Schubert calculus and algebraic combinatorics

主要研究人员：Tamvakis，Harry建议编号：DMS-0901341机构：马里兰大学学园标题：舒伯特微积分和代数组合学研究人员将继续他对量子和算术舒伯特微积分、简并轨迹和相关代数组合学的研究。以下是一些拟议的研究问题：a)完成关于各向同性Grassmannians和更一般的旗簇的经典和量子Schubert演算的工作；b)在这些空间的量子上同调环中找到合适的组合对象来解决Giambeli问题；c)发展相关的theta多项式的组合理论，这是一族缠绕Schur S和Q函数的多项式；d)在各向同性Grassmannians的上同调中找到两个Schubert类的乘积的Littlewood-Richardson型规则。研究人员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，这些课程的主题与当前的研究有直接联系。他还将撰写一本书，旨在使该领域的初学者更容易接触到现代舒伯特微积分及其各种扩展和应用。这项建议涉及计数代数几何及其与其他数学部分的相互作用，如李理论和组合学。舒伯特发明的微积分是二十世纪上同调和交集理论之前的几何学中计数技术的一个基本例子。舒伯特考虑的这类问题的一个例子是：给出欧几里得3-空间中4条直线的一般位置，有多少条直线与所有4条给定直线相交？舒伯特微积分提供了一种回答类似问题的一般方法，这些问题涉及高维的线性条件。流形的量子上同调环的最新定义包含了不变量，这些不变量计算空间中满足自然关联条件的有理曲线的数量。这一理论起源于量子物理学，它在弦理论和镜像对称性的预测中得到了应用。李群及其Schubert簇的齐次空间提供了丰富的例子，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，如等变上同调和箭形簇，它还引入了观察相当经典对象的新方法，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.