Schubert Calculus, and Degenerations to Toric Simplicial Complexes

克努森博士提出的工作涵盖了组合学和代数几何两种截然不同的联系。第一个与舒伯特微积分有关，这是一个布尔格的问题，其最小元素是格拉斯曼的交集理论（现在已经非常好地理解了）。它的扩展包括等变交理论（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），用更大的标志流形代替格拉斯曼，以及任意李群的类似物（对于后两个几乎一无所知）。自从提出这个建议以来，（由Knutson和R. Vakil）在这个晶格的另一个层次上取得了很大的进展，即格拉斯曼的等变k理论，但所有其他组合仍然存在。第二部分是表征理论中Littelmann路径模型的概括，在这种情况下，人们应该将其视为提供标志流形的平坦退化到环面变体的联合。在推广中，标志流形的唯一性质是它具有具有孤立不动点的圆的作用。因此，许多其他的变种应该有一个“路径模型”用于它们的坐标环，比如环型变种（一个试验台，在那里理论是相当琐碎的）、奇妙的紧化和希尔伯特方案。作为一个示例应用，这将为Haiman的(q,t)-Catalan数的推广提供一个正公式（其中正因为消上同调的原因而已知，但没有公式）。克努森博士的两个项目中的第一个涉及19世纪将组合学引入代数几何：计算满足若干一般相交条件的线（或平面，或由平面内线内点组成的链等）的数量。第一个有趣的问题是“给定空间中的四条一般线，有多少其他线与这四条线相连？”（答案是2。）这个问题有很多概括；其中最新和最令人兴奋的是量子交叉理论，其中可能没有所有条件的单一解决方案，而是解决方案可能在要求之间“量子隧道”，同时支付明确定义的“惩罚”。（用物理术语来说，这种惩罚意味着这种情况在经典理论中是不可能发生的，但在量子世界中却非常罕见。）已经有公式可以解决这个庞大的问题家族中的任何一个，但它们非常不令人满意，因为它们通过加减许多数字来确定计数。这样的消去式对于证明一类一般的交点问题有解是毫无用处的，而且在计算上效率非常低。克努森博士和他的合作者已经为其中一些概括提供了不可消去的公式，并对其他一些概括进行了推测。他的另一个项目与此没有直接关系，尽管它也以P. Littelmann的工作为基础，使用组合学来控制代数几何。利特尔曼展示了如何利用某些代数空间的极大对称性，比如n空间中所有k平面的集合，用高维四面体并集中的晶格点来计算它们上的函数空间。第二个建议是基于克努森博士最近的工作，他指出这种大程度的对称是不必要的——一个单一的圆形对称，加上一个技术条件（但常见且容易检查），似乎足以利用这种点状机制。这种具有对称性的代数空间在数学和物理中是特有的。

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.