Combinatorial Models in Algebra, Geometry, and Topology

这项研究计划分为三个项目，分别是代数组合学及其在其他数学领域的应用。第一个项目是关于广义旗簇上的Schubert微积分的几个具体方面。它的目的是在Schubert类的基础上找到表示两个Schubert类(上同调)的乘积的组合公式；这等价于Schubert簇的一个合适的三重交中的计点数。重点讨论复空间中完备标志的上同调中的乘法规则。在这一领域已经使用了许多不同的方法，但存在的唯一明显的正公式仅限于一些更容易的特殊情况。第二个项目是关于发展一个简单的组合模型(最近由研究者与A.Postnikov合作介绍)，用于复半单李群的表示理论，以及相应的广义旗簇的等变K-理论中的Chvalley-型乘法公式。这种构造是基于相应仿射Weyl群的分解和(非仿射)Weyl群上按Bruhat序的饱和链的计数的组合。与该领域中的其他组合结构相比，该模型具有几个优点，例如Littelmann路径模型的各种专门化。新模型将在几个方向进行扩展，如：Kac-Moody代数的表示理论、标准单项理论和FLAG变量的量子K-理论，还将寻求与其他模型的联系和对所涉及的组合子的更深入的研究。第三个项目是关于与拓扑学有关的某些形式群律的组合研究。主要应用于透镜空间和射影空间的某些浸入问题。这里概述的项目的一个统一主题是强调组合学和计算。在过去的几十年里，计算在数学研究中得到了重要的控制。这刺激了组合学的发展，因为很明显，组合结构特别适合于对复杂的数学对象进行编码，而组合方法非常适合于相关的计算。这项研究计划是正在进行的基于组合结构进行具体计算的努力的一部分。其中一个主要的研究对象是广义旗帜品种。虽然它们是经典的变种，但由于它们显著的组合复杂性和与它们相关的不同领域之间的微妙相互作用，它们在当前的数学研究中占有突出地位。与本计划中的项目相关的这些领域的例子有：列举几何(涉及满足一些一般相交条件的直线或平面的计数问题，这些条件等价于执行某些上同调计算)，以及李群(更典型地，还有Kac-Moody群)表示的几何构造。FLAG簇还为发展与某些射影簇的各种上同调理论中的计算相关的组合模型提供了有用的试验台。

This research plan is divided into three projects in algebraic combinatorics andits applications to other areas of mathematics. The first project is concernedwith several concrete aspects of Schubert calculus on generalized flagvarieties. Its goal is to find combinatorial formulas for expressing the productof two Schubert classes (in cohomology) in the basis of Schubert classes; thisis equivalent to counting points in a suitable triple intersection of Schubertvarieties. The emphasis will be on the multiplication rule in the cohomology ofthe variety of complete flags in complex space. Many different approaches havebeen used in this area, but the only manifestly positive formulas that exist arelimited to some easier special cases. The second project is concerned with thedevelopment of a simple combinatorial model (recently introduced by theinvestigator in collaboration with A. Postnikov) for the representation theoryof complex semisimple Lie groups, as well as for the Chevalley-typemultiplication formula in the equivariant K-theory of the correspondinggeneralized flag variety. The construction is based on combinatorics ofdecompositions in the corresponding affine Weyl group and enumeration ofsaturated chains in the Bruhat order on the (nonaffine) Weyl group. This modelhas several advantages over other combinatorial structures in the area, such asvarious specializations of the Littelmann path model. The new model will beextended in several directions, such as: the representation theory of Kac-Moodyalgebras, standard monomial theory, and the quantum K-theory of flag varieties.The connections with other models and a deeper study of the combinatoricsinvolved will also be pursued. The third project is concerned with acombinatorial study of certain formal group laws related to topology. The mainapplication is to certain immersion problems for lens spaces and projectivespaces.A unifying theme of the projects outlined here is the emphasis on combinatoricsand computation. During the last decades, computation has gained an importantrole in mathematical research. This stimulated the development of combinatorics,as it became clear that combinatorial structures are particularly well suitedfor encoding complex mathematical objects, while combinatorial methods are wellsuited for related computations. This research plan is part of the ongoingeffort to perform concrete computations, based on combinatorial structures. Oneof the main objects of study are generalized flag varieties. Although these areclassical varieties, they feature prominently in current mathematical researchdue to their remarkable combinatorial complexity and the subtle interplaybetween various areas related to them. Examples of such areas relevant to theprojects in this plan are: enumerative geometry (concerned with problems such ascounting the lines or planes satisfying a number of generic intersectionconditions, which are equivalent to performing certain cohomology calculations),and the geometric construction of representations of Lie groups (and, moregenerally, Kac-Moody groups). Flag varieties also provide a useful testbed forthe development of combinatorial models relevant to computations in variouscohomology theories of certain projective varieties.