Schubert Calculus, Quiver Varieties, and Kazhdan-Lusztig Coefficients

数学中的许多问题可以表述如下：如果一个人对流形(例如，向量空间或球面)中的点强加了一系列独立的条件，是否有任何点满足所有条件？这样的问题具有线性近似，更容易受到系统攻击，但仍然相当困难。这一领域的线性相交问题被称为19世纪的“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。这种追求的一个不同寻常的特点是，(长的，慢的)公式通常可以用来准确地计算解的数量，但它们不适合轻松地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机会。第一个项目将旗形上的交集理论替换为它们的余切丛，这是一个很小的变化，但随后将后者实现为“Nakajima箭图变种”的特例。圆周率派的“舒伯特微积分难题”在这些更大的箭筒变种上得到了最好的解释，箭筒变种是实际感兴趣的旗帜流形(余切丛)之间的中间地带。第二个是关于戈尔丁和圆周率最近的公式简洁地计算这个相交理论(尽管不是明显肯定的，这是该领域的长期目标)。这涉及到一些具有有趣的代数性质但没有明确的几何起源的算子的创建；该项目的一部分是寻找这种几何。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.