相关论文: Symplectic embeddings of 4-dimensional ellipsoids:…
In this survey article we describe different ways of embedding fillings of contact 3-manifolds into closed symplectic 4-manifolds.
The notion of a symplectic expansion directly relates the topology of a surface to formal symplectic geometry. We give a method to construct a symplectic expansion by solving a recurrence formula given in terms of the…
The kinetic term of the $N$-body Hamiltonian system defined on the surface of the sphere is non-separable. As a result, standard explicit symplectic integrators are inapplicable. We exploit an underlying hierarchy in the structure of the…
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…
Quite recently, McDuff showed that the existence of a symplectic embedding of one four-dimensional ellipsoid into another can be established by comparing their corresponding sequences of ECH capacities. In this note we show that these…
This paper investigates the geometry of a symplectic 4-manifold $(M,\om)$ relative to a J-holomorphic normal crossing divisor S. Extending work by Biran (in Invent. Math. 1999), we give conditions under which a homology class $A\in…
In any dimension $2n \ge 6$ we show that certain spaces of symplectic embeddings of a polydisk into a product $B^4 \times \Bbb R^{2(n-2)}$ of a $4$-ball and Euclidean space, are not path connected. We also show that any pair of such…
In this article we study proper symplectic and iso-symplectic embeddings of $4$--manifolds in $6$--manifolds. We show that a closed orientable smooth $4$--manifold admitting a Lefschetz fibration over $\C P^1$ admits a symplectic embedding…
We obtain new sharp obstructions to symplectic embeddings of four-dimensional polydisks $P(a,1)$ into four-dimensional ellipsoids $E(bc,c)$ when $1\le a< 2$ and $b$ is a half-integer. When $1 \leq a < 2-O(b^{-1})$ we demonstrate that…
We construct simply connected, minimal, symplectic 4-manifolds with exotic smooth structures and each with one Seiberg-Witten basic class up to sign, on the Noether line and between the Noether and half Noether lines by star surgeries…
We study neighborhoods of configurations of symplectic surfaces in symplectic 4-manifolds. We show that suitably `positive' configurations have neighborhoods with concave boundaries and we explicitly describe open book decompositions of the…
We solve the stabilized symplectic embedding problem for four-dimensional ellipsoids into the four-dimensional round ball. The answer is neatly encoded by a piecewise smooth function which exhibits a phase transition from an infinite…
We introduce a symplectic surgery in six dimensions which collapses Lagrangian three-spheres and replaces them by symplectic two-spheres. Under mirror symmetry it corresponds to an operation on complex 3-folds studied by Clemens, Friedman…
This article introduces a new method to construct volume-filling symplectic embeddings of 4-dimensional ellipsoids by employing polytope mutations in toric and almost-toric varieties. The construction uniformly recovers the full sequences…
The aim of this note is to present a construction of symplectic structures on orientable globally hyperbolic 4-dimensional lorentzian manifolds. Said structures are defined on the manifold itself, not on its cotangent bundle. It also…
We survey our contributions on the classification of elliptic fibrations on K3 surfaces with a non-symplectic involution. We place them in the more general framework of K3 surfaces with an involution without any hypothesis on its fixed…
We study the rigidity and flexibility of symplectic embeddings of simple shapes. It is first proved that under the condition $r_n^2 \le 2 r_1^2$ the symplectic ellipsoid $E(r_1, ..., r_n)$ with radii $r_1 \le ... \le r_n$ does not embed in…
In an earlier paper we explained how to convert the problem of symplectically embedding one 4-dimensional ellipsoid into another into the problem of embedding a certain set of disjoint balls into \CP^2 by using a new way to desingularize…
This paper aims to develop the mathematical representation of a surface generated by elliptical arcs joining the sides of a regular polygon to a point lying vertically upward on the central axis of the polygon. The volume of the…
These notes are based on a five-part minicourse on stabilized symplectic embeddings given in Les Mar\'ecottes, Switzerland during a September 2025 workshop. Our main goal is to explain the recent resolution of the (restricted) stabilized…