Study on unique vortex equations and their geometric implications.
problem Uniqueness of vortex equations involving entire functions.
method Analyzing entire functions on the complex plane and showing geometric applications.
result Uniqueness of harmonic maps and affine spherical immersions with polynomial differential constraints, but failure for non-polynomial entire functions.
Analytic formulae for point vortex dynamics on surfaces with symmetry.
problem Understanding point vortex behavior on surfaces with continuous symmetry.
method Derived analytic formulae for Green and Robin functions on surfaces with hydrodynamic Killing vector fields.
result Unified tool for detailed studies of point vortex dynamics and Euler-Arnold flows on surfaces with continuous symmetry.
The paper studies point vortex dynamics on specific Kähler twistor spaces.
problem Point vortex dynamics on Kähler twistor spaces.
method Explicit expression for Green's function, Hamiltonian determination, momentum map calculation.
result Explicit equations of motion and momentum map for point vortex dynamics.
3D gauge theories link knot polynomials to vortex partition functions.
problem Connecting knot polynomials to gauge theory partition functions.
method Construct 3D N=2 abelian gauge theories on S2imesS1. result Colored Jones polynomials derived from vortex partition functions.
The paper introduces vortex cycles and nerves, inspired by Thomson's vortex atoms.
problem Understanding vortex structures and their homology.
method Introducing and analyzing non-concentric, nesting vortex cycles and nerves.
result Whitehead CW topology and Leader uniform topology outcomes of vortex cycles.
Manton's five vortex equations derived from Yang-Mills self-duality.
problem Deriving vortex equations from Yang-Mills theory.
method Symmetry reductions of anti-self-dual Yang-Mills equations.
result Vortex equations of different types can be superposed.
Study of vortex equations' limits on growing surfaces.
problem Understanding vortex equations' behavior as surface volume increases.
method Prior estimates for Kazdan-Warner equations, relating to previous work.
result Solutions converge smoothly away from finitely many points.
Paper shows existence of vortex solutions with specific decay properties.
problem Existence of solutions to Seiberg-Witten equations with specific decay properties.
method Dimensional reduction of Seiberg-Witten equations on the plane.
result Contains both exponentially decayed and polynomial growth solutions.
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…
The paper introduces vortex nerve complexes and new Betti numbers in CW spaces.
problem Understanding the structure and properties of CW complexes and their nerves.
method Introducing vortex nerve complexes and defining new Betti numbers for CW complexes.
result New Betti numbers (vortex Bvtex, vortex nerve BvNrv, shape Bsh) are introduced and studied. New metrics help predict Brownian motion on surfaces and higher dimensions.
problem Predicting Brownian motion on complex surfaces and higher dimensions.
method Developed new metrics (Uniform Drainage Metric) for surfaces and higher dimensions.
result Uniform Drainage Metric predicts Brownian motion's narrow escape time consistently.
The paper proves estimates for vortex-type equations on compact Riemann surfaces.
problem Estimating vortex-type equations on compact Riemann surfaces.
method Proves \emph{a priori} estimates for vortex-type equations.
result Recover existing estimates for vortex bundle Monge-Ampère equation, prove existence and uniqueness for Calabi-Yang-Mills equations, and get estimates for J−vortex equation. Study vortex loops as coadjoint orbits of diffeomorphisms.
problem Understanding vortex loops in terms of coadjoint orbits.
method Analyzing vortex loops as coadjoint orbits of area-preserving diffeomorphisms.
result Vortex loops are coadjoint orbits of the diffeomorphism group.
Symplectic vortex equations link Sasakian manifolds to Kahler cones.
problem Existence and uniqueness of symplectic vortex solutions.
method Hitchin-Kobayashi correspondence and Kazdan-Warner equations.
result Construction of a map between vortex solutions and effective divisors.
Vortex solutions on flat surfaces map to harmonic spinors on Nappi-Witten space.
problem Constructing Abelian magnetic zero-modes on flat spacetime.
method Establishing a correspondence between vortex equations and harmonic spinors on the Nappi-Witten space.
result Explicit solutions of a twisted Dirac equation induce harmonic spinors on Minkowski space.
The article extends vortex filament theory to Hermitian reductive Lie algebras.
problem Investigating vortex filaments in Hermitian reductive Lie algebras.
method Develops vortex models in a purely geometric way.
result Vortex models revert to known models in Hermitian symmetric Lie algebras.
New minimal surfaces found from vortex crystals.
problem Minimal surfaces and vortex crystals.
method Gluing helicoids into minimal surfaces.
result New minimal surfaces and vortex crystals discovered.
Solved Demailly systems for Vortex bundles on manifolds.
problem Equivalence of Hartshorne ampleness and Griffiths positivity for vector bundles.
method Applied the continuity method to prove smooth solutions for the Vortex bundle.
result Smooth solutions exist for the proposed Demailly systems.
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 …
Study heat flow for gauged holomorphic maps over Kähler manifolds.
problem Understanding the moduli space of gauged holomorphic maps.
method Gradient flow approach inspired by Atiyah and Bott, heat flow method.
result Global existence of smooth solutions of gradient flow equations.
Alternative construction of quasi-Fuchsian flows using vortex equations.
problem Constructing quasi-Fuchsian flows.
method Using coupled vortex equations to construct quasi-Fuchsian flows as thermostats.
result Formulas for the marked length spectrum of quasi-Fuchsian flows.
Paper solves vortex equations on complex surfaces, linking to Higgs bundle stability.
problem Existence of solutions to doubly-coupled vortex equations on Riemann surfaces.
method Introduced doubly-coupled vortex equations and used Higgs bundle theory.
result Existence of solutions to vortex equations is equivalent to Higgs bundle stability.
Neural Networks improve incompressible flow simulations without complex kernels.
problem Simulating incompressible flows accurately and efficiently.
method Integrates Neural Networks with Random Vortex Dynamics for incompressible Navier-Stokes equations.
result Strictly enforces physical properties like incompressibility and boundary conditions.
Theory of point vortices extended to closed surfaces.
problem Extending point vortex dynamics to closed surfaces.
method Unified theory of point vortex dynamics on the plane, sphere, and closed surfaces.
result Comprehensive guide to point vortex dynamics on closed surfaces with genus zero and vanishing total vorticity.
Study how large-scale flows align small-scale vortices in 3D Euler equations.
problem Understanding how large-scale flows align small-scale vortices in 3D Euler equations.
method Constructing a Lagrangian coordinate to identify when the Lie bracket is zero and investigating the locality of the pressure term.
result Clarified conditions under which small-scale vortices are aligned by large-scale flows.
Topologically protected vortex knots and links are proposed and proven.
problem Decaying of tangled vortex structures through local reconnections and strand crossings.
method Proposed and proven topologically protected vortex structures using non-Abelian topological vortices.
result Existence of topologically protected Q8-colored links and classification using the Q-invariant. Researchers create topologically protected knots in a realizable system.
problem Creating topologically protected vortex knots in experimentally realizable systems.
method Investigated non-Abelian vortices in tetrahedral order in spin-2 Bose--Einstein condensates and bent-core nematic liquid crystals.
result Discovered the first topologically protected knots in an experimentally realizable system.
This paper uses Ghrist barcodes to track persistent shapes in video frames.
problem Detecting and tracking persistent shapes in video frames.
method Introduces Ghrist barcodes for persistent Betti numbers derived from vortex nerve complexes in triangulated video frames.
result Persistent Betti numbers of vortex nerves are k+2 for k edges. 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.
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…
Discrete model of curve deformation using discrete nonlinear Schrödinger equation.
problem Deformation of discrete space curves.
method Discrete analogue of the local induction equation using the discrete nonlinear Schrödinger equation.
result Explicit formulas for smooth and discrete curves in terms of τ functions of the two-component KP hierarchy.
The paper quantizes vortex moduli spaces on compact Kahler surfaces using determinant bundles.
problem Quantizing vortex moduli spaces on compact Kahler surfaces.
method Developed holomorphic determinant bundles and geometric quantization for vortex moduli spaces.
result Quantized vortex moduli spaces on compact Kahler surfaces using determinant bundles.
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…
Study vortex flows on Riemann surfaces, proving dominated splitting and Anosov properties.
problem Investigate flow properties on Riemann surfaces.
method Associate flow to vortex equations, investigate properties of flow.
result Show that flow always admits a dominated splitting and identify special cases of Anosov flow.
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.
problem Understanding how knotted vortices transform through reconnection.
method Using knot cobordism and Rasmussen's Invariant, the reconnection number of knots is calculated.
result The reconnection number of a positive knot is twice its Seifert genus.
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.
At critical coupling, the interactions of Ginzburg-Landau vortices are determined by the metric on the moduli space of static solutions. The asymptotic form of the metric for two well separated vortices is shown here to be expressible in terms of a Bessel function. A straightforward extension gives the metric for N vor…
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.
problem Finding invariants for colored links and understanding their topological stability.
method Equivariant bordism groups of Conner and Floyd, topological surgeries.
result Topological protection of certain tricolored knots, with specific transformations.
The paper studies the geometry of vortex-antivortex pairs on Riemann surfaces.
problem The moduli space of BPS vortex-antivortex pairs on Riemann surfaces.
method Detailed study of the Kähler metric gL2 on the moduli space, using localization formulae and numerical analysis. result The moduli space M(k+,k−)(Σ) has finite volume and is geodesically incomplete for Σ=S2. Counterexamples show no simple generalization of Hasimoto transform for higher-dimensional Euler fluids.
problem No straightforward generalization of Hasimoto transform for higher-dimensional Euler fluids.
method Derivation of evolution equations for mean curvature and torsion form for membranes.
result Existence of counterexamples implies no simple generalization of Hasimoto transform.
We introduce and illustrate a new approach to the unknotting problem via the dynamics of vortex strings in a nonlinear partial differential equation of reaction-diffusion type. To untangle a given knot, a Biot-Savart construction is used to initialize the knot as a vortex string in the FitzHugh-Nagumo equation. Remarka…
We introduce a notion of Gieseker stability for a filtered holomorphic vector bundle F over a projective manifold. We relate it to an analytic condition in terms of hermitian metrics on F coming from a construction of the Geometric Invariant Theory (G.I.T). These metrics are balanced in the sense of S.K. Donaldson.…
In differential-geometric language, vortex-lines equations on extended phase space of a system may be written as iγ˙dσ=0, where σ is a differential 1-form. This is the structure, to give a paradigmatic example, of the Hamilton equations. Here, we study equations of the same structure, where σ is a differen…
Study SO(3) vortices over orbifold Riemann surfaces and monopoles.
problem Characterize solutions of SO(3) monopole equations.
method Use moduli spaces of SO(3) vortices over orbifold Riemann surfaces.
result Compute Morse-Bott indices of a function on moduli space.