**University of York Department of Physics, April 27, 2015.**

**Title:** On vortices : a short compendium of physical concepts and mathematical techniques

**Abstract:**

Vortices are presented are presented, starting from multivalued complex functions exhibiting singularities and proceeding to consider the Hilbert space formed by orbital angular momentum (OAM) states that provides representation to ISO(2) Euclidean algebra generators identified as the relevant physical observables. Next, the three classes of physical vortices, i.e. light, electron and atom vortices, are considered and basic construction techniques such as computer generated fork holograms, spiral phase plates, beam splitters, and the paraxial wave equation Laguerre-Gauss solutions are discussed. The mechanical effects of vorticity are investigated next by studying interaction models coupling matter to light carrying OAM of fractional or irrational angular momentum or with so-called C shaped beams. The resulting dynamical equations of motion appeared to be generalizations of the well known Raman-Nath equations of light-matter interaction.