Experimentation in the Schubert Calculus

TL;DR

This paper discusses how computer-based experimentation in Schubert calculus has led to new insights, conjectures, and understanding of its intrinsic structures, demonstrating the power of computational methods in mathematical research.

Contribution

It introduces the role of large-scale computer experimentation in uncovering new phenomena and conjectures in Schubert calculus and its intrinsic structures.

Findings

Computer experimentation has inspired new conjectures in real Schubert calculus.

Insights into the Galois groups of Schubert problems are emerging from computational experiments.

Computers enable exploration of subtle phenomena in Schubert calculus that are difficult to analyze analytically.

Abstract

Many aspects of Schubert calculus are easily modeled on a computer. This enables large-scale experimentation to investigate subtle and ill-understood phenomena in the Schubert calculus. A well-known web of conjectures and results in the real Schubert calculus has been inspired by this continuing experimentation. A similarly rich story concerning intrinsic structure, or Galois groups, of Schubert problems is also beginning to emerge from experimentation. This showcases new possibilities for the use of computers in mathematical research.