arXiv research

Deformation quantization yields a new moment map on symplectic diffeomorphisms.

New flow connects symplectic maps to hyperKähler geometry.

Deform moment map on symplectic connections using star product algebras.

The symplectic representation of mapping classes is not surjective for certain types of mapping classes.

Symplectic solitons rigid if bounded, study shows.

We study iteration maps of recurrence relations arising from mutation periodic quivers of arbitrary period. Combining tools from cluster algebra theory and (pre)symplectic geometry, we show that these cluster iteration maps can be reduced to symplectic maps on a lower dimensional submanifold, provided the matrix repres…

Study shows symplectic mapping groups of K3 surfaces are infinitely generated.

The paper finds symplectic mapping class relations using pencil pairs.

Ancient symplectic solutions to mean curvature flow are flat.

The paper explores the dynamics of composite symplectic Dehn twists with nonuniform hyperbolicity.

After reviewing recent results on symplectic Lefschetz pencils and symplectic branched covers of CP^2, we describe a new construction of maps from symplectic manifolds of any dimension to CP^2 and the associated monodromy invariants. We also show that a dimensional induction process makes it possible to describe any co…

The paper extends a theorem about momentum maps to singular symplectic spaces.

New relation found in 4D symplectic mapping class group.

Study differential operators over maps and their applications in supermanifolds.

Introduces derived Lie n-groupoids with shifted symplectic structures.

In [GMPS] we proved that the moment map image of a $b$-symplectic toric manifold is a convex $b$-polytope. In this paper we obtain convexity results for the more general case of non-toric hamiltonian torus actions on $b$-symplectic manifolds. The modular weights of the action on the connected components of the exceptio…

We prove that Manolescu and Woodward's Symplectic Instanton homology, and its twisted versions are natural, and define maps associated to four dimensional cobordisms within this theory. This allows one to define representations of the mapping class group and the fundamental group of a 3-manifold, and to have a geometri…

The well-known fact that any genus $g$ symplectic Lefschetz fibration $X^{4}\to S^{2}$ is given by a word that is equal to the identity element in the mapping class group and each of whose elements is given by a positive Dehn twist, provides an intimate relationship between words in the mapping class group and 4-manif…

We give a generalization of toric symplectic geometry to Poisson manifolds which are symplectic away from a collection of hypersurfaces forming a normal crossing configuration. We introduce the tropical momentum map, which takes values in a generalization of affine space called a log affine manifold. Using this momentu…

The space of symplectic connections on a symplectic manifold is a symplectic affine space. M. Cahen and S. Gutt showed that the action of the group of Hamiltonian diffeomorphisms on this space is Hamiltonian and calculated the moment map. This is analogous to, but distinct from, the action of Hamiltonian diffeomorphism…

This paper is concerned with the rational symplectic field theory in the Floer case. For this observe that in the general geometric setup for symplectic field theory the contact manifolds can be replaced by mapping tori of symplectic manifolds with symplectomorphisms. While the cylindrical contact homology is given by …

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…

Constructs symplectic surface bundles with positive signatures.

A mapping bends Teichmüller spaces into character varieties, preserving symplectic structure.

Compute group cohomology for mapping class group with non-symplectic coefficients.

Theory of symplectic reduction in infinite dimensions developed.

The paper connects moment maps to the stability of holomorphic fibrations.

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…

Study Dehn-Seidel twists on Lagrangian spheres in K3 surfaces.

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…

Goldman symplectic form and complex structure compatible on $\mathrm{SL}(3,\mathbb R)$ Hitchin component.

We consider a connected symplectic manifold $M$ acted on by a connected Lie group $G$ in a Hamiltonian fashion. If $G$ is compact, we prove give an Equivalence Theorem for the symplectic manifolds whose squared moment map $\parallel μ\parallel^2$ is constant. This result works also in the almost-Kähler setting. Then we…

Improved non-squeezing theorem for calibrated geometries proved.

New computations show symplectic groups and mapping class groups have different properties regarding torsion.

There exist three main approaches to reduction associated to canonical Lie group actions on a symplectic manifold, namely, foliation reduction, introduced by Cartan, Marsden-Weinstein reduction, and optimal reduction, introduced by the authors. When the action is free, proper, and admits a momentum map these three appr…

The paper explores stability properties of cohomology groups and norms in symplectic and mapping class groups.

In this paper we extend the results of Kirwan et alii on convexity properties of the moment map for Hamiltonian group actions, and on the connectedness of the fibers of the moment map, to the case of non-compact orbifolds. Our motivation is twofold. First, the category of orbifolds is important in symplectic geometry b…

Study obstructs symplectic structures on Mazur manifolds.

Any nontrivial homomorphism from the mapping class group of an orientable surface of genus $g\geq 3$ to $\GL(2g,\C)$ is conjugate to the standard symplectic representation. It is also shown that the mapping class group has no faithful linear representation in dimensions less than or equal to $3g-3$.

The paper integrates quasi-Poisson manifolds into multiplicative D-valued moment maps.

We are interested in comparing properties of symplectic mapping class groups of symplectic manifolds of dimension four or higher with properties of classical mapping class groups of surfaces. For $n \geq 2$, consider a configuration of Lagrangian $S^n$s in a Weinstein domain $M^{2n}$. If it is analogous, in some sense …

Given a symplectomorphism f of a symplectic manifold X, one can form the `symplectic mapping cylinder' $X_f = (X \times R \times S^1)/Z$ where the Z action is generated by $(x,s,t)\mapsto (f(x),s+1,t)$. In this paper we compute the Gromov invariants of the manifolds $X_f$ and of fiber sums of the $X_f$ with other sympl…

The paper introduces a new form on Lie algebroids over multisymplectic manifolds.

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.

moment maps arise as a generalization of genuine moment maps on symplectic manifolds when the symplectic structure is discarded, but the relation between the mapping and the action is kept. Particular examples of abstract moment maps had been used in Hamiltonian mechanics for some time, but the abstract notion originat…

Log-symplectic structures are Poisson structures $π$ on $X^{2n}$ for which $\bigwedge^n π$ vanishes transversally. By viewing them as symplectic forms in a Lie algebroid, the $b$-tangent bundle, we use symplectic techniques to obtain existence results for log-symplectic structures on total spaces of fibration-like maps…

Functor connects symplectic and contact structures via cutting and blowups.

Minimal action of mapping class group on character variety.