Combinatorial Models in Algebra, Geometry, and Topology

本研究计划分为代数组合及其在其他数学领域的应用三个项目。第一个项目是关于广义旗种上的舒伯特微积分的几个具体方面。它的目标是在舒伯特类的基础上找到表达两个舒伯特类（上同调）的乘积的组合公式；这相当于计算舒伯特变种的一个合适的三重交集上的点。重点讨论复空间中各种完备标志在上同调中的乘法规则。在这个领域已经使用了许多不同的方法，但是存在的唯一明显肯定的公式仅限于一些更简单的特殊情况。第二个项目涉及复杂半单李群的表示理论的简单组合模型的发展（最近由研究者与a . Postnikov合作引入），以及相应广义标志品种的等变k理论中的chevalley型乘法公式。该结构是基于相应的仿射Weyl基团的分解组合和（非仿射）Weyl基团上的Bruhat序饱和链的枚举。与该领域的其他组合结构相比，该模型有几个优点，例如Littelmann路径模型的各种专门化。新模型将在几个方向上得到扩展，如：kac - moody代数的表示理论、标准单项式理论、旗变的量子k理论。与其他模型的联系以及对所涉及的组合学的更深入的研究也将继续进行。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。主要应用于镜头空间和投影空间的某些沉浸问题。这里列出的项目的一个统一主题是强调组合和计算。在过去的几十年里，计算在数学研究中占有重要的地位。这刺激了组合学的发展，因为很明显，组合结构特别适合于编码复杂的数学对象，而组合方法非常适合于相关的计算。该研究计划是基于组合结构进行具体计算的持续努力的一部分。其中一个主要研究对象是广义标志品种。虽然这些都是经典的变种，但由于它们显著的组合复杂性和与它们相关的各个领域之间微妙的相互作用，它们在当前的数学研究中占有突出地位。与本计划项目相关的这些领域的例子有：枚举几何（涉及诸如计算满足若干一般相交条件的线或面等问题，相当于执行某些上同调计算），以及李群表示的几何构造（以及更普遍的Kac-Moody群）。标志品种也为开发与某些射影品种的各种上同调理论计算相关的组合模型提供了一个有用的试验台。

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.