Schubert calculus and algebraic combinatorics

主要研究者：Tamvakis， Harry提案编号：DMS - 0901341机构：马里兰大学学院公园标题：Schubert微积分和代数组合研究员将继续他的量子和算术Schubert微积分，退化座，以及相关的代数组合的研究。以下是一些建议的研究问题：a)完成他关于各向同性格拉斯曼和更一般旗型的经典和量子舒伯特微积分的工作；b)在这些空间的量子上同调环中发现合适的组合对象来解决Giambelli问题；c)发展多项式的相关组合理论，这是一个将舒尔S和Q函数交织在一起的多项式族；d)在各向同性的Grassmannians的上同调中，找到两个Schubert类的乘积的Littlewood-Richardson型规则。研究者提出的教育活动包括开发和教授一门基于解决问题的本科课程，该课程的主题与当前的研究有直接的联系。他还将撰写一本书，旨在使该领域的初学者更容易理解现代舒伯特微积分及其各种扩展和应用。这个建议是关于枚举代数几何及其与数学的其他部分，如李论和组合的相互作用。舒伯特发明的微积分是几何学中计数技术的一个基本例子，它早于20世纪的上同论和交论。舒伯特考虑的一个问题是：给定欧几里得三维空间中一般位置上的4条直线，有多少条直线与这4条直线相交？舒伯特微积分提供了一种一般的方法来回答涉及高维线性条件的类似问题。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个数的不变量。这个理论起源于量子物理学，它可以应用于弦理论和镜像对称的预测。李群的齐次空间和它们的舒伯特变分提供了一个丰富的例子来源，其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本研究项目的重点。除了在相关领域的许多应用，如等变上同调和颤振变异，它引入了新的方法来观察相当经典的对象，并导致了重要的组合和计算挑战。

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.