Hulya Arguz: Tropical and log corals on the Tate curve with a view toward symplectic cohomology

Abstract: The Tate curve is the total space of a degeneration of a smooth elliptic curve to a nodal elliptic curve. We suggest an algebro-geometric approach to defining the symplectic cohomology ring of the Tate curve minus its central fiber, in terms of tropical geometry and the punctured log Gromov–Witten theory of Abramovich– Chen–Gross–Siebert. This is based on a proposal by Abouzaid–Siebert. Our results provide evidence for the conjectural algebro-geometric construction of the symplectic cohomology ring, in the framework of log Calabi–Yau varieties, by Gross–Hacking– Keel–Siebert.