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：如果一个人在流形（例如，向量空间或球体）中的点上强加了一系列独立的条件，是否有任何点满足所有条件？这类问题的线性近似更容易受到系统攻击，但仍然相当困难。这个领域的线性相交问题去了19世纪的名称“舒伯特演算”。PI的三个项目中有两个是舒伯特微积分领域的。这种追求的一个不寻常的特点是，（长，慢）公式通常可以精确地计算解的数量，但它们不适合轻松检查这个数字是否为正数。该项目为研究生提供了研究培训的机会。第一个项目用余切丛取代旗流形（相交理论），这是一个很小的变化，但随后将后者实现为“Nakajima流形”的特例。PI的“舒伯特演算难题”最好解释这些较大的类，一个中间地面之间（余切束）旗流形的实际利益。第二个问题涉及戈尔丁最近的一个公式，PI简洁地计算了这个相交理论（虽然不是明显的积极，但这是该领域的长期目标）。这涉及到创建一些具有有趣的代数性质的算子，但没有明确的几何起源;该项目的一部分是对这种几何的搜索。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。