Classical and Modern Schubert Calculus

古典和现代舒伯特微积分

• 批准号: 1205351
• 负责人: Anders Buch
• 金额: $ 15.49万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-08-01 至 2015-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1205351&HistoricalAwards=false
• 关键词: Classical; Modern; Schubert; Calculus

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.

枚举几何中的许多问题可以归结为旗流形X = G/P的上同调环中的计算。这个环的结构常数由深刻而美丽的组合学所支配，它们的研究福尔斯属于几个数学学科的交叉点。 例如，当X是格拉斯曼簇时，这些结构常数是Littlewood-Richardson系数，它也描述了GL（n）表示的张量积，对称多项式的乘积，并在从线性代数到统计和计算机科学的复杂性理论的许多其他领域中发挥作用。 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量。 艾伦·克努森（Allen Knutson）提出的一个更一般的组合公式指出，两步旗簇的结构常数等于三角形谜题的数量，三角形谜题的侧面有指定的整数标签。 研究人员希望证明这一猜想。 研究者的早期工作已经确定，两步旗簇的结构常数专门用于格拉斯曼的Gromov-Witten不变量，因此可以计算在一般位置满足三个舒伯特簇的固定次数的有理曲线的数量。Gromov-Witten不变量也决定了（小）量子上同调环的结构，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这个环作为事实的最精确的描述。 研究者还将研究关于旗流形的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，可以说出有多少人。 枚举几何是研究这种计数问题以及解决这些问题的方法。 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，因此解图形的数量是计算的结果。 然而，在大多数情况下，在一个公式的存在下，清楚地表明解决方案的数量是非负的，最好理解枚举问题的组合方面。 例如，这样的正公式对于证明哪些枚举问题有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，正公式倾向于用深层的组合结构和方法来包围自己，这些结构和方法比公式本身更能洞察几何问题。 研究者将试图证明一些这种类型的正公式。 他还计划编写一个计算机程序，能够计算一个大家庭的枚举问题的解决方案。这种类型的例子对于在该领域取得进展很重要，同时对于想要学习该主题的学生或其他人非常有用。 最后，研究人员将继续从事研究生和本科生在他的研究。