Classical and Modern Schubert Calculus

枚举几何中的许多问题都可以归结为在标志流形X = G/P的上同环上的计算。这个环的结构常数是由深奥而美丽的组合学决定的，它们的研究是几个数学学科的交叉。例如，当X是Grassmann变量时，这些结构常数是Littlewood-Richardson系数，它也描述GL(n)表示的张量积，对称多项式的积，并在许多其他领域发挥作用，从线性代数到计算机科学中的统计学和复杂性理论。著名的利特尔伍德-理查德森法则将任何利特尔伍德-理查德森系数表示为某些组合对象的数量，这些组合对象被称为表。艾伦·克努森（Allen Knutson）推测了一个更一般的组合公式，指出两步旗的结构常数等于边带有指定整数标签的三角形谜题的数目。研究者希望证明这个猜想。研究者早期的工作已经确定了两步标志变量的结构常数专为Grassmannians的Gromov-Witten不变量，因此计算了在一般位置满足三个Schubert变量的固定度有理曲线的个数。Gromov-Witten不变量还决定了（小）量子上同环的结构，其定义受到物理学的启发，并与镜像对称有关。因此，克努森猜想的证明将建立对这个环最精确的描述，使之成为事实。研究者还将研究旗流形的k理论和量子k理论的其他问题。经典代数几何中的一个典型问题是确定满足一系列条件的某类几何图形的完整列表。虽然很难或不可能确定具体的数字，但在许多情况下，可以说出有多少人。计数几何是研究这类计数问题及其解决方法的学科。强大的技术已经开发出来，可以将枚举几何问题转化为代数问题，因此解图的数量是计算的结果。然而，在大多数情况下，最好的理解枚举问题的组合方面，存在一个公式，使它清楚地表明，解决方案的数量是非负的。例如，这样的正公式对于证明关于枚举问题有任何解的一般陈述更有用。令人惊讶的是，正公式比非正公式更难发现和证明。作为回报，正公式倾向于用深度组合结构和方法包围自己，这些结构和方法甚至比公式本身更能洞察几何问题。研究者将试图证明这类的若干正公式。他还计划编写一个计算机程序，能够计算大量枚举问题的解。这种类型的例子对于在该领域取得进展非常重要，同时对学生或其他想要学习该主题的人非常有用。最后，研究者将继续吸引研究生和本科生参与他的研究。

Many problems in enumerative geometry can be reduced to a computation in the cohomology ring of a flag manifold X = G/P. The structure constants of this ring are ruled by deep and beautiful combinatorics, and their study falls in the intersection of several mathematical disciplines. For example, when X is a Grassmann variety, these structure constants are the Littlewood-Richardson coefficients that also describe tensor products of representations of GL(n), products of symmetric polynomials, and play a role in numerous other areas ranging from linear algebra to statistics and complexity theory in computer science. The celebrated Littlewood-Richardson rule expresses any Littlewood-Richardson coefficient as the number of certain combinatorial objects called tableaux. A more general combinatorial formula, conjectured by Allen Knutson, states that the structure constants of a two-step flag variety are equal to the number triangular puzzles with specified integer labels on the sides. The investigator hopes to prove this conjecture. Earlier work of the investigator has established that the structure constants of two-step flag varieties specialize to the Gromov-Witten invariants of Grassmannians, and therefore count the number of rational curves of a fixed degree that meet three Schubert varieties in general position. The Gromov-Witten invariants also determine the structure of the(small) quantum cohomology ring, whose definition is inspired by physics and has relations to mirror symmetry. A proof of Knutson's conjecture will therefore establish the most precise description of this ring as a fact. The investigator will also study other questions concerning the K-theory and quantum K-theory of flag manifolds.A typical question in classical algebraic geometry is to identify the complete list of geometric figures of some type that satisfy a list of conditions. While it can be difficult or impossible to identify the individual figures, it is in many cases possible to say how many there are. Enumerative geometry is the study of such counting problems as well as methods to solve them. Powerful techniques have been developed that can translate an enumerative geometric problem into an algebraic problem, so that the number of solution figures is the result of a computation. However, the combinatorial aspects of an enumerative problem are in most cases best understood in the presence of a formula that makes it clear that the number of solutions is non-negative. For example, such positive formulas are much more useful for proving general statements about which enumerative problems have any solutions at all. Surprisingly, positive formulas are significantly more difficult to discover and prove than non-positive formulas. In return the positive formulas tend to surround themselves with deep combinatorial structures and methods that provide even more insight into the geometric problem than the formulas themselves. The investigator will attempt to prove a number of positive formulas of this type. He also plans to write a computer program capable of computing the solutions of a large family of enumerative problems. Examples of this type are important for making progress in the field, and are at the same time very useful for students or others who would like to learn the subject. Finally, the investigator will continue to engage graduate and undergraduate students in his research.