Related papers: Non-Hamiltonian actions with fewer isolated fixed …

In this paper, we classify Hamiltonian $S^1$-actions on compact, four dimensional symplectic orbifolds that have isolated singular points with cyclic orbifold structure groups, thus extending the classification due to Karshon to the…

New classes of Lie-Hamilton systems are obtained from the six-dimensional fundamental representation of the symplectic Lie algebra $\mathfrak{sp}(6,\mathbb{R})$. The ansatz is based on a recently proposed procedure for constructing…

In this article, we provide an exposition about symplectic toric manifolds, which are symplectic manifolds $(M^{2n}, \omega)$ equipped with an effective Hamiltonian $\mathbb{T}^n\cong (S^1)^n$-action. We summarize the construction of $M$ as…

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…

This is the first of a series of papers devoted to the topology of symplectic Calabi-Yau $4$-manifolds endowed with certain symplectic finite group actions. We completely determine the fixed-point set structure of a finite cyclic action on…

In this paper we show that the Seiberg--Witten invariant is zero for all smooth 4--manifolds with $b_+{>}1$ which admit circle actions that have at least one fixed point. Furthermore, we show that all symplectic 4--manifolds which admit…

Given a closed, oriented surface, possibly with boundary, and a mapping class, we obtain sharp lower bounds on the number of fixed points of a surface symplectomorphism (i.e. area-preserving map) in the given mapping class, both with and…

We study Hamiltonian diffeomorphisms on symplectic Euclidean spaces that are equal to non-degenerate linear maps at infinity. Under the assumption that there exists an isolated homologically nontrivial fixed point satisfying the twist…

Let $(M,\omega)$ be a Hamiltonian $G$-space with a momentum map $F:M \to {\frak g}^*$. It is well-known that if $\alpha$ is a regular value of $F$ and $G$ acts freely and properly on the level set $F^{-1}(G\cdot \alpha)$, then the reduced…

In symplectic topology one uses elliptic methods to prove rigidity results about symplectic manifolds and solutions of Hamiltonian equations on them, where the most basic example is given by geodesics on Riemannian manifolds. Harmonic maps…

We consider compact symplectic manifolds acted on effectively by a compact connected Lie group $K$ in a Hamiltonian fashion. We prove that the squared moment map $||\mu||^2$ is constant if and only if $K$ is semisimple and the manifold is…

While symplectic manifolds have no local invariants, they do admit many global numerical invariants. Prominent among them are the so-called symplectic capacities. Different capacities are defined in different ways, and so relations between…

We construct a symplectic flow on a surface of genus g greater than one with exactly 2g-2 hyperbolic fixed points and no other periodic orbits. Moreover, we prove that a (strongly non-degenerate) symplectomorphism of a surface (with genus g…

We construct all possible Hamiltonian torus actions for which all the non-empty reduced spaces are two dimensional (and not single points) and the manifold is connected and compact, or, more generally, the moment map is proper as a map to a…

In this note we apply a 4-fold sum operation to develop an associativity rule for the pairwise symplectic sum. This allows us to show that certain diffeomorphic symplectic $4$-manifolds made out of elliptic surfaces are in fact…

Let $p$ be a prime number. We introduce symplectic actions of $p$-adic analytic Lie groups on $p$-adic symplectic manifolds. Then we show that any $p$-adic symplectic action $G\times(M,\omega)\to(M,\omega)$ has a momentum map…

A generalized moment map is proposed for arbitrary symplectic actions of compact connected Lie groups on closed symplectic manifolds, in the spirit of the circle -valued maps introduced by D. McDuff in the case of non-Hamiltonian circle…

In the breakthrough paper [V. Mu\~noz, A Smale-Barden manifold admitting K-contact but not Sasakian structure, 2024, 10.4171/JEMS/1496], it is constructed the first example of a simply connected compact 5-manifold (aka.\ Smale-Barden…

This survey focuses on strong closing lemmas in Hamiltonian dynamics that are proved using spectral invariants (also known as action selectors) in symplectic geometry. We review strong closing lemmas in low-dimensional Hamiltonian dynamics…

We give a systematic presentation of the stability theory in the non-algebraic Kaehlerian geometry. We introduce the concept of "energy complete Hamiltonian action". To an energy complete Hamiltonian action of a reductive group G on a…