舒伯特分析起源于经典的计数几何问题，是代数几何的一个分支。它是关于齐次簇上各种广义上同调理论的研究，其中包括经典上同调、环等变上同调、量子上同调、环等变量子上同调、量子K-理论。这些理论均有一个共同特性，即它们都有一组舒伯特类构成的加法基、相应的舒伯特结构常数均满足“正性”。申请人已在该领域取得了丰富的研究成果，并将致力于寻找关于部分乃至全部舒伯特结构常数的可显示正性的组合公式，这是该领域最核心的问题之一。对于经典上同调层面，申请人将研究舒伯特簇在Weyl群平移下的横截相交；对于（等变）量子上同调层面，申请人将研究C型齐次簇的（等变）量子Pieri法则以及Peterson同构的几何证明；对于量子K-理论层面，申请人将研究舒伯特结构常数的求和，这是基于申请人及其合作者研究过程中产生的一个猜测。

Schubert calculus, arising from solving problems in enumerative geometry, is a branch of algebraic geometry. It is referred to as the study of various generalized cohomology theories of homogeneous varieties, including the classical cohomology, torus-equivariant cohomology, quantum cohomology, torus-equivariant quantum cohomology, and quantum K-theory. They share a same property of having an additive basis of Schubert classes, and the corresponding Schubert structure constants all satisfy "positivity" property. The applicant has achieved lots of results in this subject, and will devote to finding a manifestly positive combinatorial formula for part or even all of the structure constants, which is one of the most central problems in this subject. For classical cohomology, the applicant will study the transversal intersections of Schubert varieties translated by Weyl group elements. For (equivariant) quantum cohomology, the applicant will study the (equivariant) quantum Pieri rules for homogeneous varieties of type C as well as a geometric proof of Peterson isomorphism. For quantum K-theory, the applicant will study the sum of Schubert structure constants, which is motivated by a conjecture proposed by the applicant and his collaborators.