In pursuit of Hofstadter's butterfly and Veselago's lens

Graphene is the prototypical atomically-thin material, and the most readily accessible condensed matter system in which the charge carriers obey a quasi-relativistic, linear energy-momentum relation. These properties have made it a remarkably fruitful system, with discoveries ranging from new water filtration mechanisms to novel quantized Hall effects, and the creation of entire fields including the discovery of topological insulators, and new phenomena and applications via arbitrary stacks of 2D layered materials. Here, I will present two ongoing explorations of intriguing phenomena that are arguably best pursued in the context of graphene: our bid to visualize the fractal structure of Hofstadter's butterfly, and a viable approach to negative refraction in the electro-optic version of Veselago's lens. Along the way I will highlight our recent advances in cryogenic infrared magneto-spectroscopy, and initial discoveries made while playing with an atomically-thin quantum spin liquid candidate.