Frontiers of realityin Schubert calculus Frank Sottile Algebraic Geometry Seminar Texas A&M University 1 December 2008 This talk is a warm up for an invited talk that I will give in the Current Events Bulletin at the AMS national meeting in January 2009. The Shapiro conjecture for Grassmannians (now a Theorem of Mukhin, Tarasov, and Varchenko) asserts that all (a priori complex-number) solutions to certain geometric problems from the Schubert calculus are actually real. Their proof is quite remarkable, using ideas from integrable systems and representation theory. Despite these advances, the full Shapiro conjecture remains open with several interesting and not quite understood generalizations that are likely true. This talk will introduce the Shapiro conjecture for Grassmannians and its links to subjects from combinatorics to complex analysis to control theory and then give an idea of its proofs and consequences, and its extensions.