arXiv research

Study symplectic embeddings of 4-manifolds using Lefschetz fibrations.

New symplectic barriers found in ball embeddings.

New symplectic caps and embeddings found in complex projective plane.

New balls smoothly fit in CP² but not symplectically.

Alternative proof of coisotropic embedding theorem for pre-symplectic manifolds.

Symplectic embeddings of balls into specific manifolds are studied, with restrictions and obstructions identified.

Embeds 3-manifolds in symplectic 4-manifolds with constraints.

New symplectic embedding obstructions found for polydisks into half-integer ellipsoids.

Symplectic embedding extended to stratified spaces.

In this paper we show that every degree 2 homology class of a 2n-dimensional symplectic manifold is represented by an immersed symplectic surface if it has positive symplectic area. Moreover, the symplectic surface can be chosen to be embedded if 2n is at least 6. We also analyze the additional conditions under which e…

The paper solves symplectic embedding problems in higher dimensions, proving new embedding conditions.

Extends symplectic reduction and theorem to Lie algebroids.

Study Markov staircases in symplectic embeddings of rational homology ellipsoids.

The paper uses Seshadri constants to construct symplectic ellipsoid embeddings.

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…

The Milnor fibre of a $\mathbb{Q}$-Gorenstein smoothing of a Wahl singularity is a rational homology ball $B_{p,q}$. For a canonically polarised surface of general type $X$, it is known that there are bounds on the number $p$ for which $B_{p,q}$ admits a symplectic embedding into $X$. In this paper, we give a recipe to…

Given an integral symplectic manifold, we construct a family of "coherent state" maps into complex projective space. The maps are built from sections of the tensor powers of a hermitian line bundle whose curvature is a multiple of the symplectic form. We show that this family is an almost-complex version of the Kodiara…

We prove symplectic hypersurfaces in Weinstein domains and give obstructions for manifold boundaries.

Let $X$ be a union of a sequence of symplectic manifolds of increasing dimension and let $M$ be a manifold with a closed $2$-form $ω$. We use Tischler's elementary method for constructing symplectic embeddings in complex projective space to show that the map from the space of embeddings of $M$ in $X$ to the cohomology …

In this paper, we demonstrate a relation among Seiberg-Witten invariants which arises from embedded surfaces in four-manifolds whose self-intersection number is negative. These relations, together with Taubes' basic theorems on the Seiberg-Witten invariants of symplectic manifolds, are then used to prove the symplectic…

We show that the pre-order defined on the category of contact manifolds by arbitrary symplectic cobordisms is considerably less rigid than its counterparts for exact or Stein cobordisms: in particular, we exhibit large new classes of contact 3-manifolds which are symplectically cobordant to something overtwisted, or to…

We compute the homotopy type of the space of T^n-equivariant symplectic embeddings from the standard 2n-dimensional ball of some fixed radius into a 2n-dimensional symplectic-toric manifold M, and use this computation to define a Z-valued step function on the positive real line which is an invariant of the symplectic-t…

We derive an obstruction to representing a homology class of a symplectic 4-manifold by an embedded, possibly disconnected, symplectic surface.

Constructs equivariant embeddings of Hermitian symmetric spaces into tangent spaces.

We obtain universal models for several types of locally conformal symplectic manifolds via pullback or reduction. The relation with recent embedding results for locally conformal Kähler manifolds is discussed.

We study the question of how many embedded symplectic or Lagrangian tori can represent the same homology class in a simply connected symplectic 4-manifold.

Defines a distance function on a manifold using symplectic embeddings and recovers the metric.

Let $X$ be any rational ruled symplectic four-manifold. Given a symplectic embedding $ι:B_{c}\into X$ of the standard ball of capacity $c$ into $X$, consider the corresponding symplectic blow-up $\tX_ι$. In this paper, we study the homotopy type of the symplectomorphism group $\Symp(\tX_ι)$, simplifying and extending t…

Dual pairs constructed for volume preserving diffeomorphisms using symplectic geometry.

The paper provides a new inequality for 4-manifolds and uses it to study knot sliceness and symplectic embeddings.

We use Donaldson's approximately holomorphic techniques to build embeddings of a closed symplectic manifold with symplectic form of integer class in the grassmannians Gr(r,N). We assure that these embeddings are asymptotically holomorphic in a precise sense. We study first the particular case of embeddings in the proje…

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 a ball of radius strictly smaller than $r_n$. We then use symplectic folding to …

The paper explores conditions for homology spheres to bound acyclic smooth manifolds and symplectic fillings.

The purpose of this paper is to present some results on the existence of homologous, nonisotopic symplectic or lagrangian surfaces embedded in a simply connected symplectic 4-dimensional manifold.

Study transverse knots and symplectic surfaces using Seiberg-Witten monopole equations.

Study shows no smooth embeddings of rational homology balls into complex projective plane.

Investigates the rotating Kepler problem for energy values ≤ -3/2.

We show that hyperelliptic symplectic Lefschetz fibrations are symplectically birational to two-fold covers of rational ruled surfaces, branched in a symplectically embedded surface. This reduces the classification of genus 2 fibrations to the classification of certain symplectic submanifolds in rational ruled surfaces…

We show that symplectically embedded $(-1)$-tori give rise to certain elements in the symplectic mapping class group of $4$-manifolds. An example is given where such elements are proved to be of infinite order.

Study shows 3D shapes can't fit in certain 4D spaces.

Study circle patterns on tori, linking symplectic forms and homeomorphisms.

We construct using Lefschetz fibrations a large family of contact manifolds with the following properties: Any bounding contact embedding into an exact symplectic manifold satisfying a mild topological assumption is non-displaceable and generically has infinitely many leaf-wise intersection points. Moreover, any Stein …

We construct the TQFT on symplectic cohomology and wrapped Floer cohomology, possibly twisted by a local system of coefficients, and prove that the TQFT respects Viterbo restriction maps and the canonical maps from ordinary cohomology. We also construct the module structure of wrapped Floer cohomology over symplectic c…

In this paper, the symplectic genus for any 2-dimensional class in a 4-manifold admitting a symplectic structure is introduced, and its relation with the minimal genus is studied. It is used to describe which classes in rational and irrational ruled manifolds are realized by connected symplectic surfaces. In particular…

Researchers describe ECC of concave lens spaces and compute symplectic capacities.

The purpose of this paper is to investigate the following problem: For a fixed 2-dimensional homology class K in a simply connected symplectic 4-manifold, up to smooth isotopy, how many connected smoothly embedded symplectic submanifolds represent K? We show that when K can be represented by a symplectic torus, there a…

New method constructs small symplectic 4-manifolds via contact gluing.

The study connects ECH capacities to Anosov flows, proving infinite capacities and obstructions.