Regular maps of high density

A regular map is a surface together with an embedded graph, having properties similar to those of the surface and graph of a platonic solid. We analyze regular maps with reflection symmetry and a graph of density strictly exceeding 1/2, and we conclude that all regular maps of this type belong to a family of maps naturally defined on the Fermat curves x^n+y^n+z^n=0, excepting the one corresponding to the tetrahedron.Comment: 13 pages, 4 figure

Lattice symmetry breaking perturbations for spiral waves

Spiral waves in two-dimensional excitable media have been observed experimentally and studied extensively. It is now well-known that the symmetry properties of the medium of propagation drives many of the dynamics and bifurcations which are experimentally observed for these waves. Also, symmetry-breaking induced by boundaries, inhomogeneities and anisotropy have all been shown to lead to different dynamical regimes as to that which is predicted for mathematical models which assume infinite homogeneous and isotropic planar geometry. Recent mathematical analyses incorporating the concept of forced symmetry-breaking from the Euclidean group of all planar translations and rotations have given model-independent descriptions of the effects of media imperfections on spiral wave dynamics. In this paper, we continue this program by considering rotating waves in dynamical systems which are small perturbations of a Euclidean-equivariant dynamical system, but for which the perturbation preserves only the symmetry of a regular square lattice

The 3D entangled structure of the proton; transverse degrees of freedom in QCD, momenta, spins and more

Light-front quantized quark and gluon states (partons) play a dominant role in high energy scattering processes. Initial state hadrons are mixed ensembles of partons, while produced pure partonic states appear as mixed ensembles of hadrons. The transition from collinear hard physics to the 3D structure including partonic transverse momenta is related to confinement which links color and spatial degrees of freedom. We outline ideas on emergent symmetries in the Standard Model and their connection to the 3D structure of hadrons. Wilson loops, including those with light-like Wilson lines such as used in the studies of transverse momentum dependent distribution functions (TMDs) may play a crucial role here, establishing a direct link between transverse spatial degrees of freedom and gluonic degrees of freedom.Comment: 8 pages, invited talk presented at the Lightcone 2017 Workshop, 18-22 Sep 2017, Mumbai, India; prepared for proceedings to be published in Few Body Physic