In this seminar I will give a brief overview of my PhD studies on the linear and nonlinear singular optics of photonic lattices, exploring deep connections between singularities in their energy-momentum spectrum such as conical intersections and flat bands, and wave singularities such as optical vortices. They are unified by the common theme of destructive interference; spectral singularities generically produce wave singularities, and vice versa.

As specific examples, I will consider nonlinear vortex dynamics and switching in ring lattices, pseudospin and vortex generation in photonic lattices with conical intersections, Fano resonances and interference in disordered flat band lattices, and entangled photon pair generation in one dimensional lattices with quadratic nonlinearity. The results I will present offer new ways to control the flow of light and its singularities, with many now being tested in experiments. The basic concepts of periodicity and singularities are ubiquitous and also applicable to other systems such as Bose-Einstein condensates, metamaterials, and cold atoms.