Title: Dual complex of genus one mapping spaces
Abstract: The dual complex of a smooth variety encodes the combinatorial structure that underlies all its possible normal crossings compactifications. We prove that the dual complexes of genus zero and genus one mapping spaces are contractible (in degrees > 0 and > 1 respectively) via an explicit deformation retraction. In genus one, the key geometric input comes from the Vakil - Zinger space and its tropical interpretation due to Ranganathan - Santos-Parker - Wise. Joint work with Siddarth Kannan (MIT). Time permitting, I will discuss ongoing work on understanding the full cohomology of the Vakil - Zinger space.