Title: Extendability of projective varieties via degeneration to ribbons

Abstract: In this talk, we will introduce an approach to the question of extendability of projective varieties via degeneration to ribbons. As an application of these methods, we will give a new proof of optimal results on the extendability of general K3 surfaces, classification of Fano threefolds and Mukai varieties, and the irreducibility of their Hilbert schemes. If time permits, we will also discuss some new results on the extendability of canonical surfaces and Calabi-Yau threefolds, and their impact on classification of Fano fourfolds.