Abstract: Schubert calculus refers to the study of cohomology theories of flag varieties and their applications to enumerative geometry. In this series of talks, I will introduce nil-Hecke rings corresponding to root systems and Coxeter groups. My goal is to explain how their algebraic structure can be used as a computational tool to address problems in Schubert calculus. While there are several combinatorial models for Schubert calculus, the Nil-Hecke ring model have the advantage of generalizing to flag varieties of any Lie-type without introducing extra computational complexity. Nil-Hecke rings can also be used to determine geometric properties of Schubert varieties, such as smoothness.
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