arXiv research

Study how large-scale flows align small-scale vortices in 3D Euler equations.

This article introduces proximal planar vortex 1-cycles, resembling the structure of vortex atoms introduced by William Thomson (Lord Kelvin) in 1867 and recent work on the proximity of sets that overlap either spatially or descriptively. Vortex cycles resemble Thomson's model of a vortex atom, inspired by P.G. Tait's …

We present a Hamiltonian framework for higher-dimensional vortex filaments (or membranes) and vortex sheets as singular 2-forms with support of codimensions 2 and 1, respectively, i.e. singular elements of the dual to the Lie algebra of divergence-free vector fields. It turns out that the localized induction approximat…

We demonstrate that the five vortex equations recently introduced by Manton ariseas symmetry reductions of the anti-self-dual Yang--Mills equations in four dimensions. In particular the Jackiw--Pi vortex and the Ambjørn--Olesen vortex correspond to the gauge group $SU(1, 1)$, and respectively the Euclidean or the $SU(2…

Study vortex loops as coadjoint orbits of diffeomorphisms.

The paper proves estimates for vortex-type equations on compact Riemann surfaces.

Vortex solutions on flat surfaces map to harmonic spinors on Nappi-Witten space.

This article introduces vortex nerve complexes in CW (Closure finite Weak) topological spaces, which first appeared in works by P. Alexandroff, H. Hopf and J.H.C. Whitehead during the 1930s. A vortex nerve is a CW complex containing one or more intersecting path-connected cycles. Each vortex nerve has its own distincti…

New minimal surfaces found from vortex crystals.

The paper studies point vortex dynamics on specific Kähler twistor spaces.

Solved Demailly systems for Vortex bundles on manifolds.

Let L --> X be a complex line bundle over a compact connected Riemann surface. We consider the abelian vortex equations on L when the metric on the surface has finitely many point degeneracies or conical singularities and the line bundle has parabolic structure. These conditions appear naturally in the study of vortex …

Alternative construction of quasi-Fuchsian flows using vortex equations.

Paper solves vortex equations on complex surfaces, linking to Higgs bundle stability.

New stretch maps minimize distortion in geometric group theory.

We obtain a Hitchin-Kobayashi-type correspondence for symplectic vortex equations, with the target a Kahler cone over a compact Sasakian manifold. We show that the correspondence reduces to studying the existence and uniqueness of Kazdan-Warner equations. Using this, we construct a map between the moduli space of solut…

Theory of point vortices extended to closed surfaces.

Topologically protected vortex knots and links are proposed and proven.

Researchers create topologically protected knots in a realizable system.

Finsler metrics with relatively non-negative (non-positive, respectively), constant and isotropic stretch curvatures are investigated in this paper. In particular, it is proved that every non-Riemannian $(α, β)$-metric with a nonzero constant flag curvature and a non-zero relatively isotropic stretch curvature over a m…

In this note we quantize the usual symplectic (Kähler) form on the vortex moduli space by modifying the Quillen metric of the Quillen determinant line bundle.

Every weak Perron number is realized as a stretch factor of a homeomorphism on a surface.

Lower bound on stretch factor for periodic maps.

We consider the vortex equations for a U(n) gauge field coupled to a Higgs field with values on the n times n square matrices. It is known that when these equations are defined on a compact Riemann surface, their moduli space of solutions is closely related to a moduli space of tau-stable holomorphic n-pairs on that su…

The Seiberg-Witten equations are defined on certain complex line bundles over smooth oriented four manifolds. When the base manifold is a complex Kahler surface, the Seiberg-Witten equations are essentially the Abelian vortex equations. Using known non-abelian generalizations of the vortex equations as a guide, we expl…

In 1974, Thurston proved that, up to isotopy, every automorphism of closed orientable surface is either periodic, reducible, or pseudo-Anosov. The latter case has lead to a rich theory with applications ranging from dynamical systems to low dimensional topology. Associated with every pseudo-Anosov map is a real number …

This article introduces an application of Ghrist barcodes in the study of persistent Betti numbers derived from vortex nerve complexes found in triangulations of video frames. A Ghrist barcode is a topology of data pictograph useful in representing the persistence of the features of changing shapes. The basic approach …

3D gauge theories link knot polynomials to vortex partition functions.

Paper shows existence of vortex solutions with specific decay properties.

Study stretch laminations in hyperbolic 3-manifolds via circle-valued maps.

The Teichmüller space $\mathcal{T}(Σ)$ of a surface $Σ$ is equipped with Thurston's asymmetric metric. Stretch lines are oriented geodesics for this metric on $\mathcal{T}(Σ)$. We give the asymptotic behavior of the lengths of the measured geodesic laminations as one follows a stretch line in the positive direction.

Study finds how periodic surfaces can bend without stretching.

Minimal stretch factor for non-orientable surfaces is small.

The paper studies maximal stretch and Lipschitz maps on negatively curved manifolds.

We derive an analytic formula for the hydrodynamic Green function and the Robin function on every orientable surface admitting a hydrodynamic Killing vector field. Closed-form expressions are provided for all fourteen canonical Riemann surfaces, covering both compact and non-compact cases; the formulae satisfy the slip…

Quadratic-time algorithm computes stretch factors and foliations for pseudo-Anosov mapping classes.

We explicitly construct pseudo-Anosov maps on the closed surface of genus $g$ with orientable foliations whose stretch factor $λ$ is a Salem number with algebraic degree $2g$. Using this result, we show that there is a pseudo-Anosov map whose stretch factor has algebraic degree $d$, for each positive even integer $d$ s…

We consider the self-dual vortex equations on a positive line bundle L --> M over a compact Kaehler manifold of arbitrary dimension. When M is simply connected, the moduli space of vortex solutions is a projective space. When M is an abelian variety, the moduli space is the projectivization of the Fourier-Mukai transfo…

Topology of vortex reconnection shows how knots transform.

The theory of the vortex filament in three-dimensional fluid dynamics, consisting mainly of the models up to the third-order approximation, is an attractive subject in both physics and mathematics. Many efforts have been devoted to the extension of the theory to higher-dimensional symmetric Lie algebras. However, such …

Giving a new form of the vortex mode equation by a proper change of parameter, our aim is to analyze the point and contact symmetries of the new equation. Fundamental invariants and a form of general solutions of point transformations along with some specific examples are also derived.

Extending work of Caffarelli-Yang and Tarantello, we present a variational existence proof for two-vortex solutions of the periodic Chern-Simons Higgs model and analyze the asymptotic behavior of these solutions as the parameter coupling the gauge field with the scalar field tends to 0.

New metrics help predict Brownian motion on surfaces and higher dimensions.

Minimal stretch factors for certain pseudo-Anosov maps are bounded.

Study on solvable Lie groups with specific Weyl connections.

In this paper, we study the Galois conjugates of stretch factors of pseudo-Anosov elements of the mapping class group of a surface. We show that - except in low-complexity cases - these conjugates are dense in the complex plane. For this, we use Penner's construction of pseudo-Anosov mapping classes. As a consequence, …

It is known that given a stable holomorphic pair $(E ,φ)$, where $E$ is a holomorphic vector bundle on a compact Kähler manifold $X$ and $φ$ is a holomorphic section of $E$, the vector bundle $E$ admits a Hermitian metric solving the vortex equation. We generalize this to pairs $(\E ,φ)$, where $\E$ is a reflexive shea…

Invariants for colored links found, with topological protection of certain knots.