Develops geometric framework for dissipative field equations.
problem Dissipative field equations and their geometric analysis.
method Canonical k-contact manifolds, k-contactifications, splitting results, regularity conditions, criteria for PDEs. result Explicit Hamiltonian descriptions for various nonlinear PDEs.
A description of classical field theories of first order in terms of Lagrangian submanifolds of premultisymplectic manifolds is presented. For this purpose, a Tulczyjew's triple associated with a fibration is discussed. The triple is adapted to the extended Hamiltonian formalism. Using this triple, we prove that Euler-…
Study on non-flat two-plectic geometry of six-sphere and its Hamiltonian dynamics.
problem Non-flat two-plectic geometry of six-sphere and Hamiltonian dynamics.
method Explicitly proving non-flatness and showing infinitesimal automorphisms via g2. result Explicit solutions of Hamilton-de Donder-Weyl equations with one- and two-dimensional sources.
In this article we present a natural generalization of Newton's Second Law valid in field theory, i.e., when the parameterized curves are replaced by parameterized submanifolds of higher dimension. For it we introduce what we have called the geodesic k-vector field, analogous to the ordinary geodesic field and which …
Geometrically describes pseudo-gauge freedom in relativistic hydrodynamics.
problem Pseudo-gauge ambiguity in relativistic hydrodynamics.
method Develops k-contact geometry to describe thermodynamic states and conservation laws. result Identifies pseudo-gauge freedom as the non-uniqueness of thermodynamic equilibrium redefinition.
New Lie systems defined on k-contact manifolds, with applications.
problem Understanding Lie systems on k-contact manifolds. method Distributional approach to k-contact manifolds and Hamiltonian vector fields. result Lie systems can be understood as Hamiltonian relative to a k-contact manifold. Develops Hamilton-Jacobi theory for non-conservative field theories in k-contact geometry.
problem Analyzes non-conservative field theories, especially dissipative systems.
method Introduces evolution k-contact k-vector fields and develops two Hamilton-Jacobi theories.
result Recover ordinary contact Hamilton-Jacobi theory as k=1, and enlarges application range.
Develops k-contact geometry theory for field theories.
problem Analyse field theories using k-contact geometry.
method Distributions maximally non-integrable with k commuting Lie symmetries.
result Established k-contact distributions and their relationships.
Defines new canonical lifts for field theories, analyzing Klein-Gordon, Polyakov string, and Einstein-Cartan gravity.
problem Analyzing natural Noether symmetries and conserved quantities in field theories.
method Defining canonical lifts to study field theories and applying Noether's theorem.
result New geometrical interpretation of Virasoro constraint in string theory.
The main purpose in the present paper is to build a Hamiltonian theory for fields which is consistent with the principles of relativity. For this we consider detailed geometric pictures of Lepage theories in the spirit of Dedecker and try to stress out the interplay between the Lepage-Dedecker (LP) description and the …
Canonical structure of the space-time symmetric analogue of the Hamiltonian formalism in field theory based on the De Donder-Weyl (DW) theory is studied. In n space-time dimensions the set of n polymomenta is associated to the space-time derivatives of field variables. The polysymplectic (n+1)-form generalizes th…
The polysymplectic (n+1)-form is introduced as an analogue of the symplectic form for the De Donder-Weyl polymomentum Hamiltonian formulation of field theory. The corresponding Poisson brackets on differential forms are constructed. The analogues of the Poisson algebra are shown to be generalized (non-commutative and…
The aim of this paper is to present the main geometrical objects on the dual 1-jet bundle J1∗(T,M) (this is the polymomentum phase space of the De Donder-Weyl covariant Hamiltonian formulation of field theory) that characterize our approach of multi-time Hamilton geometry. In this direction, we firstly intro…
Introduces a new phase space for 2D supersymmetric sigma models.
problem Developing a new Hamiltonian formulation for 2D supersymmetric sigma models.
method Introduces a phase space with spinorial momenta and derives a covariant Hamiltonian formulation.
result Shows the existence of additional supersymmetries in the new formulation.
Establishes Poincaré's lemma for formal manifolds.
problem Developing smooth relative Lie algebra homologies and cohomologies.
method Theory of formal manifolds and formal Lie groups.
result Poincaré's lemma for de Rham complexes with formal functions and generalized functions.
Foundations laid for formal manifolds in differential geometry.
problem No specific problem stated; focuses on formal manifolds.
method Introducing formal manifolds, developing their theory, and proving finite products.
result Established a fully faithful contravariant functor and finite products in the category of formal manifolds.
Explores local structure of morphisms and formal submanifolds in formal manifolds theory.
problem Understanding the local structure of morphisms and formal submanifolds in formal manifolds.
method Study of formal manifolds, including local structure of constant rank morphisms and formal submanifolds.
result Developed the local structure of constant rank morphisms and formal submanifolds.
Study non-formal pseudo-differential operators over formal ones.
problem Understanding structure of non-formal pseudo-differential operators.
method Diffeological principal bundles, smoothing connections.
result Structure of diffeological bundle of non-formal pseudo-differential operators over formal ones.
The isotropy action on certain symmetric spaces is shown to be equivariantly formal.
problem Understanding the equivariant formality of isotropy actions on symmetric spaces.
method Developed a new approach to prove equivariant formality for (Z2⊕Z2)-symmetric spaces. result Symmetric spaces with (Z2⊕Z2)-symmetry are equivariantly formal and formal in the Sullivan sense. Strong formal properties for toric and homogeneous Kähler manifolds.
problem Understanding formal properties of Kähler manifolds.
method Analyzing rationally and strongly formal properties of toric and homogeneous Kähler manifolds.
result Toric and homogeneous Kähler manifolds are both rationally and strongly formal.
A metric is formal if all products of harmonic forms are again harmonic. The existence of a formal metric implies Sullivan formality of the manifold, and hence formal metrics can exist only in presence of a very restricted topology. We show that a warped product metric is formal if and only if the warping function is c…
The study shows strong formality in certain complex manifolds.
problem Investigating strong formality in complex manifolds.
method Adapting s-strong formality from Fernandez and Muñoz to the pluripotential setting. result Compact Kähler manifolds and generalized complete intersections are strongly formal.
Formal manifolds with non-negative Ricci curvature have formal covers.
problem Formality of manifolds with non-negative Ricci curvature.
method Study of universal covers and formal properties.
result Closed non-orientable manifolds with non-negative Ricci curvature are formal.
New hierarchies derived from KP hierarchy using non-formal operators and Yang-Mills action.
problem Formal solutions of KP hierarchy and their non-formal counterparts.
method Developed new hierarchies of non-linear equations on non-formal pseudo-differential operators.
result Expressed one hierarchy as Yang-Mills action minimization.
Extended equivariant BV formalism to manifolds with boundaries.
problem Handling manifolds with boundaries in equivariant BV formalism.
method Extension of AKSZ theories to manifolds with boundaries.
result Successfully applied to manifolds with boundaries.
We define the notion of a formal connection for a smooth family of star products with fixed underlying symplectic structure. Such a formal connection allows one to relate star products at different points in the family. This generalizes the formal Hitchin connection introduced by the first author. We establish a necess…
Defines formal exponentials for graded manifolds and linearizes QP-manifolds.
problem Formal exponentials and linearizations of QP-manifolds.
method Definition of formal exponential maps, Grothendieck connections, and connections on tangent bundles.
result Linearizes QP-manifolds at points, giving formal tangent spaces L∞-algebra structures. Formality of Dolbeault DGAs on complex nilmanifolds restricted to tori.
problem Determining when Dolbeault DGAs on complex nilmanifolds are formal.
method Using quasi-isomorphisms and cohomology connections, proving formality conditions.
result Formality of Dolbeault DGAs on complex nilmanifolds is restricted to tori.
Study on geometrically formal metrics on complex manifolds.
problem Existence and properties of geometrically formal metrics on complex manifolds.
method Topological and cohomological obstructions, detailed analysis for specific manifolds, and metric constructions.
result Existence and non-existence conditions for geometrically formal metrics on various complex manifolds.
Research on formality problem for special holonomy manifolds.
problem Formality problem for manifolds with special holonomy.
method Using intersection Massey products to establish formality.
result Recent results on formality of Joyce's G_2-manifolds.
Defines formal vertex laws related to Lie conformal algebras.
problem No specific problem stated; focuses on definitions and proofs.
method Definitions and proofs of vertex/conformal versions of classical Lie theory results.
result Proves vertex/conformal versions of important Lie theory results.
Deform quantization recovers scalar curvature in complex structures.
problem Recovering scalar curvature in complex structures.
method Formal moment map construction on almost complex structures.
result Formal moment map deforms scalar curvature moment map in integrable cases.
Formal methods verify continuous auctions at exchanges.
problem Ensuring fairness and correctness in continuous auctions.
method Formal specification, design, and verification of continuous double auctions.
result A verified algorithm satisfies natural properties of auctions.
We describe a canonical form for linear differential operators that are formally self-adjoint or formally skew-adjoint.
There are solved standard problems related to Formal (Holomorphic) Segre preserving Mappings of non-trivial Real-Formal Hypersurfaces in C2.
Derives localization formulas in Batalin-Vilkovisky formalism.
problem Localization in Batalin-Vilkovisky formalism.
method Equivariant localization formulas in Batalin-Vilkovisky formalism.
result Derives localization formulas in Batalin-Vilkovisky formalism.
Proves formal self-adjointness of certain differential operators.
problem Verifying conjectures about differential operators.
method Proving formal self-adjointness through mathematical proof.
result Proves two conjectures about differential operators.
New findings on complex manifold properties under deformations.
problem Properties of Dolbeault and Bott-Chern formalities are not preserved under holomorphic deformations.
method Construction of a complex manifold to demonstrate non-preservation of properties.
result Existence of a manifold satisfying ∂∂-lemma but with non-vanishing Aeppli-Bott-Chern-Massey product. A Riemannian manifold is called geometrically formal if the wedge product of harmonic forms is again harmonic, which implies in the compact case that the manifold is topologically formal in the sense of rational homotopy theory. A manifold admitting a Riemannian metric of positive sectional curvature is conjectured to …
We prove that for a fibration of simply-connected spaces of finite type F↪E→B with F being positively elliptic and $H^*(F,\qq)$ not possessing non-trivial derivations of negative degree, the base B is formal if and only if the total space E is formal. Moreover, in this case the fibration map i…
This paper formalizes manifolds in positive characteristic varieties.
problem Establishing l-adic formal manifold structures on positive characteristic varieties.
method Develops and proves the existence of l-adic formal manifold structures and abelianized Galois symmetries.
result Proves l-adic homotopic equivalence and l-local lifting for simply-connected varieties.
In this paper, we study the formal solution space of a nonlinear PDE in a fiber bundle. To this end, we start with foundational material and introduce the notion of a pfd structure to build up a new concept of profinite dimensional manifolds. We show that the infinite jet space of the fiber bundle is a profinite dimens…
This paper explores formal verification for autonomous systems, identifying limitations and proposing improvements.
problem Ensuring safety of autonomous systems like self-driving cars and drones.
method Formal verification techniques based on formal methods, analyzing three assumptions and their limitations.
result Preliminary work to improve the strength of evidence provided by formal verification.
We investigate harmonic forms of geometrically formal metrics, which are defined as those having the exterior product of any two harmonic forms still harmonic. We prove that a formal Sasakian metric can exist only on a real cohomology sphere and that holomorphic forms of a formal Kähler metric are parallel w.r.t. the L…
We discuss the question of geometric formality for rationally elliptic manifolds of dimension 6 and 7. We prove that a geometrically formal six-dimensional biquotient with b2=3 has the real cohomology of a symmetric space. We also show that a rationally hyperbolic six-dimensional manifold with b2≤2 and …
We recall the construction of non-formal deformation quantization of the Poincare Group ISO(1,1) on its coadjoint orbit and exhibit the associated non-formal star-exponentials.
An action of a compact Lie group is called equivariantly formal, if the Leray--Serre spectral sequence of its Borel fibration degenerates at the E_2-term. This term is as prominent as it is restrictive. In this article, also motivated by the lack of junction between the notion of equivariant formality and the concept o…
Paper formalizes multi-dimensional FSD using geometric methods.
problem Complex measure theory and calculus barriers to formalization in proof assistants.
method Geometric framework for first-order stochastic dominance in N dimensions.
result Geometric approach bypasses complex integration theory for direct comparison of survival probabilities.