Quantum cohomology of homogeneous spaces

格罗莫夫-维腾理论起源于弦理论的物理模型。Kontsevich和Manin在稳定映射模空间上的制地工作为这一理论提供了严格的数学基础，并为解决代数几何中突出的计数问题做出了贡献。因此，他们开创了一个新的研究分支，在代数、辛几何和理论物理中有着广泛的应用。粗略地说，Gromov-Witten不变量计算光滑射影簇上固定次数和亏格的曲线。考虑的曲线必须满足特定的偶然性，这些偶然性迫使得到的曲线数目是有限的。Gromov-Witten不变量可以概括为所谓的量子上同调环。这是一个分次代数，它的量子积结构是普通杯积的变形。这个乘积的结合性产生了枚举解之间的非平凡关系。本项目是关于齐次空间上的Gromov-Witten理论-一类具有传递群作用的光滑射影簇。在这种情况下，稳定映射的模空间具有特别令人满意的性质，这些性质允许直观地接近Gromov-Witten理论。特别是，有可能对量子产品中的最小度数做出更尖锐的陈述。首先，我们将研究这些极小度的分布，并试图计算出所有极小度的一个极小上界。此外，我们的目的是证明稳定映射模空间的比不可约性更好的性质。模空间的拟齐性等性质简化了Gromov-Witten不变量的描述，并为从理论上理解它们的结构铺平了道路。因此，该项目最终将是关于计算Gromov-Witten不变量的新方法。

Gromov-Witten theory has its origins in physical models of string theory. Groundbraking work of Kontsevich and Manin on the moduli space of stable maps gave the theory a rigorous mathematical foundation and contributed to solve outstanding enumerative problems in algebraic geometry. Thereby, they initiated a new branch of research with far-reaching applications in algebraic and symplectic geometry and theoretical physics.Roughly speaking, Gromov-Witten invariants count curves of fixed degree and genus on a smooth projective variety. The considerer curves have to satisfy specific incidences which force the resulting number of curves to be finite. Gromov-Witten invariants can be summarized in the so-called quantum cohomology ring. This is a graded algebra whose quantum product structure is a deformation of the ordinary cup product. The associativity of this product yields non-trivial relations among the enumerative solutions.The present project is about Gromov-Witten theory on homogeneous spaces - a class of smooth projective varieties which carry a transitive group action. In this case, the moduli space of stable maps has particularly desirable properties which allow an intuitive approach to Gromov-Witten theory. In particular, it is possible to make sharper statements about the minimal degrees in quantum products. First of all, we will study in this project the distribution of these minimal degrees and try to compute a minimal upper bound of all minimal degrees.Moreover, the aim of the project is to prove finer properties than irreducibility of the moduli space of stable maps. Properties of the moduli space such as quasi-homogeneity simplify the description of Gromov-Witten invariants and prepare the path for a theoretical understanding of their structure. Thus, the project will be finally about new methods for computing Gromov-Witten invariants.