Combinatorial algebraic geometry: Modern Schubert calculus and generalized splines

每当方程模拟我们周围的世界时-无论是描述国内生产总值如何依赖于不同的经济部门，还是预测航天器或飓风的轨迹-这些方程也描述了一个几何物体。 对于一个简单的方程组，比如直线x+y=1，几何学似乎没有多大用处。 当我们有复杂的系统，有几十个方程，几百个变量时，几何方法可以是分析系统重要的定量和定性方面的少数方法之一，比如它有一个最大值或它有多少块。 这个研究项目为描述复杂的方程组开发了重要的几何工具。 PI提出了解决现代Schubert微积分中三个相互关联的问题：第一，研究仿射Grassmannian的上同调;第二，研究Schubert簇的交同调;第三部分研究了旗簇的一个子簇族Hessenberg簇的上同调，以回答关于旗簇的主要公开问题。 为此，PI将使用GKM理论，一种描述等变上同调的组合和代数算法，以及GKM理论的最近扩展，称为广义样条。 该项目的目标是：1）开发广义样条的理论工具和算法，2）应用这些工具在我们研究的各种上同调环中执行特定的计算和组合计算。

Whenever equations model the world around us---whether describing how the gross domestic product depends on different sectors of the economy or predicting the trajectory of a spacecraft or a hurricane---those equations also describe a geometric object. For a simple system of equations like the line x+y=1, geometry might not seem very useful. When we have complicated systems with dozens of equations in hundreds of variables, methods from geometry can be one of the few ways to analyze important quantitative and qualitative aspects of the system, like where it has a maximum or how many pieces it has. This research project develops important geometric tools for describing complicated systems of equations. It then applies these tools to specific systems of equations with important applications in mathematics, physics, computer science, and other fields.The PI proposes to solve three linked questions in modern Schubert calculus: the first studies the cohomology of the affine Grassmannian; the second studies the intersection homology of Schubert varieties; and the third studies the cohomology of a family of subvarieties of the flag variety called Hessenberg varieties, in order to answer major open questions about the flag variety. To do this, the PI will use GKM theory, a combinatorial and algebraic algorithm to describe equivariant cohomology, as well as a recent extension of GKM theory called generalized splines. The goals of the project will be 1) to develop theoretical tools and algorithms in generalized splines and 2) to apply these tools to perform specific computational and combinatorial calculations in the various cohomology rings we study.