Variations on a Theme of Schubert Calculus

TL;DR

This tutorial provides an accessible overview of combinatorial and algebraic tools in Schubert calculus, focusing on Grassmannians, flag varieties, and orthogonal Grassmannians, including less-covered topics like type B Schubert calculus.

Contribution

It introduces the orthogonal Grassmannian and type B Schubert calculus, filling a gap in existing educational resources and connecting combinatorial and geometric perspectives.

Findings

Comprehensive overview of Schubert calculus tools

Introduction to orthogonal Grassmannians and type B calculus

Educational resource based on graduate lecture notes

Abstract

In this tutorial, we provide an overview of many of the established combinatorial and algebraic tools of Schubert calculus, the modern area of enumerative geometry that encapsulates a wide variety of topics involving intersections of linear spaces. It is intended as a guide for readers with a combinatorial bent to understand and appreciate the geometric and topological aspects of Schubert calculus, and conversely for geometric-minded readers to gain familiarity with the relevant combinatorial tools in this area. We lead the reader through a tour of three variations on a theme: Grassmannians, flag varieties, and orthogonal Grassmannians. The orthogonal Grassmannian, unlike the ordinary Grassmannian and the flag variety, has not yet been addressed very often in textbooks, so this presentation may be helpful as an introduction to type B Schubert calculus. This work is adapted from the author's lecture notes for a graduate workshop during the Equivariant Combinatorics Workshop at the Center for Mathematics Research, Montreal, June 12-16, 2017.