Solves Einstein vacuum equations gluing problem for close Minkowski data.
problem Solving the characteristic gluing problem for Einstein vacuum equations.
method Derived infinite-dimensional and 10-dimensional gauge-dependent and gauge-invariant charges; constructed null lapse function and conformal geometry.
result Obstructions to gluing problem are gauge-dependent charges, modulo gauge-invariant charges.
Maximally hyperbolic solutions contain future neighborhoods of intersecting hypersurfaces.
problem Maximally globally hyperbolic solutions of higher-dimensional vacuum Einstein equations.
method Analyzing intersections of characteristic hypersurfaces.
result Contains a future neighborhood of intersecting hypersurfaces.
We give a local integral formula, valid on general curved space-times, for the characteristic Cauchy problem for the Dirac equation with arbitrary spin using the method developed by Friedlander in his book "the wave equation on a curved spacetime" (1975). The results obtained by Penrose in the flat case in "Null hypers…
Paper solves Einstein vacuum equations gluing problem with applications.
problem Solving characteristic gluing problem for Einstein vacuum equations.
method Codimension-10 gluing construction, relating charges to ADM energy, and localized version.
result Asymptotically flat data can be glued to Kerr spacetime with 10 charges.
Study geometric singular solutions of generalized Monge-Ampère equations.
problem Solving generalized Monge-Ampère equations on a plane.
method Using exterior differential systems and Cauchy characteristics.
result Criteria for geometric singular solutions to be equivalent to specific types.
Decides undecidability of equations and first-order theory for Seifert 3-manifold groups.
problem Decidability of equations and first-order theory in Seifert 3-manifold groups.
method Encoding Hilbert's tenth problem and using it to show undecidability.
result Undecidability of equations and first-order theory in Seifert 3-manifold groups with non-negative Euler characteristic.
The paper extends macroscopic market making to stochastic games, revealing properties and solving equations.
problem Price competition among market makers in a stochastic game setting.
method Extension of macroscopic market making framework to stochastic games, introducing multidimensional characteristic equations.
result New well-posedness results for forward-backward stochastic differential equations.
Solves characteristic problem in general relativity for null data.
problem Characteristic Cauchy problem in Einstein vacuum field equations.
method Abstract data formalism and tangential components of the ambient Ricci tensor.
result Formulates and solves the characteristic problem completely abstractly.
Paper glues characteristic data to Kerr spacetime, proving spacelike gluing.
problem Solving characteristic gluing problem for Einstein vacuum equations.
method Detailed characteristic gluing of strongly asymptotically flat data to Kerr spacetime.
result Alternative proof of spacelike gluing construction for strongly asymptotically flat spacelike initial data.
Conservation laws vanishing along characteristic directions of a given system of PDEs are known as characteristic conservation laws, or characteristic integrals. In 2D, they play an important role in the theory of Darboux-integrable equations. In this paper we discuss characteristic integrals in 3D and demonstrate that…
An unusual formula for the Euler characteristics of even dimensional triangulated manifolds is deduced from the generalized Dehn-Sommerville equations.
We present a connection between the Killing fields that arise in the loop-group approach to integrable systems and conservation laws viewed as elements of the characteristic cohomology. We use the connection to generate the complete set of conservation laws (as elements of the characteristic cohomology) for the Tzitzei…
It has been recently shown that rough volatility models, where the volatility is driven by a fractional Brownian motion with small Hurst parameter, provide very relevant dynamics in order to reproduce the behavior of both historical and implied volatilities. However, due to the non-Markovian nature of the fractional Br…
We generalize the notion of involutivity to systems of differential equations of different orders and show that the classical results due to Guillemin and Quillen relating involutivity, restrictions, characteristics and characteristicity, known for first order systems, extend to the general context, though in a modifie…
Paper defines and proves geometric uniqueness of Einstein field equations.
problem Einstein field equations characteristic Cauchy problem
method Covariant definition of double null data, proving geometric uniqueness
result Double null data fully covariant and geometrically unique
The paper extends gluing theorems for linearized gravitational fields in static spacetimes with cosmological constant.
problem Establishing gluing theorems for linearized vacuum gravitational fields on characteristic surfaces.
method Analyzing linearised Einstein equations in Bondi gauge on static four-dimensional spacetimes with cosmological constant.
result Generalization and extension of gluing theorems to include cosmological constant and arbitrary topology.
The infinitesimal period relation (also known as Griffiths' transversality) is the system of partial differential equations constraining variations of Hodge structure. This paper presents a study of the characteristic cohomology associated with that system of pde.
Proves stability of Minkowski space for specific initial data.
problem Stability of Minkowski space under spacelike-characteristic initial data.
method Vectorfield method and bootstrapping argument, with new geometric constructions.
result Global nonlinear stability of Minkowski space proved for the spacelike-characteristic Cauchy problem.
We extend the Mason-Newman Lax pair for the elliptic complex Monge-Ampère equation so that this equation itself emerges as an algebraic consequence. We regard the function in the extended Lax equations as a complex potential. We identify the real and imaginary parts of the potential, which we call partner symmetries, w…
Unified framework for constructing kernels for transport equations and Koopman eigenfunctions.
problem Constructing kernels for transport equations and Koopman eigenfunctions.
method Three methods: variational principle, Green's function, and resolvent operator.
result Kernels constructed via these methods are identical under mild assumptions.
We classify quasilinear systems in Riemann invariants whose characteristic webs are linearizable on every solution. Although the linearizability of an individual web is a rather nontrivial differential constraint, the requirement of linearizability of characteristic webs on all solutions imposes simple second-order con…
We calculate relations on characteristic classes which are obstructions preventing closed Kähler manifolds from carrying holomorphic Cartan geometries. We apply these relations to give global constraints on the phase spaces of complex analytic determined and underdetermined systems of differential equations.
We study four distinct second-order nonlinear equations of Rabelo which describe pseudospherical surfaces. By transforming these equations to the constant-characteristic form we relate them to some well-studied integrable equations. Two of the Rabelo equations are found to be related to the sine-Gordon equation. The ot…
There are (at least) two different approaches to define equivariant analogue of the Euler charateristic for a space with a finite group action. The first one defines it as an element of the Burnside ring of the group. The second approach emerged from physics and includes the orbifold Euler characteristic and its higher…
We classify 7-dimensional cocalibrated $\G_2$-manifolds with parallel characteristic torsion and non-abelian holonomy. All these spaces admit a metric connection ∇c with totally skew-symmetric torsion and a spinor field Ψ1 solving the equations in the common sector of type II superstring theory. T…
This paper is a natural companion of [Alekseevsky D.V., Alonso Blanco R., Manno G., Pugliese F., Ann. Inst. Fourier (Grenoble) 62 (2012), 497-524, arXiv:1003.5177], generalising its perspectives and results to the context of third-order (2D) Monge-Ampère equations, by using the so-called "meta-symplectic structure" ass…
In this paper, we study the structure of the singular set for a C1 smooth surface in the 3-dimensional Heisenberg group H1. We discover a Codazzi-like equation for the p-area element along the characteristic curves on the surface. Information obtained from this ordinary differential equation …
The Einstein equations in wave map gauge are a geometric second order system for a Lorentzian metric. To study existence of solutions of this hyperbolic quasi diagonal system with initial data on a characteristic cone which are not zero in a neighbourhood of the vertex one can appeal to theorems due to Cagnac and Dossa…
An almost contact metric structure is parametrized by a section of an associated homogeneous fibre bundle, and conditions for this to be a harmonic section, and a harmonic map, are studied. These involve the characteristic vector field, and the almost complex structure in the contact subbundle. Several examples are giv…
These lecture notes are a systematic and self-contained exposition of the cohomological theories naturally related to partial differential equations: the Vinogradov C-spectral sequence and the C-cohomology, including the formulation in terms of the horizontal (characteristic) cohomology. Applications to computing invar…
We study the geometry of multidimensional scalar 2nd order PDEs (i.e. PDEs with n independent variables) with one unknown function, viewed as hypersurfaces E in the Lagrangian Grassmann bundle M(1) over a (2n+1)-dimensional contact manifold (M,C). We develop the theory of character…
We determine the expected curvature polynomial of random real projective varieties given as the zero set of independent random polynomials with Gaussian distribution, whose distribution is invariant under the action of the orthogonal group. In particular, the expected Euler characteristic of such random real projective…
The paper explores conditions for homothetic Killing vectors on spacetime hypersurfaces.
problem Conditions for the existence of homothetic Killing vectors on spacetime hypersurfaces.
method General identities relating deformation tensor and tensor on hypersurfaces, applied to specific settings.
result Necessary and sufficient conditions for homothetic Killing vectors on spacetime hypersurfaces.
The Conclusive Theorem has been established to determine the dependence of the three-axes positive-definite Finsleroid metric functions F on the Finsleroid azimuthal angle θ in the three-dimensional case N=3, provided that the condition of the angle-separation in the involved characteristic functions is implied. …
We consider hypersurfaces in Einstein-Sasaki 5-manifolds which are tangent to the characteristic vector field. We introduce evolution equations that can be used to reconstruct the 5-dimensional metric from such a hypersurface, analogous to the (nearly) hypo and half-flat evolution equations in higher dimensions. We use…
Paper improves American option valuation in complex models.
problem Valuation of American options in time-dependent jump-diffusion models.
method Integral equations and characteristic functions for explicit exercise boundary determination.
result Efficient and accurate pricing method for American options in various models.
A normal form for edge metrics is derived under the necessary conditions that the metric be normalized and exact. The normal forms for such an edge metric are shown to be in 1-1 correspondence with representative metrics for a reduced conformal infinity on the boundary. The normal form is constructed via solution of a …
In this paper we study the spacelike-characteristic Cauchy problem for the Einstein vacuum equations. We prove that given initial data on a maximal compact spacelike hypersurface Σ≃B(0,1)⊂R3 and the outgoing null hypersurface H emanating from ∂Σ, the time of ex…
Valence two Killing tensors in the Euclidean and Minkowski planes are classified under the action of the group which preserves the type of the corresponding Killing web. The classification is based on an analysis of the system of determining partial differential equations for the group invariants and is entirely algebr…
Formula conjectured for rational cuspidal curves in projective plane.
problem Counting rational cuspidal curves in projective plane.
method Extending Kontsevich's recursion formula and using geometric input about tangency of curves at nodal points.
result Conjectural formula agrees with earlier computations and extends to rational quartics with E6 singularity.
I consider the existence and structure of conservation laws for the general class of evolutionary scalar second-order differential equations with parabolic symbol. First I calculate the linearized characteristic cohomology for such equations. This provides an auxiliary differential equation satisfied by the conservatio…
Paper develops a framework for hyperbolic Monge-Ampère equation on strips, proving well-posedness and stability.
problem Addressing the rigidity-flexibility dichotomy for wrinkled patterns in thin elastic sheets.
method Develops hodograph transformation and parametrix-corrector decomposition to handle corner singularities and prove well-posedness.
result Proves existence and uniqueness of hodograph weak solutions and derives energy estimates for stability.
Perimeter on manifolds leads to new symmetrization methods.
problem Applying symmetrization methods to quasilinear elliptic problems on RN. method Generalization of perimeter to manifolds, using hear kernel regularization.
result New symmetrization method on spheres for quasilinear elliptic problems.
Study on 3D spacetimes, focusing on vacuum data and energy bounds.
problem Existence and properties of solutions for Einstein equations in 3D spacetimes.
method Analysis of 2D general relativistic initial data sets, construction of vacuum, spacelike data, and review of global Hamiltonian charges.
result Established lower bounds for energy in terms of angular momentum, linear momentum, and center of mass.
We prove that the existence of a dispersionless Lax pair with spectral parameter for a nondegenerate hyperbolic second order partial differential equation (PDE) is equivalent to the canonical conformal structure defined by the symbol being Einstein-Weyl on any solution in 3D, and self-dual on any solution in 4D. The fi…
The Euler characteristic is the only additive topological invariant for spaces of certain sort, in particular, for manifolds with some finiteness properties. A generalization of the notion of a manifold is the notion of a V-manifold. Here we discuss a universal additive topological invariant of V-manifolds: the univers…
NOGaP uses neural operators and GPs to solve PDEs with uncertainty quantification.
problem Lack of uncertainty measures in neural operator solutions for PDEs.
method NOGaP combines neural operators with Gaussian Processes to provide probabilistic solutions.
result NOGaP offers improved prediction accuracy and uncertainty quantification.
For a principal $\rmSU(n)$-bundle over a compact manifold of dimension 2,3,4, we determine the orbit types of the action of the gauge group on the space of connections modulo pointed local gauge transformations. We find that they are given by Howe subgroups of $\rmSU(n)$ for which a certain characteristic equation is…