arXiv research

Method resolves 4D symplectic orbifolds using complex geometry.

New flows introduced for symplectic geometry.

Abstract collects open problems in billiards and symplectic geometry.

Criterion found for blowing down in 6D symplectic geometry.

A dictionary connects symplectic to contact geometry, with applications to complex and G-structures.

Introduces systolic inequalities in Riemannian and symplectic geometry.

Symplectic and Poisson structures proved for information geometry's Frobenius manifold.

The paper explores symplectic geometry of Cartan-Hartogs domains.

Quantizes symplectic manifolds with bounded geometry using Berezin-Toeplitz method.

Characterizes Anosov flows in 3D using symplectic and contact geometry.

Proves a vanishing property for symplectic manifold cohomology.

Symplectic structures simplified for compact manifolds.

We will survey some aspects of the smooth topology, algebraic geometry, symplectic geometry and contact geometry of anti-canonical pairs in complex dimension two.

New flow connects symplectic maps to hyperKähler geometry.

We study the geometry of manifolds carrying symplectic pairs consisting of two closed 2-forms of constant ranks, whose kernel foliations are complementary. Using a variation of the construction of Boothby and Wang we build contact-symplectic and contact pairs from symplectic pairs.

Survey and generalization of implosion and contraction in symplectic and hyperkähler geometry.

The paper tackles isotropy of symplectic forms using Hodge flows.

This paper uses a generalization of symplectic geometry, known as $n$-symplectic geometry and developed by Norris, to find observables on three-dimensional manifolds. It will be seen that for the cases considered, the $n$-symplectic observables are derivable from the symplectic observables of $C^2$. The quantization of…

A new method simplifies contact Hamiltonian mechanics.

The paper extends symplectic techniques to generalized complex geometry.

Study contact geometry of symplectic divisors, invariant under specific transformations.

In this paper we define a new category of almost complex riemannian 4- manifolds and discuss some basic properties of such pseudo symplectic manifolds. Some motivation based on the Seiberg - Witten theory is imposed.

This article investigates the complex symplectic geometry of the deformation space of complex projective structures on a closed oriented surface of genus at least 2. The cotangent symplectic structure given by the Schwarzian parametrization is studied carefully and compared to the Goldman symplectic structure on the ch…

We introduce the symplectic twistor operator $T_s$ in symplectic spin geometry, as a symplectic analogue of the twistor operator in Riemannian spin geometry. We focus on the real dimension 2 and compute the space of its solutions on ${\mathbb R}^2$. Our analysis is based on the techniques of metaplectic Howe duality.

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 capacities often lead to surprising relations between different aspects of sympl…

Study reveals new geometric structures for magnetic field Hamiltonian systems.

For a symplectic manifold admitting a metaplectic structure (a symplectic analogue of the Riemannian spin structure), we construct a sequence consisting of differential operators using a symplectic torsion-free affine connection. All but one of these operators are of first order. The first order ones are symplectic ana…

We study the geometry of a family of Lie groups, which contained the classical affine Lie groups, endowed with an exact left invariant symplectic form. We show that this family is closed by symplectic reduction and symplectic double extension in the sense of Dardié and Medina. We prouve also that these groups are endow…

Paper proves a symplectic inequality using trisections and contact geometry.

The paper explores complex geometries of 3-forms on symplectic 6-manifolds.

We study noncommutative generalizations of such notions of the classical symplectic geometry as degenerate Poisson structure, Poisson submanifold and quotient manifold, symplectic foliation and symplectic leaf for associative Poisson algebras. We consider these structures for the case of the endomorphism algebra of a v…

Symplectic structure found on projective structures on surfaces with boundary.

We study the fields of endomorphisms intertwining pairs of symplectic structures. Using these endomorphisms we prove an analogue of Moser's theorem for simultaneous isotopies of two families of symplectic forms. We also consider the geometric structures defined by pairs and triples of symplectic forms for which the squ…

Lectures on symplectic and Poisson geometry, quantization, and quantum field theory.

The paper extends the functional geometry of the visual cortex to more complex architectures using contactization and symplectization.

This is a survey on symplectic birational geometry. In arbitrary dimension, this subject is centered around the notion of uniruledness. In low dimensions, we will also discuss Kodaira dimension and minimality.

We prove a version of the Arnol'd conjecture for Lagrangian submanifolds of conformal symplectic manifolds: a Lagrangian $L$ which has non-zero Morse-Novikov homology for the restriction of the Lee form $β$ cannot be disjoined from itself by a $C^0$-small Hamiltonian isotopy. Furthermore for generic such isotopies the …

This article sketches various ideas in contact geometry that have become useful in low-dimensional topology. Specifically we (1) outline the proof of Eliashberg and Thurston's results concerning perturbations of foliatoins into contact structures, (2) discuss Eliashberg and Weinstein's symplectic handle attachments, an…

Study complex lines in symplectic geometry, generalizing previous results.

Auxiliary equations improve bounds in symplectic geometry.

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

We extend the AKSZ formulation of the Poisson sigma model to more general target spaces, and we develop the general theory of graded geometry for poly-symplectic and poly-Poisson structures. In particular we prove a Schwarz-type theorem and transgression for graded poly-symplectic structures, recovering the action func…

Diffeology extends differential geometry to complex spaces.

New method constructs symplectic structures on 4-manifolds from trisections.

Let (M,w,L) be a symplectic manifold endowed with a lagrangian foliation L. Liberman and Weinstein have shown that the leaves of L are endowed with an affine structure. In this paper we provide links between the theories of affine manifolds and symplectic geometry. Using the work of Donaldson who have shown the existen…

New Poisson structures on hypersurface algebroids discovered.

The goal of this note is to give an introduction to locally conformally symplectic and Kähler geometry. In particular, Sections 1 and 3 aim to provide the reader with enough mathematical background to appreciate this kind of geometry. The reference book for locally conformally Kähler geometry is "Locally conformal Kähl…

In this article we study Ammann tilings from the perspective of symplectic geometry. Ammann tilings are nonperiodic tilings that are related to quasicrystals with icosahedral symmetry. We associate to each Ammann tiling two explicitly constructed highly singular symplectic spaces and we show that they are diffeomorphic…