Frontiers of reality in Schubert calculus

24Citations
Citations of this article
10Readers
Mendeley users who have this article in their library.

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. © 2009 American Mathematical Society.

Cite

CITATION STYLE

APA

Sottile, F. (2010). Frontiers of reality in Schubert calculus. Bulletin of the American Mathematical Society, 47(1), 31–71. https://doi.org/10.1090/S0273-0979-09-01276-2

Register to see more suggestions

Mendeley helps you to discover research relevant for your work.

Already have an account?

Save time finding and organizing research with Mendeley

Sign up for free