Symplectic packing constructions

Toric symplectic ball packing

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...

Packing stability for irrational symplectic 4-manifolds

The packing stability in symplectic geometry was first noticed by Biran [Bir97]: the symplectic obstructions to embed several balls into a manifold disappear when their size is small enough. This phenomenon is known to hold for all closed manifolds with rational symplectic class (see [Bir99] for the 4-dimensional case, and [BH11, BH13] for higher dimensions), as well as ellipsoids [BH13]. In th...

Lecture 5: Some Basic Constructions in Symplectic Topology

In complex geometry the blowing-up operation amounts to replace a point in a space by the space of complex tangent lines through that point. It is a local operation which can be explicitly written down as follows. Consider the blow-up of Cn at the origin. This is the complex submanifold of Cn × CPn−1 C̃ ≡ {((z1, z2, · · · , zn), [w1, w2, · · · , wn])|(z1, z2, · · · , zn) ∈ [w1, w2, · · · , wn]}....

Some New Constructions of Authentication Codes with Arbitration and Multi-Receiver from Singular Symplectic Geometry

Two Constructions of Multireceiver Authentication Codes from Singular Symplectic Geometry over Finite Fields

Multireceiver authentication codes allow one sender to construct an authenticated message for a group of receivers such that each receiver can verify authenticity of the received message. In this paper, two constructions of multireceiver authentication codes from singular symplectic geometry over finite fields are given. The parameters and the probabilities of success for different types of dec...