Geometry, Arithmetic, and Combinatorics of Schubert Calculus

研究人员将研究经典的Schubert微积分在李群的齐次空间上的推广及其在计数代数几何、算术几何、量子上同调和代数组合学中的相关应用。所提出的研究问题包括：(A)建立并证明旗形上同调中两个Schubert类的乘积的计算规则；(B)在齐次空间的上同调和量子上同调环上建立Pieri型规则；(C)找到关于正交李群和辛李群的量子Schubert多项式的良好理论；(D)研究theta和eta多项式的组合理论，特别是它们的双重形式，及其在等变上同调中的应用；(E)在旗簇的Arakelov几何中实现更有效的计算。这位研究人员提出的教育活动包括将他的研究中产生的问题纳入马里兰大学高中数学竞赛。他还在写一本关于舒伯特微积分及其各种现代扩展的书，这本书将以统一的方式强调该理论的几何方面，跨越不同的谎言类型。19世纪末，赫尔曼·舒伯特首次系统地研究了可数射影几何，并发明了一种微积分，使他能够以系统的方式解决过多的计数问题。他讨论的这类问题的一个例子是：给定欧几里得3-空间中随机生成的四条直线，有多少条直线会与所有这四条给定的直线相交？舒伯特的基本工作是20世纪上同调和交集理论的先驱，以及它们今天在量子物理(通过弦理论)和数论(通过算术交集理论)中的应用。此外，整个故事可以被放置在一个更对称的框架中，并使用李群及其表示理论的语言来讲述，组合的复杂性和重要性越来越高。研究人员对这些理论进行了多年的研究，并展示了如何实现组合显式计算和公式，这些计算和公式给出了每种情况下环结构的整体图景。最近，他成功地以一种统一的方式在所有经典李氏类型中做到了这一点--在上同调的情况下--从而在快速发展的现代舒伯特微积分领域开辟了新的研究途径。该奖项由组合学计划共同资助。

The investigator will study the generalization of the classical Schubert calculus to the homogeneous spaces of Lie groups and related applications to enumerative algebraic geometry, arithmetic geometry, quantum cohomology, and algebraic combinatorics. The proposed research problems include the following: (a) Formulate and prove a rule for computing the product of two Schubert classes in the cohomology of flag manifolds; (b) establish a Pieri type rule in the cohomology and quantum cohomology ring of homogeneous spaces; (c) find a good theory of quantum Schubert polynomials for the orthogonal and symplectic Lie groups; (d) study the combinatorial theory of theta and eta polynomials, in particular their double versions, with applications to equivariant cohomology; (e) achieve more efficient computations in the Arakelov geometry of flag varieties. The educational activities proposed by the investigator include incorporating problems stemming from his research into the University of Maryland High School Mathematics Competition. He also is working on a book on Schubert calculus and its various modern extensions which will emphasize the geometric aspects of the theory, in a uniform manner across the different Lie types.In the late 19th century, Hermann Schubert made a first systematic study of enumerative projective geometry and invented a calculus which enabled him to solve a plethora of enumerative problems in a systematic fashion. An example of the kind of question he addressed is: given four randomly generated lines in Euclidean 3-space, how many lines will intersect all four of the given lines? Schubert's fundamental work was a precursor of the cohomology and intersection theories of the 20th century, and their modern extensions today, with applications to quantum physics (via string theory) and number theory (via arithmetic intersection theory). Moreover, the entire story can be placed in a more symmetric framework, and told using the language of Lie groups and their representation theory, at an increasing level of combinatorial complexity and importance. The investigator has studied these theories over a period of many years and has shown how to achieve combinatorially explicit computations and formulas which give a global picture of the ring structure in each case. Recently, he has succeeded in doing this in a uniform way across all classical Lie types -- in the case of cohomology -- and has thus opened new avenues of research in the rapidly growing field of modern Schubert calculus.This award is co-funded by the Combinatorics Program.