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。第一个涉及舒伯特微积分，这是一个布尔格的问题，其最小元素是(现在已经非常清楚的)关于Grassmannians的交集理论。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)，用更大的旗形取代Grassmanian，以及任意Lie群的类似(对于后两者，几乎什么都不知道)。自从提案提交以来，Knutson和R.Vakil已经取得了很大的进展，朝着这个格子中的另一个层次--Grassmannians的等变K-理论，但所有其他组合仍然存在。第二部分是对Littelmann的路径模型在表征理论中的推广，在这一背景下，人们应该认为它提供了旗帜流形到环面变种的联合的平坦退化。在推广中，旗形使用的唯一性质是它带有孤立不动点的圆的作用。因此，许多其他变种应该有其坐标环的“路径模型”，例如Toric变种(一个试验台，其中的理论相当琐碎)，奇妙的紧凑化，和希尔伯特方案。作为一个实例应用，这将为Haiman对(q，t)-Catalan数的推广提供一个正定公式(其中正性因上同调消失而闻名，但没有公式)。Knutson博士的两个项目中的第一个涉及19世纪组合学对代数几何的侵入：计算满足一些一般相交条件的直线(或平面，或由平面内直线内一点组成的链等)的数量。第一个有趣的问题是“假设空间中有四条一般的线，有多少条其他线能接触到这四条线？”(答案是2。)对这个问题有很多概括；最新、最令人兴奋的理论之一是量子交集理论，该理论认为，对于所有条件，可能没有单一的解决方案，而是解决方案可能会在不同要求之间形成“量子隧道”，同时支付定义明确的“惩罚”。(从物理上讲，这种惩罚意味着这种情况在经典意义上是不可能发生的，但在量子世界中是非常罕见的。)已经有了解决这一大类问题中的任何一个的公式，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。这种抵消公式对于证明一般的交集问题是有答案的基本上是无用的，而且在计算上是非常低效的。克努森博士和他的合作者为其中的一些推广提供了不可抵消的公式，并对其他的进行了猜想。他的另一个项目没有直接关系，尽管它也使用组合学来控制代数几何，建立在P.Littelmann的工作基础上。Littelmann展示了如何利用某些代数空间的非常大的对称性，例如n-空间中所有k平面的集合，通过大维四面体并中的格点来计算它们上的函数空间。第二个建议是基于Knutson博士最近的工作，他指出这种高度的对称性是不必要的--一个单一的圆形对称性，加上一个技术性的(但常见的和容易检查的)条件，似乎足以利用这种格点机制。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.