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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 are now three proofs 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.
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