Quantum cohomology of homogeneous spaces

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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.