Minimal surfaces and hypersurfaces in the unit spheres and their index

In the first part of the talk (introduction) I will recall classical and semi-classical results and conjectures. In the second part I will review the current status of PDE gluing constructions of minimal surfaces. In the last (main) part of the talk I will present the recent paper arXiv:2512.07734 (Kapouleas-Zou) on the index of minimal surface doublings, in as much detail as time permits.