Schubert Calculus, Quiver Varieties, and Kazhdan-Lusztig Coefficients

舒伯特微积分、箭袋品种和 Kazhdan-Lusztig 系数

• 批准号: 1953948
• 负责人: Allen Knutson
• 金额: $ 33万
• 依托单位: Cornell University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-08-01 至 2024-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1953948&HistoricalAwards=false
• 关键词: Schubert; Calculus; Quiver; Varieties; Kazhdan

Many questions in mathematics can be phrased as follows: if one imposes a certain list of independent conditions on the points in a manifold (for example, a vector space or sphere), are there any points that satisfy all the conditions? Such questions have linear approximations more prone to systematic attack, but yet still quite difficult. This field of linear intersection questions goes by the 19th-century name "Schubert calculus". Two of the PI's three projects are in the realm of Schubert calculus. One of the unusual features of this pursuit is that (long, slow) formulae are generally available to count the number of solutions exactly, but they are ill-suited to easily check whether this number is positive. The project provides research training opportunities for graduate students.The first project replaces (intersection theory on) flag manifolds with their cotangent bundles, a small change, but then realizes the latter as special cases of "Nakajima quiver varieties". The PI's "Schubert calculus puzzles" are best interpreted on these larger quiver varieties, an intermediate ground between the (cotangent bundles of) flag manifolds of actual interest. The second concerns a recent formula of Goldin and the PI computing this intersection theory succinctly (although not manifestly positively, a long-term goal in the field). This involved the creation of some operators with intriguing algebraic properties, but no clear geometric origin; part of the project is a search for this geometry.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

数学中的许多问题可以表述如下：如果对流形（例如向量空间或球体）中的点施加一系列独立条件，是否有任何点满足所有条件？此类问题具有线性近似，更容易受到系统攻击，但仍然相当困难。这一线性相交问题领域在 19 世纪被称为“舒伯特微积分”。 PI 的三个项目中有两个属于舒伯特微积分领域。这种追求的不寻常特征之一是（长、慢）公式通常可用于精确计算解的数量，但它们不适合轻松检查该数字是否为正。该项目为研究生提供了研究培训机会。第一个项目用余切束替换了旗形流形（交集理论），这是一个很小的变化，但随后将后者实现为“中岛箭袋品种”的特例。 PI 的“舒伯特微积分难题”在这些较大的箭袋变种上得到了最好的解释，它们是实际感兴趣的旗流形（余切丛）之间的中间基础。第二个涉及 Goldin 和 PI 的最新公式，该公式简洁地计算了该交叉理论（尽管不是明显积极的，但这是该领域的长期目标）。这涉及创建一些具有有趣代数性质的运算符，但没有明确的几何起源；该项目的一部分是寻找这种几何形状。该奖项反映了 NSF 的法定使命，并且通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。