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学两个相当不同的联系。第一个涉及舒伯特演算，一个布尔格的价值问题，其最小元素是（现在非常好理解）交叉理论的格拉斯曼。它的扩展包括等变相交理论（最近解决了克努森和T。Tao），K理论（最近由A. Buch）、量子上同调（未解决，但存在一个非常可靠的猜想）、用更大的旗形流形取代格拉斯曼流形以及任意李群的类似物（对于最后两个几乎一无所知）。自该提案提交以来，已经取得了很大进展（由Knutson和R。Vakil）在这个晶格中的一个更高的水平，格拉斯曼的等变K理论，但所有其他的组合仍然存在。第二部分是Littelmann的路径模型在表示论中的推广，在这种情况下，人们应该把它看作是提供一个平坦的退化的旗流形的一个联盟的环面品种。在推广中，旗流形的唯一性质是它带有孤立不动点的圆的作用。因此，许多其他的变种应该有一个“路径模型”为他们的坐标环，如环面变种（一个试验平台，在那里理论是相当平凡的），精彩的紧化，和希尔伯特计划。作为一个应用实例，这将为Haiman推广的（q，t）-Catalan数提供一个p正公式（其中正性是已知的消失上同调的原因，但没有公式）。克努森博士的两个项目中的第一个涉及到19世纪组合学对代数几何的入侵：计算线（或平面，或由平面内的线内的点组成的链等）的数量满足多个通用相交条件。第一个有趣的问题是“给定空间中的四条通用线，有多少其他线与这四条线接触？”（答案是2）这个问题有许多概括;最新和最令人兴奋的一个是量子交叉理论，其中可能没有单一的解决方案来解决所有的条件，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”。（从物理学的角度来看，这种惩罚意味着经典上不可能发生，但在量子世界中非常罕见。已经有公式来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，因为它们通过添加和减去许多数字来确定计数。这样的可消公式对于证明一般类的相交问题有答案基本上是无用的，而且它们在计算上非常低效。克努森博士和他的合作者已经为其中一些推广提供了不可取消的公式，并对其他的推广提出了质疑。他的另一个项目没有直接相关，但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展示了如何使用非常大的对称性，某些代数空间，如一套所有的k-平面在n-空间，计算空间的职能，他们在格点内的工会大规模的四面体。第二个建议是基于Knutson博士最近的工作，他指出这种大程度的对称性是不必要的--一个单一的圆对称，加上一个技术条件（但很常见，也很容易检查），似乎就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学中是特有的。

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.