arXiv research

Symplectic forms match on circle pattern space.

Bielavsky introduced and investigated the class of symmetric symplectic spaces, that is, symmetric spaces endowed with a symplectic form invariant with respect to symmetries. Since the theory of symmetric spaces has generalizations, we ask a question about their possible symplectic versions. We do construct such genera…

Study of symplectic cross-ratios in moduli spaces of line configurations.

Symplectic embedding extended to stratified spaces.

In this paper we introduce the notion of a smooth structure on a stratified space, the notion of a Poisson smooth structure and the notion of a weakly symplectic smooth structure on a stratified symplectic space, refining the concept of a stratified symplectic Poisson algebra introduced by Sjamaar and Lerman. We show t…

Study on smooth moduli spaces of branes in symplectic 4-manifolds.

Let (M,ω) be a symplectic manifold, and (Σ,σ) a closed connected symplectic 2-manifold. We construct a weakly symplectic form {ω^{D}}_{(Σ, σ)} on the space of immersions Σ\to M that is a special case of Donaldson's form. We show that the restriction of {ω^{D}}_{(Σ,σ)} to any orbit of the group of Hamiltonian symplectom…

Anti-symplectic involutions connect a sphere in a symplectic surface.

Study joint invariants on symplectic spaces, extending group and space variations.

The paper classifies symplectic fillings of lens spaces and constructs cobordisms.

We prove that the kernels of the restrictions of symplectic Dirac or symplectic Dirac-Dolbeault operators on natural subspaces of polynomial valued spinor fields are finite dimensional on a compact symplectic manifold. We compute those kernels for the complex projective spaces. We construct injections of subgroups of t…

Deform moment map on symplectic connections using star product algebras.

Constructs curves in log-symplectic manifolds, classifying and obstructing certain structures.

Symplectic structure found on projective structures on surfaces with boundary.

Symplectic forms from two phase spaces are proven equivalent.

We consider the global symplectic classification problem of plane curves. First we give the exact classification result under symplectomorphisms, for the case of generic plane curves, namely immersions with transverse self-intersections. Then the set of symplectic classes form the symplectic moduli space which we compl…

The paper explores symplectic connections on homogeneous spaces, finding a unique invariant connection.

Teichmüller space realized as symplectic quotient.

The paper explores symplectic geometry of Cartan-Hartogs domains.

For a finite dimensional symplectic manifold $(M,ω)$ with a symplectic form $ω$, corresponding loop space ($LM=C^\infty(S^1,M)$) admits a weak symplectic form $Ω^ω$. We prove that the loop space over $\mbr^n$ admits Darboux chart for the weak symplectic structure $Ω^ω$. Further, we show that inclusion map from the symp…

The harmonic cohomology of a Donaldson symplectic submanifold and of an Auroux symplectic submanifold are compared with that of its ambient space. We also study symplectic manifolds satisfying a weakly Lefschetz property, that is, the $s$-Lefschetz propery. In particular, we consider the symplectic blow-ups of the comp…

The paper explores spaces of Kähler and symplectic forms on 4-manifolds.

Develops a finite construction for self-duality and related moduli spaces over Riemann surfaces.

Let $M$ be a symplectic symmetric space, and let $\imath : M \to V$ be an extrinsic symplectic symmetric immersion, i.e., $(V, Ω)$ is a symplectic vector space and $\imath$ is an injective symplectic immersion such that for each point $p \in M$, the geodesic symmetry in $p$ is compatible with the reflection in the affi…

Develops a correspondence between symplectic orbits and Grassmannians.

Study of symplectic manifolds degenerating into singular spaces.

Simplified presentation of symplectic fillings of lens spaces.

New framework for symplectic reduction on stacky spaces.

Constructs a Morse-Bott function on symplectic Grassmannians.

Extends AKSZ to poly-symplectic structures for more complex spaces.

Study connects lens spaces' fundamental group to their symplectic fillings' second Betti numbers.

Found a new connected component in symplectic structures.

Introduces a new method for symplectic reduction along submanifolds.

We give finiteness results and some classifications up to diffeomorphism of minimal strong symplectic fillings of Seifert fibered spaces over S^2 satisfying certain conditions, with a fixed natural contact structure. In some cases we can prove that all symplectic fillings are obtained by rational blow-downs of a plumbi…

Classifies symplectic fillings of specific torus bundles.

Characterizes symplectic and variational operators for scalar evolution equations.

During the last thirty years, symplectic or Marsden--Weinstein reduction has been a major tool in the construction of new symplectic manifolds and in the study of mechanical systems with symmetry. This procedure has been traditionally associated to the canonical action of a Lie group on a symplectic manifold, in the pr…

The paper constructs a moduli space for opers and proves its symplectic properties.

Let $(M,ω)$ be a Hamiltonian $G$-space with a momentum map $F:M \to {\frak g}^*$. It is well-known that if $α$ is a regular value of $F$ and $G$ acts freely and properly on the level set $F^{-1}(G\cdot α)$, then the reduced space $M_α:=F^{-1}(G\cdot α)/G$ is a symplectic manifold. We show that if the regularity assumpt…

Solves symplectic and conformal symplectic group actions equivalence problem.

Symplectic coordinates found on projective structures on orbifolds.

New method integrates Poisson homogeneous spaces to symplectic groupoids.

Let $(V, \Om)$ be a symplectic vector space and let $φ: M \ra V$ be a symplectic immersion. We show that $φ(M) \subset V$ is (locally) an extrinsic symplectic symmetric space (e.s.s.s.) in the sense of \cite{CGRS} if and only if the second fundamental form of $φ$ is parallel. Furthermore, we show that any symmetric spa…

The abstract discusses applications of Menke's JSJ decomposition to symplectic fillings of various 3-manifolds.

Study on symplectic Dirac operators on foliations, estimating eigenvalues.

The Pontryagin forms on 1-jet bundle of Riemannian metrics, are shown to provide, in a natural way, diffeomorphism-invariant pre-symplectic structures on the space of Riemannian metrics for dimensions $n=4r-2$. The equivariant Pontryagin forms provide canonical moment maps for these structures. In dimension two, the sy…

The aim of our article is the study of solution space of the symplectic twistor operator $T_s$ in symplectic spin geometry on standard symplectic space $({\mathbb R}^{2n},ω)$, which is the symplectic analogue of the twistor operator in (pseudo)Riemannian spin geometry. In particular, we observe a substantial difference…

New models for symplectic structures on classifying stacks.