Characterizes winding of braided vector fields in tubular domains.
problem Understanding the topology of braided vector fields in complex domains.
method Defines field line winding as a measure of entanglement, proving its uniqueness in classifying vector field topology.
result Field line winding uniquely classifies the topology of braided vector fields.
The paper examines asymptotic lines of plane fields in 3D space.
problem Qualitative properties of asymptotic lines in plane fields.
method Analysis of null directions and Gaussian curvature.
result Asymptotic lines coincide with classical ones in completely integrable fields.
Generic singularities of line fields have been studied for lines of principal curvature of embedded surfaces. In this paper we propose an approach to classify generic singularities of general line fields on 2D manifolds. The idea is to identify line fields as bisectors of pairs of vector fields on the manifold, with re…
Finite type curves are asymptotic lines of plane fields, with examples.
problem Characterizing finite type curves as asymptotic lines of plane fields.
method Proving finite type curves are asymptotic lines of suitable plane fields, providing examples.
result Generalizations of V. Arnold's results for plane fields.
Discrete line fields on surfaces generalize vector fields and model curvature dynamics.
problem Modeling geometric and physical properties on surfaces using line fields.
method Discretization of Morse-Smale line fields on surfaces, defining critical elements and their indices.
result Euler theorem and homotopy type characterization hold for discrete line fields.
New framework for analyzing line fields on surfaces, proving stability under specific conditions.
problem Understanding structural stability and generic transitions of line fields on surfaces.
method Developed a new topological framework and introduced representations of complete invariants for line fields and their transitions.
result Line fields with 1-prong and 3-prong singularities are generic under an incompressibility condition.
Study curvature lines of a vector field on surfaces.
problem Behavior of curvature lines at umbilical points.
method Analyzes transversal eqüiaffine vector fields on surfaces.
result Behavior of curvature lines at isolated umbilical points.
Study on complex line fields on almost-complex manifolds, proving existence conditions.
problem Existence of linearly independent complex line fields on almost-complex manifolds.
method Prove necessary and sufficient conditions for the existence of one, two, or three fields over certain manifolds.
result Necessary and sufficient condition for the existence of complex line fields over certain manifolds.
A Poincaré-Hopf Theorem for line fields with point singularities on orientable surfaces can be found Hopf's 1956 Lecture Notes on Differential Geometry. In 1955 Markus presented such a theorem in all dimensions, but Markus' statement only holds in even dimensions 2k≥4. In 1984 Jänich presented a Poincaré-Hopf th…
Floer constructs homology from flow lines in generalized dynamical systems and combinatorial vector fields.
problem Computing homology in discrete and smooth dynamical systems.
method Counting flow lines between orbits and critical points.
result Directly recovers Z2 homology from flow lines. Classifies singularities of smooth vector fields on the line.
problem Classifying singularities of smooth vector fields on the line.
method Local classification with respect to C1-conjugacy, including normal forms and unfoldings. result Complete description of the 1-d case achieved.
Stable knots and links can exist in electromagnetic fields.
problem Stability of knots and links in electromagnetic fields.
method Proving the existence of electromagnetic fields preserving link topology.
result Every link can be realized as stable field lines in electromagnetic fields.
Inverts rank m symmetric tensor fields using line integrals.
problem Recovering symmetric tensor fields from line integrals.
method Computes normal operator and presents inversion formula.
result Recovering rank m tensor fields from data (Nm0f,…,Nmmf). A new method interprets astrophysical spectra using geometric paths to distinguish line profiles.
problem Tackling the indistinguishability of spectral line profiles under scalar summaries.
method Introduces a geometric representation of line profiles using rough path theory, mapping profiles to a common velocity grid and defining descriptors from path properties.
result Compact descriptors separate morphologies with similar scalar summaries, revealing ordered line structures.
Electromagnetic fields can be knotted and stable, linked to quasipositive links.
problem Understanding the topology of electromagnetic fields and their stability.
method Constructing complex algebraic plane curves to create quasipositive links containing the original link.
result Electromagnetic fields can be knotted and stable, corresponding to Legendrian knots.
Formula derived for a magnetic line invariant.
problem Solving MHD problems with two-scale mean fields.
method Combinatorial definition of M3 invariant for three-component links. result Proven formula for M invariant verified for simple cases. We call two Engel structures isotopic if they are homotopic through Engel structures by a homotopy that fixes the characteristic line field. In the present paper we define an isotopy invariant of Engel structures on oriented circle bundles over closed oriented three-manifolds and apply it to give an isotopy classificat…
The paper simplifies proofs and characterizes contact structures in 3D.
problem Contact structures induced by geodesic vector fields in 3D.
method New proofs and characterizations of contact structures.
result Contact structures in 3D are universally tight under certain conditions.
In this paper, we compute the derivatives of the line segment energy for a symmetric tensor field and apply them to obtain slightly more general log-concavity estimates for positive solutions of heat equations and first eigenfunctions on bounded strictly convex domains.
Novel multisymplectic framework for pseudo-Fueter curves in Hamiltonian field theory.
problem Generalizing Floer theory to multisymplectic geometry.
method Introducing pseudo-Fueter curves in a compatible almost hyperkähler structure.
result Gradient lines of multisymplectic action functional are pseudo-Fueter curves.
Study weak geodesic lines in Kähler metric space, disproving a conjecture.
problem Disproving a conjecture about weak geodesic lines in Kähler metrics.
method Establish Ross-Witt Nyström correspondence, construct weak geodesic lines.
result Some weak geodesic lines are smooth, disproving a popular conjecture.
This paper explores ML in power line communications, from modeling to diagnostics.
problem Improving efficiency and diagnostics in power line communications.
method Discusses classical ML models and their application to PLC at various layers.
result Demonstrates how ML can enhance various aspects of PLC.
We recall the definition of the quadratic helicity invariant and of the higher asymptotic ergodic M-invariant. We present a simpler new proof (in part) that the M-invariant is ergodic. The M-invariant is a higher invariant, this means that for the magnetic field with closed magnetic lines the invariant is not a f…
The geodesic flow on the tangent bundle is the flow of a certain vector field which is called the spray S:TM→TTM. The flow lines of the vector field $\ka_{TM}øTS:TTM\to TTTM$ project to the Jacobi fields on TM. This could be called the Jacobi flow.
Holomorphic vector fields and anti-canonical divisors on complex manifolds are studied.
problem Existence of non-trivial holomorphic vector fields on compact complex manifolds.
method Vanishing result for measure preserving holomorphic vector fields, Gibbs stability, and log terminal singularities.
result No non-trivial holomorphic vector fields on compact complex manifolds with big anti-canonical line bundle.
We first describe the numerical invariants attached to the second fundamental form of a spacelike surface in four-dimensional Minkowski space. We then study the configuration of the nu-principal curvature lines on a spacelike surface, when the normal field nu is lightlike (the lightcone configuration). Some observation…
In this paper, on the first, we prove Δr=2H where Δ is the Laplacian operator, r=(r1,r2,r3) the position vector field and H is the mean curvature vector field of a surface S in the 3-dimensional Heisenberg group H3. In the second, we classify the ruled surfaces by straight…
Proves existence of Killing fields in smooth spacetimes with compact Cauchy horizons.
problem Existence of Killing fields in smooth spacetimes with compact Cauchy horizons.
method Normalized surface gravity of compact non-degenerate Cauchy horizons in smooth vacuum spacetimes.
result Proves the Isenberg-Moncrief conjecture on the existence of Killing fields.
Study finds conditions for minimum of Donaldson's functional on Higgs bundles with non-holomorphic Higgs fields.
problem Finding conditions for minimum of Donaldson's functional on Higgs bundles with non-holomorphic Higgs fields.
method Restriction of Donaldson's functional to diagonal metrics on Higgs bundles with non-holomorphic Higgs fields.
result Provides necessary and sufficient conditions for the functional to attain a minimum.
Here are described the axiumbilic points that appear in generic one parameter families of surfaces immersed in R4. At these points the ellipse of curvature of the immersion, Little, Garcia - Sotomayor has equal axes. A review is made on the basic preliminaries on axial curvature lines and the associated axiumbilic poin…
We construct a 2+1 dimensional classical gauge theory on manifolds with spin structure whose action is a refinement of the Atiyah-Patodi- Singer eta-invariant for twisted Dirac operators. We investigate the properties of the Lagrangian field theory for closed, spun 3-manifolds and compact, spun 3-manifolds with boundar…
The paper explores a B-field transform of complex structures on complex tori.
problem Deforming complex structures on complex tori using B-field transformations.
method Constructing holomorphic line bundles with integrable connections and interpreting them as deformations of complex tori by flat gerbes.
result Homological mirror symmetry between deformed and original complex tori.
An NQ-manifold is a non-negatively graded supermanifold with a degree 1 homological vector field. The focus of this paper is to define the Wilson loops/lines in the context of NQ-manifolds and to study their properties. The Wilson loops/lines, which give the holonomy or parallel transport, are familiar objects in usual…
The paper studies flow lines on Higgs bundle moduli spaces, classifying them via secant varieties.
problem Classifying flow lines on moduli spaces of Higgs bundles.
method Gradient flow lines for L2 norm of Higgs field, Morse-theoretic compactification, secant varieties. result Flow lines have an algebro-geometric classification via secant varieties.
We study real lines on certain Moishezon threefolds which are potentially twistor spaces of 3CP^2. Here, line means a smooth rational curve whose normal bundle is O(1)^2 and the reality implies the invariance under an anti-holomorphic involution on the threefolds. Our threefolds are birational to double coverings of CP…
New method uses resurgent analysis to determine growth rate of quantum field theory coefficients.
problem Determining the growth rate of quantum field theory coefficients.
method Resurgence analysis on the Stokes line, leading to transseries decomposition and continued across natural boundary.
result Essential exponent of growth has Cardy-like interpretation as effective central charge.
In this technical note we give a purely geometric understanding of discrete torsion, as an analogue of orbifold Wilson lines for two-form tensor field potentials. In order to introduce discrete torsion in this context, we describe gerbes and the description of certain type II supergravity tensor field potentials as con…
Study families of flat connections with nilpotent Higgs fields, showing similar monodromy to regular Higgs bundles.
problem Investigate Cimes-families of flat connections with nilpotent Higgs fields. method Analyze families of flat connections including real twistor lines and conformal limits, deducing monodromy similarities.
result Traces of holonomies are asymptotically exponential in rational powers of the parameter of the family.
Kauffman knot polynomial invariants are discovered in classical abelian Chern-Simons field theory. A topological invariant tI(L) is constructed for a link L, where I is the abelian Chern-Simons action and t a formal constant. For oriented knotted vortex lines, tI satisf…
A venerable problem in combinatorics and geometry asks whether a given incidence relation may be realized by a configuration of points and lines. The classic version of this would ask for algebraic lines over some field or possibly real pseudolines: embedded circles (isotopic to RP1) in the real projective plane. In…
Study eigenvalues of Bochner Laplacian on symplectic manifolds.
problem Understanding low-lying eigenvalues of Bochner Laplacian on symplectic manifolds.
method Analyzes high tensor powers of positive line bundles on symplectic manifolds.
result Asymptotic expansions for low-lying eigenvalues.
Contact structures are induced by nondegenerate skew fibrations of R^3.
problem Understanding contact structures induced by skew fibrations of R^3.
method Characterizing nondegenerate fibrations and using a recent result on tightness in contact metric 3-manifolds.
result The plane field induced by any nondegenerate fibration is a tight contact structure.
Abstract study shows Milnor metric from BF theory partition function.
problem Milnor metric on manifold determinant line.
method Batalin--Vilkovisky formalism, BV pushforward to cohomology.
result Recovery of Milnor metric from BF theory partition function. LA-MOKA computes braid monodromy for line arrangements over complex projective plane.
problem Computing braid monodromy for algebraic curves in P2. method Discretization of continuous geometry into a discrete combinatorial structure.
result Explicit conditions ensure topological correctness of computed braid monodromy.
Extends RT TQFT to include surface defects and line defects.
problem Distinguishing non-isotopic embeddings of surfaces in 3D bordisms.
method Uses modular tensor categories, symmetric Frobenius algebras, and equivariant multi-modules.
result Invariant distinguishes non-isotopic embeddings of 2-tori but not 2-spheres.
We analyse the asymptotical growth of Vassiliev invariants on non-periodic flow lines of ergodic vector fields on domains of R3. More precisely, we show that the asymptotics of Vassiliev invariants is completely determined by the helicity of the vector field. As an application, we determine the asymptotic Alexander…
Explains a property of algebras related to quantum field theories.
problem Explains a property of algebras encoding line defects in quantum field theories.
method Physical explanation of a property of quantized algebras using dualities and field theories.
result Physical explanation of a large center in quantized algebras when the deformation parameter is a root of unity.
New invariant detects non-homeomorphic arrangements with similar coefficients.
problem Detecting non-homeomorphic arrangements with similar coefficients.
method Loop linking number, braid monodromy, and Rybnikov's arrangements.
result Fundamental groups of complements are not isomorphic.