Introduction to the real spectrum. (pdf)
Notes from the second talk I gave at Colorado College in early December 2004 (the first was on the discrete geometry of Chicken McNuggets). The purpose of these notes is to convince a general mathematical audience that the collection of all orderings on a field is an interesting geometric object. Moreover, it develops some of the background material that one needs to appreciate "real arithmetic geometry," a topic of some very interesting recent papers of Colliot-Thelene, Parimala, and Scheiderer.