Related papers: Lagrangian isotopies and symplectic function theor…
Using methods from symplectic topology, we prove existence of invariant variational measures associated to the flow $\phi_H$ of a Hamiltonian $H\in C^{\infty}(M)$ on a symplectic manifold $(M,\omega)$. These measures coincide with Mather…
We prove that there are at most two possibilities for the base of a Lagrangian fibration from a complex projective irreducible symplectic fourfold.
A C-symplectic structure is a complex-valued 2-form which is holomorphically symplectic for an appropriate complex structure. We prove an analogue of Moser's isotopy theorem for families of C-symplectic structures and list several…
We generalize Voisin's theorem on deformations of pairs of a symplectic manifold and a Lagrangian submanifold to the case of Lagrangian normal crossing subvarieties. Partial results are obtained for arbitrary Lagrangian subvarieties. We…
A description of Lagrangian and Hamiltonian formalisms naturally arisen from the invariance structure of given nonlinear dynamical systems on the infinite--dimensional functional manifold is presented. The basic ideas used to formulate the…
Can a given Lagrangian submanifold be realized as the fixed point set of an anti-symplectic involution? If so, it is called \emph{real}. We give an obstruction for a closed Lagrangian submanifold to be real in terms of the displacement…
This article gives a classification, up to symplectic equivalence, of singular Lagrangian foliations given by a completely integrable system of a 4-dimensional symplectic manifold, in a full neighbourhood of a singular leaf of focus-focus…
In this article, we generalize the theory of discrete Lagrangian mechanics and variational integrators in two principal directions. First, we show that Lagrangian submanifolds of symplectic groupoids give rise to discrete dynamical systems,…
We propose a way to define and compute invariants of general smooth 4-manifolds based on topological twists of non-Lagrangian 4d N=2 and N=3 theories in which the problem is reduced to a fairly standard computation in topological A-model,…
In this paper, we study a series of $L^2$-torsion invariants from the viewpoint of the mapping class group of a surface. We establish some vanishing theorems for them. Moreover we explicitly calculate the first two invariants and compare…
The special geometry of calibrated cycles, closely related to mirror symmetry among Calabi--Yau 3-folds, is itself a real form of a new subject, which we call slightly deformed algebraic geometry. On the other hand, both of these geometries…
We introduce the notion of a symplectic Lie affgebroid and their Lagrangian submanifolds in order to describe the Lagrangian (Hamiltonian) dynamics on a Lie affgebroid in terms of this type of structures. Several examples are discussed.
Pairs (Hamiltonian system, Lagrangian distribution), called dynamical Lagrangian distributions, appear naturally in Differential Geometry, Calculus of Variations and Rational Mechanics. The basic differential invariants of a dynamical…
n this paper we define an invariant of a pair of 6 dimensional symplectic %optional manifold with vanishing 1st Chern class and its Lagrangian submanifold with vanishing Maslov index. This invariant is a function on the set of the path…
Given a smooth Tonelli Hamiltonian on the torus $\mathbb{T}^{n}$ and a $C^{2}$ Lagrangian graph $W \subset T^{*}\mathbb{T}^{n}$ that is invariant under the Hamiltonian flow and contained within a Ma\~n\'e supercritical energy level, we…
A Lagrangian field on a symplectic manifold $M$ is a family $\Lambda=\{\Lambda_x|x \in M\}$ of pointed Lagrangian submanifolds of $M$. This notion is a generalization of a real Lagrangian polarization for which each $\Lambda_x$ is the leaf…
We consider the equivalence problem for symplectic and conformal symplectic group actions on submanifolds and functions of symplectic and contact linear spaces. This is solved by computing differential invariants via the Lie-Tresse theorem.
In a previous article, we proved that symplectic homeomorphisms preserving a coisotropic submanifold C, preserve its characteristic foliation as well. As a consequence, such symplectic homeomorphisms descend to the reduction of the…
We introduce the Euler-Lagrange cohomology to study the symplectic and multisymplectic structures and their preserving properties in finite and infinite dimensional Lagrangian systems respectively. We also explore their certain difference…
We provide some constructions using Lagrangian cobordisms which improve known examples for some symplectic squeezing problems. Additionally, we prove a flexibility result that Lagrangian submanifolds which are Lagrangian isotopic are also…