Urs Frauenfelder
We study Furuta's finite dimensional approximations for the flow lines of the Moment action functional for toric symplectic orbifolds and show their relation to Givental's toric map space.
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In this paper we define invariants of Hamiltonian group actions for central regular values of the moment map. The key hypotheses are that the moment map is proper and that the ambient manifold is symplectically aspherical. The invariants are based on the symplectic vortex equations. Applications include an existence theorem for relative periodic orbits, a computation for circle actions on a complex vector space, and a theorem about the relation between the invariants introduced here and the Seiberg--Witten invariants of a product of a Riemann surface with a two-sphere.
The nth symmetric product of a Riemann surface carries a natural family of Kaehler forms, arising from its interpretation as a moduli space of abelian vortices. We give a new proof of a formula of Manton-Nasir for the cohomology classes of these forms. Further, we show how these ideas generalise to families of Riemann surfaces. These results help to clarify a conjecture of D. Salamon on the relationship between Seiberg-Witten theory on 3-manifolds fibred over the circle and symplectic Floer homology.
Making use of theory of differentiable stacks, we study symplectic vortex equations over a compact orbifold Riemann surface. We discuss the category of representable morphisms from a compact orbifold Riemann surface to a quotient stack. After that we define symplectic vortex equations over a compact orbifold Riemann surface. We also discuss the moduli space of solutions to the equations for linear actions of the circle group on the complex plane.
We consider the symplectic vortex equations for a linear Hamiltonian torus action. We show that the associated genus zero moduli space itself is homotopic (in the sense of a homotopy of regular G-moduli problems) to a toric manifold with combinatorial data directly obtained from the original torus action. This allows to view the wall crossing formula of Cieliebak and Salamon for the computation of vortex invariants as a consequence of a generalized Jeffrey-Kirwan localization formula for integrals over symplectic quotients.
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 approximation (LIA) of the hydrodynamical Euler equation describes the skew-mean-curvature flow on vortex membranes of codimension 2 in any dimension, which generalizes the classical binormal, or vortex filament, equation in 3D. This framework also allows one to define the symplectic struc…
Vortex blob methods are typically characterized by a regularization length scale, below which the dynamics are trivial for isolated blobs. In this article we observe that the dynamics need not be trivial if one is willing to consider distributional derivatives of Dirac delta functionals as valid vorticity distributions. More specifically, a new singular vortex theory is presented for regularized Euler fluid equations of ideal incompressible flow in the plane. We determine the conditions under which such regularized Euler fluid equations may admit vorticity singularities which are stronger than…
The moduli space of solutions to the vortex equations on a Riemann surface are well known to have a symplectic (in fact Kähler) structure. We show this symplectic structure explictly and proceed to show a family of symplectic (in fact, Kähler) structures $Ω_{Ψ_0}$ on the moduli space, parametrised by $Ψ_0$, a section of a line bundle on the Riemann surface. Next we show that corresponding to these there is a family of prequantum line bundles ${\mathcal P}_{Ψ_0} $on the moduli space whose curvature is proportional to the symplectic forms $Ω_{Ψ_0}$.
Integrability of point-vortex dynamics via symplectic reduction: a survey
本文基于辛约化理论建立统一框架，证明二维流形上 N=2、3、4 个点涡及更一般 Hamilton 系统的可积性，并在球面、平面、双曲平面和平坦环面加以说明。
We derive a wall crossing formula for the symplectic vortex invariants of toric manifolds. As an application, we give a proof of Batyrev's formula for the quantum cohomology of a monotone toric manifold with minimal Chern number at least two.
Let $(X,ω)$ be a compact symplectic manifold with a Hamiltonian action of a compact Lie group $G$ and $μ: X\to \mathfrak g$ be its moment map. In this paper, we study the $L^2$-moduli spaces of symplectic vortices on Riemann surfaces with cylindrical ends. We studied a circle-valued action functional whose gradient flow equation corresponds to the symplectic vortex equations on a cylinder $S^1\times \mathbb R$. Assume that $0$ is a regular value of the moment map $μ$, we show that the functional is of Bott-Morse type and its critical points of the functional form twisted sectors of the symplec…
Holomorphic approximation for Hamiltonian diffeomorphism groups via the Symplectic Density Property
该研究证明全纯灵活Stein流形的余切丛具有辛密度性质，进而推导得出实余切丛的哈密顿微分同胚可由复化余切丛的复辛自同构实现Carleman意义下的全纯逼近。
We give an intrinsic proof that Vorobjev's first approximation of a Poisson manifold near a symplectic leaf is a Poisson manifold. We also show that Conn's linearization results cannot be extended in Vorobjev's setting.
Assume $(X, ω)$ is a compact symplectic manifold with a Hamiltonian compact Lie group action and the zero in the Lie algebra is a regular value of the moment map $μ$. We prove that a finite energy symplectic vortex exponentially converges to (un)twisted sectors of the symplectic reduction at cylinder ends whose metrics grow up at least cylindrically fast, without assuming the group action on the level set $μ^{-1}(0)$ is free. It generalizes the corresponding results by Ziltener [23, 24] under the free action assumption. The result of this paper is the first step in setting up the quotient morp…
Gauge theory and symplectic structures
该研究针对四维以上流形辛结构存在性障碍问题，在殆厄米特流形上引入椭圆方程组，证明其指标性质、解的横截性与唯一性，相关结果为高维规范理论提供了证据。
Symplectic groupoid multiplication and Poisson integrators
本文研究局部辛群胚乘法在泊松流形哈密顿流近似方法构造中的作用，证明其可用于构造拉格朗日二分法的新型乘法型近似序列，并给出理论结果、实现方式及示例。
In this paper we show that the dimensionally reduced Seiberg-Witten equations lead to a Higgs field and study the resulting moduli spaces. The moduli space arising out of a subset of the equations, shown to be non-empty for a compact Riemann surface of genus g >= 1, gives rise to a family of moduli spaces carrying a hyperkahler structure. For the full set of equations the corresponding moduli space does not have the aforementioned hyperkahler structure but has a natural symplectic structure. For the case of the torus, g=1, we show that the full set of equations has a solution, different from t…
We prove the invariant of the symplectic capacity for the Zakharov system on a torus. If the Zakharov solution map is well-defined, then it can be regarded as a symplectomorphism. Thus, we first show the global well-posedness via the local well-posedness and the conservation law. The invariant of the symplectic capacity can be obtained using an approximation method. Many authors use an approximation method to obtain the nonsqueezing theorem, instead of an invariant of the symplectic capacity. However, the conditions of the Hamiltonian system introduced by Kuksin can be relaxed by a new modifie…
Even Order Explicit Symplectic Geometric Algorithms for Solving Quaternions in Guidance Navigation and Control via Diagonal Padé Approximation and Cayley Transform
本文提出基于对角Padé逼近与Cayley变换的偶数阶显式辛几何算法，通过四步策略统一构造可配置阶数的高阶辛差分格式，用于求解制导导航与控制中的四元数运动学微分方程，并证明其辛性、正交性与可逆性。
Symplectic filtering of Hamiltonian dynamics with moving sensors
提出一种在线辛滤波算法，利用移动传感器测量重构参数化哈密顿偏微分方程的解，保证辛结构和能量守恒，并证明动态传感器布置优于静态布置。
On some rigidity results for high-dimensional confoliations
该研究针对任意奇数维环境空间的c-辛共叶状结构，证明了接触区域存在性的阻碍，推广了3维Reeb稳定性定理，给出两种接触逼近新概念并构造了无法被接触逼近的例子，依托伪全纯曲线理论完成证明。
In the first part of the article we study Hamiltonian diffeomorphisms of $\mathbb{R}^{2n}$ which are generated by sub-quadratic Hamiltonians and prove a middle dimensional rigidity result for the image of coisotropic cylinders. The tools that we use are Viterbo's symplectic capacities and a series of inequalities coming from their relation with symplectic reduction. In the second part we consider the nonlinear string equation and treat it as an infinite-dimensional Hamiltonian system. In this context we are able to apply Kuksin's approximation by finite dimensional Hamiltonian flows and prove …
Our aim in this work is to study a system of equations which generalises at the same time the vortex equations of Yang-Mills-Higgs theory and the holomorphicity equation in Gromov theory of pseudoholomorphic curves. We extend some results and definitions from both theories to a common setting. We introduce a functional generalising Yang-Mills-Higgs functional, whose minima coincide with the solutions to our equations. We prove a Hitchin-Kobayashi correspondence allowing to study the solutions of the equations in the Kaehler case. We give a structure of smooth manifold to the set of (gauge equi…
Let A be the space of irreducible connections (vector potentials) over a SU(n)-principal bundle on a three-dimensional manifold M. Let T be the fiber product of the tangent and cotangent bundles of A. We endow T with a symplectic structure Ωwhich is represented by a vortex formula. The corresponding Poisson bracket will give a parallel formula discussed by Marsden-Weinstein in case of the electric-magnetic field, U(1)-gauge. We shall prove the Maxwell equations on T. The first two equations are the Hamilton equations of motion derived from the symplectic structure on T, and the second equation…
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