New algorithms predict spatio-temporal data without assuming its structure.
problem Predicting high-dimensional spatio-temporal data without assuming its structure.
method Light cone decompositions and three simple algorithms for predictive state reconstruction.
result Good predictive performance and distributions over spatio-temporal data.
Decomposes moduli space of Riemann surfaces into convex polytopes.
problem Decomposing moduli space of Riemann surfaces into cells.
method Using Nakamura graphs to define cell decomposition, parametrized by graphs or permutations.
result Cells are convex polytopes defined by light-cone string parameters.
Classifies surfaces with zero mean curvature in a light cone.
problem Classifying surfaces with zero mean curvature in a light cone.
method Examined geodesics and screw motions, used Weierstrass representations.
result Complete classification of ruled zero mean curvature surfaces.
Solves surface problem in 3D light cone.
problem Björling problem for zero mean curvature surfaces in the three-dimensional light cone.
method Solves the Björling problem for zero mean curvature surfaces in the three-dimensional light cone.
result Constructs and classifies all rotational zero mean curvature surfaces.
The paper connects curves on a light cone to KdV equations.
problem Understanding differential invariants of curves on a light cone.
method Poisson equivalence and centro-affine action of Lorentzian group.
result Solutions of KdV equations as flows of curves on the cone.
The paper studies hypersurface evolution in a light-cone and curvature flow.
problem Investigating the evolution of hypersurfaces in a light-cone.
method Exploring variational problems associated with hypersurfaces and curvature flow.
result Established perpetual existence and smooth convergence of curvature flow to a circle.
Paper proves rigidity of certain manifold immersions into specific spacetimes.
problem Rigidity of isometric immersions into light cones.
method Analyzes Riemannian manifolds of dimension n-1 into light cones of n+1 spacetimes.
result Shows rigidity of isometric immersions for n ≥ 3.
The study proves that certain constant mean curvature surfaces in a specific cone are either spheres or horospheres.
problem Characterizing constant mean curvature surfaces in a three-dimensional light cone.
method Analyzing entire constant mean curvature graphs in the light cone Q + 3 \mathbb{Q}^3_+ Q + 3 under the condition of bounded Gaussian curvature. result Entire constant mean curvature graphs in the light cone are either horospheres or spheres.
Reconstructing manifold structure from boundary light observations.
problem Reconstructing Lorentzian manifold structure from boundary light observations.
method Constructive proof using Snell's law for reflections at the boundary.
result Topological, differentiable, and conformal structure of subsets of sources uniquely determined.
Curvature flow and inverse curvature flow solutions on 2D light cone identified.
problem Identifying self-similar solutions to curvature flow and inverse curvature flow on 2D light cone.
method Proved correspondence between CF and ICF solutions, analyzed ellipses and hyperboles, and characterized self-similar solutions.
result Ellipses and hyperboles are the only curves evolving under homotheties on the 2D light cone.
New inequality shows all special submanifolds in light cone are totally umbilical spheres.
problem Characterizing submanifolds with parallel mean curvature in Lorentz-Minkowski spacetime.
method Established an integral inequality and used it to derive a rigidity result.
result All compact submanifolds with parallel mean curvature in light cone are totally umbilical spheres.
The paper studies volumes of conformally flat manifolds in light-cone geometry.
problem Volume maximization of conformally flat manifolds in light-cone geometry.
method Computes variational formulas for the volume of hypersurfaces in light-cone.
result Hypersurfaces of conformally flat manifolds maximize volume in certain null hypersurfaces.
Extends gauge conditions for superparticle to conic neighbourhood.
problem Applying gauge conditions to superparticle in momentum space.
method Patching gauge conditions over different parts of field space.
result Extension of light-cone gauge to conic neighbourhood.
The paper classifies periodic solitons in curve flows on the light-cone.
problem Investigating periodic solitons in curve flows on the light-cone.
method Deriving Harnack inequality for heat flow, classifying space-periodic solitons for a third-order curvature flow.
result Closed soliton solutions form a family of transcendental curves with specific rotation indices.
Proves compatibility of light cones and projective structures.
problem Clarifying different concepts of compatibility between conformal and projective structures.
method Analyzes compatibility criteria introduced by Ehlers-Pirani-Schild and Trautman-Scholz.
result Proves that the compatibility criterion introduced by Ehlers-Pirani-Schild is correct.
Measuring supernova neutrinos removes spacetime's conformal freedom.
problem Determining the conformal factor of spacetime's visible part.
method Measuring neutrino cones in addition to light cones.
result The conformal factor can now be determined.
The paper classifies orbits of S O ( 3 , 1 ) SO(3,1) S O ( 3 , 1 ) in a 4D Minkowski space.
problem Classifying orbits of S O ( 3 , 1 ) SO(3,1) S O ( 3 , 1 ) in a 4D Minkowski space. method Analyzing the stabilizer and r-slice of L ( ⋀ 2 E 1 4 ) L(\bigwedge^2 E^4_1 ) L ( ⋀ 2 E 1 4 ) . result Each S O ( 3 , 1 ) SO(3,1) S O ( 3 , 1 ) -orbit in L ( ⋀ 2 E 1 4 ) L(\bigwedge^2 E^4_1 ) L ( ⋀ 2 E 1 4 ) is either a neutral hypersurface homothetic to L ± \mathcal{L}_{\pm} L ± or a hypersurface with a two-dimensional involutive distribution. Study submanifolds in null hypersurfaces of generalized Robertson-Walker spacetimes.
problem Characterize submanifolds in null hypersurfaces of generalized Robertson-Walker spacetimes.
method Analyze light cones, lightlike cylinders, and null cones in specific spacetimes; provide conditions for conformal diffeomorphism.
result Conditions guaranteeing conformal diffeomorphism to hyperbolic space, round cylinder, and sphere.
New examples of Calabi-Yau metrics on cones with irregular smooth links.
problem Finding new Calabi-Yau metrics on cones with irregular smooth links.
method Explicit computation of Reeb field and Minkowski decompositions of toric Calabi-Yau cones.
result Examples of complete Calabi-Yau metrics on cones with irregular smooth links.
Study connects contact structures to cone geodesics and contactomorphisms.
problem Understanding contact structures on cone geodesics.
method Review and generalize cone geodesics to contact manifolds, establish correspondence with contactomorphisms.
result Established correspondence between contactomorphisms and cone structures.
This paper mainly aims to establish the well-posedness on time interval [ 0 , ε − 1 2 T ] [0,\varepsilon^{-\frac{1}{2}}T] [ 0 , ε − 2 1 T ] of the classical initial problem for the bosonic membrane in the light cone gauge. Here ε \varepsilon ε is the small parameter measures the nonlinear effects. In geometric, the bosonic membrane are timelike submanifo…
Study light ray transform in pseudo-Euclidean space, derive inversion formula, and prove stability.
problem Analyzing light ray transform in pseudo-Euclidean space.
method Investigate normal operator, derive inversion formula, analyze as Fourier Integral Operator.
result Derive an inversion formula and prove stability estimates.
Existence and uniqueness in R n , 1 {\Bbb R}^{n,1} R n , 1 of entire spacelike hypersurfaces contained in the future of the origin O O O and asymptotic to the light-cone, with scalar curvature prescribed at their generic point M M M as a negative function of the unit vector O m → \overrightarrow{Om} O m pointing in the direction of $\overrighta…
Researchers compute the index of a specific operator on contact manifolds.
problem Computing the index of a twisted Dolbeault operator on toric contact manifolds.
method Using equivariant techniques, they localized the symbol to Reeb orbits and applied polytope decomposition.
result They derived an Atiyah-Bott-Lefschetz type formula for the index.
The paper classifies submanifolds in a specific Lorentz-Minkowski space.
problem Characterizing submanifolds in a Lorentz-Minkowski space.
method Constructing a global frame field and analyzing extrinsic invariants.
result Local classification theorems for specific submanifold classes.
A hex sphere is a singular Euclidean sphere with four cones points whose cone angles are (integer) multiples of 2*pi/3 but less than 2*pi. Given a hex sphere M, we consider its Voronoi decomposition centered at the two cone points with greatest cone angles. In this paper we use elementary Euclidean geometry to describe…
Study on extremals in sub-Lorentzian geometry defined by antinorm.
problem Characterizing extremals in sub-Lorentzian structures.
method Deriving Hamiltonian system and conditions for extremal trajectories.
result Conditions for normal extremal trajectories and properties of abnormal extremals.
Complex geometry and symplectic geometry are mirrors in string theory. The recently developed generalised complex geometry interpolates between the two of them. On the other hand, the classical and quantum mechanics of a finite number of degrees of freedom are respectively described by a symplectic structure and a comp…
The paper studies flat metrics with cone singularities on surfaces, decomposing moduli spaces into polyhedra.
problem Moduli spaces of flat metrics with cone singularities on surfaces.
method Geometry of Euclidean and Minkowski polyhedra, foliations, hyperbolic and spherical convex polyhedra.
result Moduli spaces of flat metrics with cone singularities have natural decompositions into polyhedra.
Injectivity result for light ray transform on Lorentzian manifolds.
problem Injectivity of light ray transform on Lorentzian manifolds.
method Explicit relationship between geodesic and magnetic vector fields.
result Injectivity up to natural obstruction under certain conditions.
The paper explores Cholesky decompositions for symmetric matrices and their geometric properties.
problem Understanding the structure and properties of symmetric matrices through Cholesky decompositions.
method Introducing cones of symmetric matrices, proving Cholesky-type factorizations, and showing geometric properties.
result Each symmetric matrix admits an uncountable family of Cholesky-type factorizations, and these cones are isometric Riemannian manifolds.
Study the geometry and holonomy of indecomposable cones.
problem Classify the holonomy of indecomposable cones.
method Computation of cocycles in s o ( 1 , n − 1 ) \mathfrak{so}(1,n-1) so ( 1 , n − 1 ) for indecomposable subalgebras. result Structure theorems for the base manifold and description of cone holonomy in specific cases.
We are generalizing to higher dimensions the Bavard-Ghys construction of the hyperbolic metric on the space of polygons with fixed directions of edges. The space of convex d-dimensional polyhedra with fixed directions of facet normals has a decomposition into type cones that correspond to different combinatorial types …
We construct and study a family of double-periodic almost entire solutions of the maximal surface equation. The solutions are parameterized by a submanifold of 3 × 3 3\times 3 3 × 3 -matrices (the so-called generating matrices). We show that the constructed solutions are either space-like or of mixed type with the light-cone type…
New mappings in Minkowski spacetime classified under mild conditions.
problem Characterizing mappings in Minkowski spacetime.
method Analyzing mappings under minimal assumptions.
result Mappings fall into three categories based on specific conditions.
Solves portfolio selection with constraints using martingale theory.
problem Portfolio selection with constraints on wealth and portfolio.
method Transformed into a mean-variance problem without constraints, solved using martingale theory.
result Directly presents semi-analytical expressions of efficient policy.
Study Hilbert's projective metric for bounded growth functions leading to Sinkhorn's algorithm convergence.
problem Optimal transport in unbounded settings with heavy-tailed distributions.
method Hilbert's projective metric for integrable functions of bounded growth, kernel integral operators as contractions.
result Exponential convergence of Sinkhorn's algorithm for light-tailed marginal distributions.
In two former papers, the authors independently proved that the space of hyperbolic cone-3-manifolds with cone angles less than 2π and fixed singular locus is locally parametrized by the cone angles. In this sequel, we investigate the local shape of the deformation space when the singular locus is no longer fixed, i.e.…
New bounds found for eigenvalues of Laplace operator in Lorentz-Minkowski space.
problem Eigenvalue bounds for spacelike submanifolds in Lorentz-Minkowski space do not match Euclidean space results.
method Developed a new integral formula on compact spacelike sections of the light cone in L m \mathbb{L}^m L m to prove extrinsic upper bounds. result Eigenvalue achieves upper bounds if and only if submanifold lies minimally in certain hypersphere.
A general theory of rigid completely integrable analytic partial differential equations is endeavoured. The tube over the light cone in C^3 is shown to be the unique model (up to biholomorphisms) having CR automorphism group of maximal dimension equal to 10. Explicit formulas for the Lie prolongation of vector fields t…
Study left-invariant pseudo-Riemannian metrics on Lie groups focusing on null cone Lie algebras.
problem Characterize left-invariant pseudo-Riemannian metrics on Lie groups in the null cone.
method Use bracket flow on Lie algebra to study metrics on Lie groups.
result Classify all cases of null cone Lie algebras in signatures (1,q) and (2,q).
We construct a cut-off version of nonpertubative closed Bosonic string field theory in the light-cone gauge with imaginary string coupling constant. We show that the partition function is a continuous function of the string coupling constant, and conjecture a relation between the formal power series expansion of this p…
We show that finiteness of the Lorentzian distance is equivalent to the existence of generalised time functions with gradient uniformly bounded away from light cones. To derive this result we introduce new techniques to construct and manipulate achronal sets. As a consequence of these techniques we obtain a functional …
Simulates null geodesics for a massive dipole metric using Sage and Python.
problem Modeling null geodesics for a Bonnor massive dipole metric.
method Symbolic-numerical algorithm in Sage and Python for visualization.
result Visualizes 3D geodesics and their bending due to curvature.
We propose a method to compute complex volume of 2-bridge link complements. Our construction sheds light on a relationship between cluster variables with coefficients and canonical decompositions of link complements.
Study shows unique tangent cones for area-minimizing currents at boundary points.
problem Uniqueness of tangent cones for area-minimizing currents with arbitrary multiplicity.
method Analysis of area minimizing currents in C 2 C^2 C 2 submanifolds with arbitrary boundary multiplicity. result Tangent cones are unique at density Q / 2 Q/2 Q /2 boundary points. Decomposes nonconvex polynomials into convex parts using algebraic methods.
problem Decomposing multivariate polynomials into the difference of two convex polynomials.
method Reduces problem to linear, second order cone, and semidefinite programming.
result Optimizing over subsets of valid difference of convex decompositions speeds up CCP.
Invariants for 3-manifolds with toral boundaries, related by sutured decompositions.
problem Invariants for 3-manifolds with toral boundaries and non-degenerate Thurston norm.
method Constructing an invariant called guts and proving its invariance under sutured decompositions.
result The guts of different homology classes are related by sutured decompositions.