Tuesday 16 July 2019 |

Events for day: Wednesday 10 July 2019 |

15:30 - 16:30 Geometry and Topology Weekly SeminarModular Vector Fields and Calabi-Yau Modular Forms School MATHEMATICS In this lecture we introduce a spacial moduli space $sf T$ of the pairs formed by definite Calabi-Yau $n$-folds (arising from the Dwork family) along with $n+ 1$ differential $n$-forms. We observe that there exists a unique vector field $ extsf{R}$ on $sf T$, called modular vector field, satisfying a certain equation involving the Gauss-Manin connection. It turns out that the $q$-expansion (Fourier series) of the components of a solution of $sf R$, which are called Calabi-Yau modular forms, has integer coefficients, up to multiplying by a constant rational number. In particular, in the case of elliptic curves and $K3$-surfaces, where $n= ... |