Geometry of surface group homomorphisms

The space of homomorphisms from the fundamental group of a compact surface to a Lie group is a remarkably rich and versatile object, playing a key role in mathematical developments spanning disciplines of algebra, analysis, geometry, and mathematical physics. In this talk I will discuss and weave together two major threads of research within this larger story: The study of manifolds that are obtained by taking quotients of symmetric spaces (the "inside view") and those obtained as quotients of domains in flag varieties (the "boundary view"). This discussion will start with classical objects--hyperbolic structures on surfaces---and continue into topics of ongoing research.