Related papers: Formal symplectic groupoid of a deformation quanti…

We introduce the notion of symplectic microfolds and symplectic micromorphisms between them. They form a monoidal category, which is a version of the "category" of symplectic manifolds and canonical relations obtained by localizing them…

Using a basic idea of Sullivan's rational homotopy theory, one can see a Lie groupoid as the fundamental groupoid of its Lie algebroid. This paper studies analogues of Lie algebroids with non-trivial higher homotopy. Using various homotopy…

Let M be a symplectic 4-manifold. A semitoric integrable system on M is a pair of real-valued smooth functions J, H on M for which J generates a Hamiltonian S^1-action and the Poisson brackets {J,H} vanish. We shall introduce new global…

These are lecture notes for the course "Poisson geometry and deformation quantization" given by the author during the fall semester 2020 at the University of Zurich. The first chapter is an introduction to differential geometry, where we…

Let $G$ be a compact connected semisimple Lie group. We extend the techniques of Weinstein [W] to give a construction in group cohomology of symplectic forms $\omega$ on \lq twisted' moduli spaces of representations of the fundamental group…

Motivated by the study of symplectic Lie algebroids, we study a describe a type of algebroid (called an $E$-tangent bundle) which is particularly well-suited to study of singular differential forms and their cohomology. This setting…

In this paper we provide a quantization via formality of Poisson actions of a triangular Lie algebra $(\mathfrak g,r)$ on a smooth manifold $M$. Using the formality of polydifferential operators on Lie algebroids we obtain a deformation…

We develop differential and symplectic geometry of differentiable Deligne-Mumford stacks (orbifolds) including Hamiltonian group actions and symplectic reduction. As an application we construct new examples of symplectic toric DM stacks as…

A surjective submersion $\pi : M \to B$ carrying a field of simplectic structures on the fibres is symplectic if this Poisson structure is minimal. A symplectic submersion may be interpreted as a family of mechanical systems depending on a…

We investigate the uniqueness of so-called exotic structures on certain exact symplectic manifolds by looking at how their symplectic properties change under small nonexact deformations of the symplectic form. This allows us to distinguish…

A presymplectic structure on odd dimensional manifold is given by a closed 2-form which is nondegenerate, i.e., of maximal rank. We investigate geometry of presymplectic manifolds. Some basic theorems analogous to those in symplectic and…

We consider formal deformations of the Poisson algebra of functions (with singularities) on $T^*M$ which are Laurent polynomials of fibers. Tn the case: $\dim M=1$ ($M=S^1, {\bf R}$), there exists a non-trivial $\star$-product on this…

Groupoids are mathematical structures able to describe symmetry properties more general than those described by groups. They were introduced (and named) by H. Brandt in 1926. Around 1950, Charles Ehresmann used groupoids with additional…

We review recent works concerning deformation quantization of abelian supergroups. Indeed, we expose the construction of an induced representation of the Heisenberg supergroup and an associated pseudodifferential calculus by using…

We show that the symplectic contraction map of Hilgert-Manon-Martens -- a symplectic version of Popov's horospherical contraction -- is simply the quotient of a Hamiltonian manifold $M$ by a "stratified null foliation" that is determined by…

We give an intrinsic definition of toric symplectic stacks, and show that they are classified by simple convex polytopes equipped with some additional combinatorial data. This generalizes Delzant's classification of toric symplectic…

In this note we introduce the notion of a smooth structure on a conical pseudomanifold $M$ in terms of $C^\infty$-rings of smooth functions on $M$. For a finitely generated smooth structure $C^\infty (M)$ we introduce the notion of the Nash…

Let $(X,\omega)$ be a symplectic rational 4 manifold. We study the space of tamed almost complex structures $\mathcal{J}_{\omega}$ using a fine decomposition via smooth rational curves and a relative version of the infinite-dimensional…

A regular Poisson manifold can be described as a foliated space carrying a tangentially symplectic form. Examples of foliations are produced here that are not induced by any Poisson structure although all the basic obstructions vanish.

For a monotone symplectic manifold and a smooth anticanonical divisor, there is a formal deformation of the symplectic cohomology of the divisor complement, defined by allowing Floer cylinders to intersect the divisor. We compute this…