functions can be used as generating potentials of relevant geometric structures for a Riemannian manifold such as the Riemannian metric and affine connections. We study wether this procedure can also be applied to tensors of rank four and find a negative answer. We study this from the perspective of solving t…
Solves open problems for fully nonlinear elliptic equations on manifolds.
problem Solving fully nonlinear elliptic equations on manifolds.
method Analytic slope invariant and Nakai-Moishezon criterion.
result Solves open problems including hessian and hessian quotient equations.
For every Finsler metric F we associate a Riemannian metric gF (called the Binet-Legendre metric). The transformation F↦gF is C0-stable and has good smoothness properties, in contrast to previous constructions. The Riemannian metric gF also behaves nicely under conformal or bilipshitz deformation …
New maximal surfaces solve Bernstein problems.
problem Bernstein problems in centroaffine geometry.
method Calabi affine maximal surfaces and orthonormal frame fields.
result Complete centroaffine extremal hypersurfaces solve all Bernstein problems.