Schubert calculus and algebraic combinatorics

主要研究者：Tamvakis，Harry 提案编号：DMS -0901341机构：马里兰州大学公园标题：舒伯特微积分和代数组合学研究者将继续他的量子和算术舒伯特微积分，简并轨迹，和相关的代数组合学的研究。 以下是一些建议的研究问题：a）完成他的工作，对各向同性格拉斯曼和更一般的旗品种的经典和量子舒伯特演算; B）发现正确的组合对象，以解决这些空间的量子上同调环中的Giambelli问题; c）发展相关的组合理论的theta多项式，一个家庭的多项式交织的舒尔S和Q功能; d）在迷向Grassmannian的上同调中找到两个Schubert类乘积的Littlewood-Richardson型定则。研究者提出的教育活动包括开发和教授基于问题解决的本科课程，这些课程与当前研究有直接联系。他还将致力于一本书，旨在使现代舒伯特微积分及其各种扩展和应用更容易接触到初学者在该地区。这一建议是有关枚举代数几何及其与其他部分的数学，如李理论和组合学的相互作用。舒伯特发明的微积分是一个基本的例子，枚举技术在几何，早于上同调和交叉理论的二十世纪。一个例子的那种问题所考虑的舒伯特是：鉴于4线在欧几里德3-空间的一般立场，有多少线相交的所有4个给定的线？舒伯特演算提供了一个通用的方法来回答类似的问题，涉及线性条件在更高的维度。流形的量子上同调环的最新定义包含了计算空间中满足自然关联条件的有理曲线的数量的不变量。这个理论起源于量子物理学，在那里它可以应用于弦理论和镜像对称的预言。李群的齐性空间和它们的舒伯特变种提供了丰富的例子，在这些例子中，量子上同调环的显式计算是可能的。由此产生的现代舒伯特微积分是这个研究项目的重点。除了许多相关领域的应用，如等变上同调和多品种，它引入了新的方式来看待相当经典的对象，并导致重要的组合和计算的挑战。

ABSTRACTPrincipal Investigator: Tamvakis, Harry Proposal Number: DMS - 0901341Institution: University of Maryland College ParkTitle: Schubert calculus and algebraic combinatoricsThe investigator will continue his study of quantum and arithmetic Schubert calculus, degeneracy loci, and related algebraic combinatorics. The following are some of the proposed research problems: a) complete his work on the classical and quantum Schubert calculus for isotropic Grassmannians and more general flag varieties; b) discover the right combinatorial objects to address the Giambelli problem in the quantum cohomology rings of these spaces; c) develop the related combinatorial theory of theta polynomials, a family of polynomials which intertwines the Schur S and Q functions; d) find a Littlewood-Richardson type rule for the product of two Schubert classes in the cohomology of isotropic Grassmannians. The educational activities proposed by the investigator include developing and teaching an undergraduate course based on problem solving with topics that have a direct connection to current research. He also will work on a book which aims to make the modern Schubert calculus and its various extensions and applications more accessible to beginners in the area.This proposal is concerned with enumerative algebraic geometry and its interaction with other parts of mathematics such as Lie theory and combinatorics. The calculus invented by Schubert is a fundamental example of enumerative techniques in geometry which predate the cohomology and intersection theories of the twentieth century. An example of the kind of question considered by Schubert is: given 4 lines in Euclidean 3-space in general position, how many lines intersect all 4 of the given lines? Schubert calculus provides a general method to answer similar questions involving linear conditions in higher dimensions. The more recent definition of the quantum cohomology ring of a manifold incorporates invariants that count the number of rational curves in the space which satisfy natural incidence conditions. This theory has its origin in quantum physics, where it has applications to string theory and the predictions of mirror symmetry. The homogeneous spaces of Lie groups and their Schubert varieties provide a rich source of examples where explicit computations of quantum cohomology rings are possible. The resulting modern Schubert calculus is the focus of this research project. Besides the many applications to related areas such as equivariant cohomology and quiver varieties, it introduces fresh new ways of looking at rather classical objects, and leads to important combinatorial and computational challenges.