## Frontiers of reality in Schubert calculus

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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.## References

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## Additional Information

**Frank Sottile**- Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
- MR Author ID: 355336
- ORCID: 0000-0003-0087-7120
- Email: sottile@math.tamu.edu
- 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
- © Copyright 2009
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2566445