Schubert Calculus, and Degenerations to Toric Simplicial Complexes

舒伯特微积分和环面单纯复形的退化

• 批准号: 0636154
• 负责人: Allen Knutson
• 金额: $ 6.01万
• 依托单位: University of California-San Diego
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-10-01 至 2008-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0636154&HistoricalAwards=false
• 关键词: Schubert; Calculus; Degenerations; Toric; Simplicial

Dr. Knutson's proposed work covers two rather different connections ofcombinatorics and algebraic geometry. The first concerns Schubert calculus, a boolean lattice's worth of problems whose minimal element is the (now extremely well-understood) intersection theory on Grassmannians. Its extensions include equivariant intersection theory (recently solved by Knutson and T. Tao), K-theory (recently solved by A. Buch), quantum cohomology (unsolved, but a very solid conjecture exists), replacing the Grassmannian by larger flag manifolds, and analogues for arbitrary Lie groups (for these last two almost nothing is known). Since the submission of the proposal, much progress has been made (by Knutson and R. Vakil) towards one more level in this lattice, the equivariant K-theory of Grassmannians, but all other combinations remain. The second part is a generalization of Littelmann's path model in representation theory, which one should regard in this context as providing a flat degeneration of the flag manifold to a union of toric varieties. In the generalization, the only property used of the flag manifold is that it carries an action of the circle with isolated fixed points. Many other varieties should thus have a "path model" for their coordinate ring, such as toric varieties (a testbed, where the theory is rather trivial), wonderful compactifications, and Hilbert schemes. As an example application, this would provide a ppositive formula for Haiman's generalization of the (q,t)-Catalan numbers (where positivity is known for vanishing-cohomology reasons, but there is no formula).The first of Dr. Knutson's two projects concerns a 19th-century intrusionof combinatorics into algebraic geometry: counting the number of lines (or planes, or chains consisting of a point inside a line inside a plane etc.)satisfying a number of generic intersection conditions. The first interesting such question is ``Given four generic lines in space, how many other lines touch all four?'' (The answer is 2.) There are many generalizations of this problem; one of the newest and most exciting is quantum intersection theory, in which there may be no single solution to all the conditions, but rather the solution may "quantum tunnel" between the requirements, while paying a well-defined "penalty."(In physical terms, this penalty means the occurrence is impossible classically, but in the quantum world is only very rare.) There already exist formulae to solve any one of this huge family of problems, but they are extremely unsatisfying, as they determine the count by adding and subtracting many numbers. Such cancelative formulae are essentially useless for proving that a general class of intersection problems has an answer, and moreover they are computationally very inefficient. Dr. Knutson and his collaborators have provided noncancelative formulae for some of these generalizations, and have conjectures about others. His other project is not directly related, though it also uses combinatorics to control algebraic geometry, building on work of P. Littelmann. Littelmann showed how to use the very great symmetry of certain algebraic spaces, such as the set of all k-planes in n-space, to compute the space of functions on them in terms of lattice points inside a union of large-dimensional tetrahedra. This second proposal is based on recent work of Dr. Knutson's indicating that this large degree of symmetry is unnecessary -- a single circular symmetry, plus a technical (but common and easily checked) condition, seem to be enough to be able to make use of this lattice-point machinery. Such algebraic spaces with symmetry are endemic in mathematics and physics.

克纳森博士提出的工作涵盖了组合学和代数几何的两种截然不同的联系。第一个涉及舒伯特微积分，这是一个布尔格问题，其最小元素是（现在已经非常容易理解的）格拉斯曼函数的交集理论。它的扩展包括等变交集理论（最近由 Knutson 和 T.Tao 解决）、K 理论（最近由 A.Buch 解决）、量子上同调（未解决，但存在一个非常可靠的猜想），用更大的标志流形取代格拉斯曼，以及任意李群的类似物（对于最后两个几乎一无所知）。自提交提案以来，（Knutson 和 R. Vakil）在该格子的另一个层次（格拉斯曼的等变 K 理论）方面取得了很大进展，但所有其他组合仍然存在。第二部分是表示论中利特曼路径模型的概括，在这种情况下，人们应该将其视为向复曲面簇的并集提供标志流形的平坦退化。概括来说，标志流形使用的唯一属性是它承载具有孤立固定点的圆的作用。因此，许多其他变体应该有一个针对其坐标环的“路径模型”，例如环面变体（一个测试平台，其中的理论相当琐碎）、奇妙的紧化和希尔伯特方案。作为一个示例应用，这将为 Haiman 对 (q,t)-Catalan 数的推广提供一个正公式（其中正性因消失上同调原因而闻名，但没有公式）。Knutson 博士的两个项目中的第一个涉及 19 世纪组合数学对代数几何的入侵：计算线（或平面，或由平面内线内的点组成的链）的数量等）满足许多通用交叉条件。第一个有趣的此类问题是“给定空间中的四条通用线，有多少条其他线接触所有四条线？”（答案是2。）这个问题有很多概括；最新和最令人兴奋的理论之一是量子交叉理论，其中可能没有针对所有条件的单一解决方案，而是该解决方案可以在要求之间“量子隧道”，同时付出明确的“惩罚”。（在物理术语中，这种惩罚意味着这种情况在经典中是不可能发生的，但在量子世界中非常罕见。）已经存在解决这一庞大问题系列中任何一个问题的公式，但它们非常不令人满意，因为它们通过以下方式确定计数：添加和减去许多数字。这样的抵消公式对于证明一般类型的交叉问题有答案基本上是无用的，而且它们的计算效率非常低。克纳森博士和他的合作者为其中一些概括提供了非抵消公式，并对其他概括做出了猜想。他的另一个项目没有直接相关，尽管它也基于 P. Littelmann 的工作，使用组合学来控制代数几何。 Littelmann 展示了如何利用某些代数空间（例如 n 空间中所有 k 平面的集合）的极大对称性，以大维四面体并内的格点来计算其上的函数空间。第二个提议基于克努森博士最近的工作，表明这种大程度的对称性是不必要的——单个圆形对称性，加上技术（但常见且易于检查）条件，似乎足以能够利用这种格点机械。这种具有对称性的代数空间在数学和物理学中很常见。