Lecture notes on quantum cohomology of the flag manifold

TL;DR

This paper provides an overview of recent combinatorial methods for computing genus 0 Gromov-Witten invariants of flag manifolds, emphasizing quadratic algebra techniques and collaborative research insights.

Contribution

It introduces combinatorial computation techniques and quadratic algebra approaches for quantum cohomology of flag manifolds, based on recent joint research.

Findings

Development of combinatorial formulas for Gromov-Witten invariants

Application of quadratic algebra methods to quantum cohomology

Connections to joint work with Gelfand, Kirillov, and Postnikov

Abstract

This is an exposition of some recent developments related to the object in the title, particularly the combinatorial computation of the (genus 0) Gromov-Witten invariants of the flag manifold and the quadratic algebra approach. The notes are largely based on my joint papers with S.Gelfand, A.N.Kirillov, and A.Postnikov. This is by no means an exhaustive survey of the subject, but rather a casual introduction to its combinatorial aspects.