Classical and Modern Schubert Calculus

计数几何中的许多问题都可以归结为一个旗形X=G/P的上同调环上的计算。这个环的结构常数是由深奥而优美的组合学决定的，它们的研究属于几个数学学科的交叉。例如，当X是Grassmann变种时，这些结构常数是Littlewood-Richardson系数，它还描述了GL(N)的表示的张量积，对称多项式的乘积，并在从线性代数到计算机科学中的统计和复杂性理论的许多其他领域发挥作用。著名的Littlewood-Richardson规则将任何Littlewood-Richardson系数表示为称为Tableaux的某些组合对象的数目。由Allen Knutson猜想的一个更一般的组合公式指出，一个两步旗簇的结构常数等于边上有指定整数标号的三角形谜题的个数。调查人员希望证明这一猜想。研究者的早期工作证明了两步标志簇的结构常数专用于Grassmannians的Gromov-Witten不变量，从而计算出在一般位置满足三个Schubert簇的定次有理曲线的个数。Gromov-Witten不变量还决定了(小)量子上同调环的结构，该环的定义受到物理学的启发，并与镜像对称性有关。因此，对克努森猜想的证明将建立对这个环作为事实的最准确描述。研究人员还将研究与旗形的K理论和量子K理论有关的其他问题。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种类型的几何图形的完整列表。虽然很难或不可能确定个别数字，但在许多情况下，可以说有多少人。计数几何是研究这类计数问题及其解决方法的学科。已经开发出强大的技术，可以将枚举几何问题转化为代数问题，从而使解图的数量是计算的结果。然而，在大多数情况下，枚举问题的组合方面最好是在存在一个公式的情况下才能理解，该公式清楚地表明解的数目是非负的。例如，这样的正公式对于证明关于枚举问题有任何解决方案的一般陈述要有用得多。令人惊讶的是，正公式比非正公式更难发现和证明。反过来，正公式往往被深层的组合结构和方法所包围，这些结构和方法比公式本身提供了对几何问题更多的洞察。调查者将尝试证明许多这种类型的正公式。他还计划编写一种计算机程序，能够计算一大类枚举问题的解。这种类型的例子对于在该领域取得进展很重要，同时对学生或其他想要学习这门学科的人来说也非常有用。最后，研究人员将继续让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.