Formulas for Quiver Varieties and Quantum Schubert Calculus

调查员将研究计数计量学这一广泛领域的课题。箭牌品种是一类广义的退化轨迹，包括确定性品种和舒伯特品种等许多重要品种。作者将尝试推广已知的关于A型箭图簇的几何和组合公式和构造，使之适用于更一般的动态箭图簇。他还将研究广义旗形的各种上同调理论，重点是Schubert类的乘法。特别是，他将试图理解等向格拉斯曼尼亚人(与A.Kresch和H.Tamvakis)的量子上同调。他还希望确定Grassmannians和其他齐次空间上的Gromov-Witten不变量是否由正性性质决定。作为研究的一个单独目标，研究人员将开发计算所研究的公式和不变量的计算机程序。代数几何的大部分发展都是由列举几何问题推动的，即要求满足一定条件的特定类型的几何对象的数量的问题。例如，量子上同调是一种计算满足若干固定亚簇的簇中给定度曲线的数目的理论。量子上同调的发明是由物理学推动的，在物理学中，计算曲线的能力应用于镜像对称。一般而言，计数问题可以通过构造一个模空间来解决，该模空间对每个可计数的几何对象有一个点，使得对象上的条件对应于模空间的子簇。这将一个计数问题简化为理解一个更大空间的亚类的交集。相关的亚簇往往可以被构造为箭图簇的特例，因此这类簇为计数几何提供了有力的工具。此外，箭图簇的研究在代数几何和组合数学之间产生了许多新的关系，并在这两个领域产生了强有力的方法和结构。

The investigator will study topics in the broad area of enumerativegeometry. Quiver varieties form a general type of degeneracy locus,which includes many important varieties such as determinantalvarieties and Schubert varieties. The investigator will attempt togeneralize known geometric and combinatorial formulas andconstructions concerning equioriented quiver varieties of type A towork for more general quiver varieties of Dynkin type. He will alsostudy various cohomology theories of generalized flag manifolds withan emphasis on the multiplication of Schubert classes. In particular,he will attempt to understand the quantum cohomology of isotropicGrassmannians (with A. Kresch and H. Tamvakis). He also hopes todetermine if the Gromov-Witten invariants on Grassmannians and otherhomogeneous spaces are determined by positivity properties. As aseparate goal in his research, the investigator will develop computerprograms for computing the studied formulas and invariants.Much development in algebraic geometry has been motivated byenumerative geometric problems, that is, problems that ask for thenumber of geometric objects of a specified type that satisfy certainconditions. For example, quantum cohomology is a theory for countingthe number of curves of given degree in a variety that meet a numberof fixed subvarieties. The invention of quantum cohomology wasmotivated by physics, where the ability to count curves hasapplications to mirror symmetry. In general, enumerative problems canbe attacked by constructing a moduli space that has one point for eachgeometric object that could be counted, such that the conditions onthe objects correspond to subvarieties of the moduli space. Thisreduces an enumerative problem to understanding the intersection ofsubvarieties of a larger space. The relevant subvarieties can oftenbe constructed as special cases of quiver varieties, so this type ofvarieties provide a powerful tool in enumerative geometry. Inaddition, the study of quiver varieties has given rise to many newrelations between algebraic geometry and combinatorics, as well aspowerful methods and constructions in both fields.