arXiv research

New approach uses isotropic geometry to solve Euclidean problems.

Solves index problem for curved BGG sequences in parabolic geometry.

Survey on automating geometry problem solving with large models.

PINNs solve differential geometry problems in complex shapes.

Simpler method derived for path geometries on surfaces, characterizing projective path geometries.

New heat equation method solves intertwining problems in CR geometry.

Lecture notes on geodesics in differential geometry.

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.

Researchers solve a Riemannian geometry problem using warped products.

Study domination between non-Fuchsian surface group representations and anti-de Sitter geometry.

Paper solves degenerated circle pattern metric problem in spherical geometry.

We study the geometry of fanning curves in the Grassmann manifold of n-dimensional subspaces of $\mathbb{R}^{kn}$; 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.

Solves Tian's stabilization problem for toric Fano manifolds.

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.

Solves a specific Calabi conjecture on special nilmanifolds.

Paper solves open problem in complex Finsler geometry.

The paper classifies hypersurfaces in a specific 4D geometry.

Solves Lie's 3D metric problem for projective vector fields.

Solves multi-class imbalanced data problem with geometry-based sampling and synthetic data.

Solves capillary $L_p$-Christoffel-Minkowski problem in half-space.

Survey solves curvature problems with hyperbolic spaces.

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.

Deep NURBS improves PINNs for solving PDEs on arbitrary geometries.

Geometrization Theorem solves complex geometry problems.

Complementing the previous paper in the series, this paper classifies $|2|$-graded parabolic geometries, listing their important properties: the group $G_0$, 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.

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.

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…

We present an analytic approach on how to solve the problem $|\nabla u|=f(u)$, $Δu = g(u)$, in connected domains $Ω\subseteq\mathbb{R}^n$.

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.

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.

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.

For every Finsler metric $F$ we associate a Riemannian metric $g_F$ (called the Binet-Legendre metric). The transformation $F \mapsto g_F$ is $C^0$-stable and has good smoothness properties, in contrast to previous constructions. The Riemannian metric $g_F$ also behaves nicely under conformal or bilipshitz deformation …

New maximal surfaces solve Bernstein problems.