Related papers: Volume and symplectic structure for l-adic local s…

We present the Hamiltonian formalism for the Euler equation of symplectic fluids, introduce symplectic vorticity, and study related invariants. In particular, this allows one to extend D.Ebin's long-time existence result for geodesics on…

The classical Arnold-Liouville theorem describes the geometry of an integrable Hamiltonian system near a regular level set of the moment map. Our results describe it near a nondegenerate singular level set: a tubular neighborhood of a…

We renormalize the Chern-Simons invariant for convex-cocompact hyperbolic 3-manifolds by finding the asymptotics along an equidistance foliation. We prove that the metric Chern-Simons invariant has an exponentially divergent term given by…

The (local) invariant symplectic action functional $\A$ is associated to a Hamiltonian action of a compact connected Lie group $\G$ on a symplectic manifold $(M,\omega)$, endowed with a $\G$-invariant Riemannian metric $<\cdot,\cdot>_M$. It…

We construct locally defined symplectic torus actions on ribbon graph complexes. Symplectic reduction techniques allow for a recursive formula for the symplectic volumes of these spaces. Taking the Laplace transform results in the…

In the spirit of recent work of Harada-Kaveh and Nishinou-Nohara-Ueda, we study the symplectic geometry of Popov's horospherical degenerations of complex algebraic varieties with the action of a complex linearly reductive group. We…

This paper studies the action of symplectic homeomorphisms on smooth submanifolds, with a main focus on the behaviour of symplectic homeomorphisms with respect to numerical invariants like capacities. Our main result is that a symplectic…

We introduce non-smooth symplectic forms on manifolds and describe corresponding Poisson structures on the algebra of Colombeau generalized functions. This is achieved by establishing an extension of the classical map of smooth functions to…

Let $G$ be a Lie group, with an invariant metric on its Lie algebra $\mathfrak{g}$. Given a surface $\Sigma$ with boundary, and a collection of base points $\mathcal{V}\subset \Sigma$ meeting every boundary component, the moduli space…

A polysymplectic structure is a vector-valued symplectic form, that is, a closed nondegenerate 2-form with values in a vector space. We first outline the polysymplectic Hamiltonian formalism with coefficients in a vector space $V$, then…

We show there is a class of symplectic Lie algebra representations over any field of characteristic not 2 or 3 that have many of the exceptional algebraic and geometric properties of both symmetric three forms in two dimensions and…

Let $X$ be a smooth connected algebraic curve over an algebraically closed field $k$. We study the deformation of $\ell$-adic Galois representations of the function field of $X$ while keeping the local Galois representations at all places…

We prove one deformation theoretic extension of the Gromov non-squeezing phenomenon to $lcs$ structures, or locally conformally symplectic structures, which suitably generalize both symplectic and contact structures. We also conjecture an…

Let $\mathcal{O}$ be a closed orientable 2-orbifold of negative Euler characteristic. Huebschmann constructed the Atiyah-Bott-Goldman type symplectic form $\omega$ on the deformation space $\mathcal{C}(\mathcal{O})$ of convex projective…

We study the symplectic geometry of the moduli space of closed n-gons with fixed side-lengths in hyperbolic 3-space. We prove that these moduli spaces have a symplectic structure coming from Poisson Lie theory. We construct completely…

Three dimensional unsteady flow of fluids in the Lagrangian description is considered as an autonomous dynamical system in four dimensions. The condition for the existence of a symplectic structure on the extended space is the frozen field…

A symplectic structure on the space of nondegenerate and nonparametrized curves in a locally affine manifold is defined. We also consider several interesting spaces of nondegenerate projective curves endowed with Poisson structures. This…

Given a symplectic manifold $M$, we may define an operad structure on the the spaces $\op^k$ of the Lagrangian submanifolds of $(\bar{M})^k\times M$ via symplectic reduction. If $M$ is also a symplectic groupoid, then its multiplication…

We study deformations of symplectic structures on a smooth manifold $M$ via the quasi-Poisson theory. By a fact, we can deform a given symplectic structure $\omega $ to a new symplectic structure $\omega_t$ parametrized by some element $t$…

We make use of $\mathcal{F}$-structures and technology developed by Paternain - Petean to compute minimal entropy, minimal volume, and Yamabe invariant of symplectic 4-manifolds, as well as to study their collapse with sectional curvature…