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Frontiers of reality in Schubert calculus


Author: Frank Sottile
Journal: Bull. Amer. Math. Soc. 47 (2010), 31-71
MSC (2010): Primary 14M15, 14N15
DOI: https://doi.org/10.1090/S0273-0979-09-01276-2
Published electronically: November 2, 2009
MathSciNet review: 2566445
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Abstract: 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.


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Additional Information

Frank Sottile
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
Email: sottile@math.tamu.edu

DOI: https://doi.org/10.1090/S0273-0979-09-01276-2
Keywords: Schubert calculus, Bethe ansatz, Wronskian, Calogero-Moser space
Received by editor(s): July 6, 2009
Received by editor(s) in revised form: July 22, 2009
Published electronically: November 2, 2009
Additional Notes: The work of Sottile is supported by NSF grant DMS-0701050
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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