Frontiers of Reality in Schubert Calculus

TL;DR

This paper discusses the proof of the Mukhin-Tarasov-Varchenko theorem in Schubert calculus, highlighting its implications, alternative proofs, and the ongoing open questions related to the original Shapiro conjecture.

Contribution

It reviews the proof of the real solutions theorem in Schubert calculus and explores its broader mathematical implications and unresolved conjectures.

Findings

All solutions to certain Schubert calculus problems are real.

The theorem has multiple proofs, including one using integrable systems.

The original Shapiro conjecture remains unproven and is subject to modifications.

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.