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Symplectic U7, U8 and U9 singularities

Trębska Ż.

Demonstratio Mathematica

2015

Vol. 48, nr 2

322--347

We use the method of algebraic restrictions to classify symplectic U7, ;U8 and U9 singularities. We use discrete symplectic invariants to distinguish symplectic singularities of the curves. We also give the geometric description of symplectic classes.

Symplectic singularities and solvable Hamiltonian mappings

Fukuda T., Janeczko S.

Demonstratio Mathematica

2015

Vol. 48, nr 2

118--146

We study singularities of smooth mappings (…) of R2n into symplectic space (…) by their isotropic liftings to the corresponding symplectic tangent bundle (…). Using the notion of local solvability of lifting as a generalized Hamiltonian system, we introduce new symplectic invariants and explain their geometric meaning. We prove that a basic local algebra of singularity is a space of generating functions of solvable isotropic mappings over (…) endowed with a natural Poisson structure. The global properties of this Poisson algebra of the singularity among the space of all generating functions of isotropic liftings are investigated. The solvability criterion of generalized Hamiltonian systems is a strong method for various geometric and algebraic investigations in a symplectic space. We illustrate this by explicit classification of solvable systems in codimension one.

A note on groups of symplectomorphisms

Rybicki T.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

1998

[Z] 38

115-126

It is shown that the groups of Hamiltonian diffeomorphisms of a symplectic manifold determine uniquely the smooth and symplectic structures themselves. An analogous result is true for the Lie algebras of Hamiltoniam vector fields.