The connections between Maxwell’s equations and symplectic matrix are studied. First, we analyze the continuous-time Maxwell’s differential equations in free space and verify its time evolution matrix (TEMA) is symplectic-unitary matrix for complex space or symplectic-orthogonal matrix for real space. Second, the spatial differential operators are discretized by pseudo-spectral (PS) approach wi...

We consider Riemannian metrics compatible with the symplectic structure on T 2 ×M , where T 2 is a symplectic 2-Torus and M is a closed symplectic manifold. To each such metric we attach the corresponding Laplacian and consider its first positive eigenvalue λ1. We show that λ1 can be made arbitrarily large by deforming the metric structure, keeping the symplectic structure fixed. This extends a...

We define and solve the toric version of the symplectic ball packing problem, in the sense of listing all 2n-dimensional symplectic–toric manifolds which admit a perfect packing by balls embedded in a symplectic and torus equivariant fashion. In order to do this we first describe a problem in geometric–combinatorics which is equivalent to the toric symplectic ball packing problem. Then we solve...

It is shown how one can do symplectic reduction for locally conformal symplectic manifolds, especially with an action of a Lie group. This generalizes well known procedures for symplectic manifolds to the slightly larger class of locally conformal symplectic manifolds. The whole setting is very conformally invariant.

This paper uses a generalization of symplectic geometry, known as n-symplectic geometry and developed by Norris1, to find observables on three-dimensional manifolds. It will be seen that for the cases considered , the n-symplectic observables are derivable from the symplectic observables of C2. The quantization of these observables, as well as those on the frame bundle of R, is also examined.

We introduce an analogue of the inflation technique of LalondeMcDuff, allowing us to obtain new symplectic forms from symplectic surfaces of negative self-intersection in symplectic four-manifolds. We consider the implications of this construction for the symplectic cones of Kähler surfaces, proving along the way a result which can be used to simplify the intersections of distinct pseudoholomor...

We show how to construct a resolution of symplectic orbifolds obtained as quotients of presymplectic manifolds with a torus action. As a corollary, this allows us to desingularise generic symplectic quotients. Given a manifold with a Hamiltonian action of a compact Lie group, symplectic reduction at a coadjoint orbit which is transverse to the moment map produces a symplectic orbifold. If moreo...

In [17], we considered the local classification of plane curves on the symplectic plane. In particular, we introduced the number “symplectic defect”, which represents the difference of two natural equivalence relations on plane curves, the equivalence by diffeomorphisms and that by symplectomorphisms. For an immersion, two equivalence relations coincide, so the symplectic defect is null. For co...

We study differential equations that lead to extremal points in symplectic pseudospectra. In a two-level approach, where on the inner level we compute extremizers of the symplectic ε-pseudospectrum for a given ε and on the outer level we optimize over ε, this is used to solve symplectic matrix nearness problems such as the following: For a symplectic matrix with eigenvalues of unit modulus, we ...

We study the generic properties of symplectic relations. Local models of symplectic relations are described and the corresponding local symplectic invariants are derived. A stratification of the Lagrangian Grassmannian in the product symplectic space (N ×M, π∗ MωM − π∗ NωN ) is constructed and global homological properties of the strata are investigated.