  The theorem of Mukhin, Tarasov, and Varchenko (formerly the Shapiro
conjecture for Grassmannians) asserts that all (a priori complex) solutions to
certain geometric problems in the Schubert calculus are actually real. Their
proof is quite remarkable, using ideas from integrable systems, Fuchsian
differential equations, and representation theory. There is now a second proof
of this result, and it has ramifications in other areas of mathematics, from
curves to control theory to combinatorics. Despite this work, the original
Shapiro conjecture is not yet settled. While it is false as stated, it has
several interesting and not quite understood modifications and generalizations
that are likely true, and the strongest and most subtle version of the Shapiro
conjecture for Grassmannians remains open.
