辛几何

A section of a Hamiltonian system is a hypersurface in the phase space of the system, usually representing a set of one-sided constraints (e.g. a boundary, an obstacle or a set of admissible states). In this paper we give local…

We construct a special class of semiclassical Fourier integral operators whose wave fronts are symplectic micromorphisms. These operators have very good properties: they form a category on which the wave front map becomes a functor into the…

We show that a closed weakly-monotone symplectic manifold of dimension $2n$ which has minimal Chern number greater than or equal to $n+1$ and admits a Hamiltonian toric pseudo-rotation is necessarily monotone and its quantum homology is…

Using a simplified version of Kuranishi perturbation theory that we call semi-global Kuranishi structures, we give a definition of the equivariant Lagrangian Floer cohomology of a pair of Lagrangian submanifolds that are fixed under a…

It was proven in the first author's paper "Contact 3-manifolds twenty years since J. Martinet's work" (Ann. Inst. Fourier, 42(1992), 165--192) that any tight contact structure on the 3-sphere is diffeomorphic to the standard one. It was…

The Cox construction presents a toric variety as a quotient of affine space by a torus. The category of coherent sheaves on the corresponding stack thus has an evident description as invariants in a quotient of the category of modules over…

We are proposing Tulczyjew's triple for contact dynamics. The most important ingredients of the triple, namely symplectic diffeomorphisms, special symplectic manifolds, and Morse families, are generalized to the contact framework. These…

We prove that Legendrian and transverse links in overtwisted contact structures having overtwisted complements can be classified coarsely by their classical invariants. We further prove that any coarse equivalence class of loose links has…

We compute the Picard group of a stable b-symplectic manifold $M$ by introducing a collection of discrete invariants $\mathfrak{Gr}$ which classify $M$ up to Morita equivalence.

We show in this work that the restricted three-body problem is in general not of contact type and zero is the energy value where the contact property breaks down. More explicitly, sequences of generating orbits with increasingly negative…

We present a geometric construction of central S^1-extensions of the quantomorphism group of a prequantizable, compact, symplectic manifold, and explicitly describe the corresponding lattice of integrable cocycles on the Poisson Lie…

We present techniques, inspired by monodromy considerations, for constructing compact monotone Lagrangians in certain affine hypersurfaces, chiefly of Brieskorn-Pham type. We focus on dimensions 2 and 3, though the constructions generalise…

We study a class of Poisson tensors on a fibered manifold which are compatible with the fiber bundle structure by the so-called almost coupling condition. In the case of a $5$-dimensional orientable fibered manifolds with $2$-dimensional…

Kostant gave a model for the real geometric quantization associated to polarizations via the cohomology associated to the sheaf of flat sections of a pre-quantum line bundle. This model is well-adapted for real polarizations given by…

In this short note we observe that the boundary of a properly embedded compact exact Lagrangian sub-manifolds in a subcritical Weinstein domain $X$ necessarily admits Reeb chords. The existence of this Reeb chords either follows from an…

For a simply-connected compact semisimple Lie group $G$ and its maximal torus $T$, we study the $A_{\infty}$-functor associated to the moment Lagrangian correspondence from the cotangent bundle $T^*G$ to the square $G/T^{-} \times G/T$. In…

We study the combinatorial Reeb flow on the boundary of a four-dimensional convex polytope. We establish a correspondence between "combinatorial Reeb orbits" for a polytope, and ordinary Reeb orbits for a smoothing of the polytope,…

In this paper we study submanifolds of contact manifolds. The main submanifolds we are interested in are contact coisotropic submanifolds. Based on a correspondence between symplectic and contact coisotropic submanifolds, we can show…

The graph complex acts on the spaces of Poisson bi-vectors $P$ by infinitesimal symmetries. We prove that whenever a Poisson structure is homogeneous, i.e. $P = L_{\vec{V}}(P)$ w.r.t. the Lie derivative along some vector field $\vec{V}$,…

We (re)consider how the Fukaya category of a Lefschetz fibration is related to that of the fibre. The distinguishing feature of the approach here is a more direct identification of the bimodule homomorphism involved.