Towards Parallel Algorithms for Gromov-Witten Invariants of Elliptic Curves

We present a parallel enhanced algorithm for exploring mirror symmetry for elliptic curves through the correspondence of algebraic and tropical geometry, focusing on Gromov-Witten invariants of elliptic curves and, in particular, Hurwitz numbers. We present a new highly efficient algorithm for computing generating series for these numbers. A sequential version of the algorithm has been implemented using Singular and OSCAR. The implementations in [1] outperform by far the previous methods [3] provided in Singular. In this note, we describe work towards the natural next step, which is parallelization. We have integrated our algorithm with GPI-Space, a workflow management system for high-performance computing developed at Fraunhofer ITWM. This allows us to run our algorithm simultaneously on a large number of cores. This facilitates computation of quasi-modular representations of the respective generating series.