Related papers: Orbifold adjunction formula and symplectic cobordi…
A cobordism between links in thickened surfaces consists of a surface $ S $ and a $3$-manifold $M $, with $ S $ properly embedded in $ M \times I $. We show that there exist links in thickened surfaces such that if $(S,M) $ is a cobordism…
In this paper, we compute contact homology of some quasi-regular contact structures, which admit Hamiltonian actions of Reeb type of Lie groups. We will discuss the toric contact case, (where the torus is of Reeb type), and the case of…
The deformation problem for pseudoholomorphic curves and related geometrical properties of the total moduli space of pseudoholomorphic curves are studied. A sufficient condition for the saddle point property of the total moduli space is…
We first study symplectically embedded curves in symplectic surfaces with high self-intersection numbers compared to their genus. We prove in two different ways that such a curve completely determines both the diffeomorphism type of the…
R Kerr's Ricci-flat Lorentz 4-manifold was shown by B Carter in 1968 to support a classically completely integrable system of geodesics; here we interpret this system in terms of a locally conformally symplectic structure, which we identify…
Let $K$ be a compact Lie group of positive dimension. We show that for most unitary $K$-modules the corresponding symplectic quotient is not regularly symplectomorphic to a linear symplectic orbifold (the quotient of a unitary module of a…
This paper develops a rigorous mathematical framework for light propagation by constructing the optical phase space with its symplectic structure and the extended phase space with its contact structure. We prove that light rays in…
This paper studies linear relations and their self-adjoint realizations arising from 2d-dimensional canonical systems, with a focus on how the symplectic structure interacts with boundary conditions. Understanding this interplay allows us…
We compute cohomology of the moduli space of genus three curves with level two structure and some related spaces. In particular, we determine the cohomology groups of the moduli space of plane quartics with level two structure as…
In this article, we construct countably many mutually non-isotopic diffeomorphisms of some closed non simply-connected 4-manifolds that are homotopic to but not isotopic to the identity, by surgery along $\Theta$-graphs. As corollaries of…
We introduce the notion of asymptotically finitely generated contact structures, which states essentially that the Symplectic Homology in a certain degree of any filling of such contact manifolds is uniformly generated by only finitely many…
We study two different actions on the moduli spaces of logarithmic connections over smooth complex projective curves. Firstly, we establish a dictionary between logarithmic orbifold connections and parabolic logarithmic connections over the…
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.…
The geometric quantization of a symplectic manifold endowed with a prequantum bundle and a metaplectic structure is defined by means of an integrable complex structure. We prove that its semi-classical limit does not depend on the choice of…
We compute symplectic cohomology for Milnor fibres of certain compound Du Val singularities that admit small resolution by using homological mirror symmetry. Our computations suggest a new conjecture that the existence of a small resolution…
We study the space of pseudo-holomorphic spheres in compact symplectic manifolds with convex boundary. We show that the theory of Gromov-Witten invariants can be extended to the class of semi-positive manifolds with convex boundary. This…
We discuss Witten's formulas for the symplectic volumes of moduli spaces of flat connections on 2-manifolds from the viewpoint of Hamiltonian cobordism as introduced by Ginzburg-Guillemin-Karshon.
We consider strong symplectic fillings of the unit cotangent bundle of a hyperbolic surface, equipped with its canonical contact structure. We show that every finitely presentable group can be realised as the fundamental group of such a…
We introduce symplectic Calabi-Yau caps to obtain new obstructions to exact fillings. In particular, it implies that any exact filling of the standard unit cotangent bundle of a hyperbolic surface has vanishing first Chern class and has the…
These notes are based on a course that took place at the Universit\'e de Nantes in June 2011 during the "Trimester on Contact and Symplectic Topology". We will explain how holomorphic curves can be used to study symplectic fillings of a…