Schedule, Notes/Handouts & Videos

We first give a general introduction to harmonic maps, especially from (universal cover of) a Riemann surface into hyperbolic surfaces, symmetric spaces, and singular spaces. We then explain why they are useful to study moduli space of surface group representations into PSL(2,R) and general Lie groups, and to study compactification of these moduli spaces. This talk will be introductory and expository. We emphasize on PSL(2,R) case to give a guiding picture and then explain other examples.

Like violets in spring, hyperbolic geometry sprouted up in several different places in the early 19th century. But the seeds had been germinating for quite some time before then. We will explore some of the first investigations into the parallel postulate and its negations, from ancient Greece to medieval Persia to Italy in the early modern age. What were people able to prove about hyperbolic geometry before they even acknowledged that it could exist?