Abstract: I will first concentrate on doubling gluing constructions for minimal surfaces, including a recent construction for free boundary minimal surfaces in the unit ball (with D. Wiygul: arXiv:1711.00818).
I will then discuss the Linearized Doubling methodology and its applications so far (J. Differential Geom. 106:393-449, 2017; and with P. McGrath: arXiv:1707.08526),
and some further ongoing work expanding the scope of these methods to new cases.
Finally depending on time availability I will briefly discuss or mention recent progress in some other gluing constructions, for example a recent general construction for Constant Mean Curvature hypersurfaces (with C. Breiner: arXiv:1707.04008), a construction for ancient solutions of the Ricci flow related to an old question of Page (with S. Brendle: Comm. Pure Appl. Math., 70(7):1366-1401, 2017), and desingularization gluing constructions for minimal surfaces including a recent construction for free boundary minimal surfaces in the unit ball (with M. Li: arXiv:1709.08556) and a construction in the round three-sphere (with D. Wiygul: