Formalism for superfield theory problems via Poincaré-Cartan form.
problem Formalism for first-order Berezinian variational problems in superfield theory.
method Intrinsic description of Hamilton-Cartan formalism through Poincaré-Cartan form.
result Noether theorem and examples from superfield theory and supermechanics discussed.
We study the fractional gravity for spacetimes with non-integer dimensions. Our constructions are based on a geometric formalism with the fractional Caputo derivative and integral calculus adapted to nonolonomic distributions. This allows us to define a fractional spacetime geometry with fundamental geometric/physical …
Let C→M be the bundle of connections of a principal bundle on M. The solutions to Hamilton-Cartan equations for a gauge-invariant Lagrangian density Λ on C satisfying a weak condition of regularity, are shown to admit an affine fibre-bundle structure over the set of solutions to Euler-Lagrange equations for …
This paper develops a Hamiltonian reduction method for field theories over affine principal bundles.
problem Developing a Hamiltonian reduction theory for field theories over affine principal bundles.
method Introducing a canonical identification to describe the reduced multisymplectic space without a connection.
result Derivation of reduced Hamilton-Cartan equations and a reduced covariant bracket.
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.
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…
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.
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.
The paper examines geometric formality on specific surfaces under the Chern-Ricci flow.
problem Understanding geometric formality on class VII surfaces.
method Analysis of Chern-Ricci flow on specific surfaces.
result Evolution of geometric formality under the Chern-Ricci flow.
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.
New equivariant formality concepts solve the toral rank conjecture.
problem Toral rank conjecture and equivariant formality of actions.
method Rational homotopy theory, Hirsch-Brown models, A-infinity algebras.
result Actions with new properties satisfy the toral rank conjecture.
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.
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.
Surveying fairness formalizations in machine learning for prediction.
problem Preventing algorithmic discrimination in critical decisions.
method Survey and critique of existing fairness formalizations from machine learning and social sciences.
result Limitations of existing fairness formalizations and suggestions for new ones.
Study formal properties of SU(2) and SO(3) bundles over Wolf spaces.
problem Formality of total spaces of SU(2) and SO(3) bundles over Wolf spaces.
method Formality analysis using symmetric positive quaternionic Kähler manifolds.
result All 3-Sasakian homogeneous spaces are formal, but some Wolf spaces are not.
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 …
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.
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.
Introduces formal frames for manifolds and their properties.
problem Understanding and generalizing frames and connections on manifolds.
method Introduces formal frames, canonical forms, and torsions.
result Equivalence of vanishing torsions to realizability of formal frames as ordinary frames.
New infinite family of hyperbolic L-space knots with specific semigroups.
problem Characterizing semigroups of L-space knots.
method Defined formal semigroups from Alexander polynomials and analyzed hyperbolic knots.
result Found an infinite family of hyperbolic L-space knots with semigroups generated by five elements.
Models close to formal for Vaisman and quasi-Sasakian manifolds, including nilmanifolds.
problem Formality of quasi-Sasakian and Vaisman manifolds.
method Providing models that are as close as possible to being formal for compact manifolds with transversely Kaehler structures.
result Classification of corresponding nilmanifolds.
We generalize the Lagrangian-Hamiltonian formalism of Skinner and Rusk to higher order field theories on fiber bundles. As a byproduct we solve the long standing problem of defining, in a coordinate free manner, a Hamiltonian formalism for higher order Lagrangian field theories. Namely, our formalism does only depend o…
Formal Normal Form created for special CR singularities.
problem CR singularities in complex manifolds.
method Formal Normal Form construction for real-smooth submanifolds.
result A new Formal Normal Form established.
Formally proves machine learning for simple classifiers.
problem Proving PAC learnability for decision stumps.
method Formal proof in Lean, separating deterministic and probabilistic proofs.
result Formal proof of PAC learnability for decision stumps.
The article confirms Joyce's examples of G2-holonomy are formal spaces.
problem Formality of special holonomy manifolds and nearly Kähler manifolds.
method Analyzing rational homotopy theory and cohomology algebra.
result Joyce's examples of G2-holonomy are formal spaces.
Several large classes of homogeneous spaces are known to be formal---in the sense of Rational Homotopy Theory. However, it seems that far fewer examples of non-formal homogeneous spaces are known. In this article we provide several construction principles and characterisations for non-formal homogeneous spaces, which w…