Fano 4-folds in weighted Grassmannians

A weighted flag variety is a weighted projective analog of the usual flag variety. In this talk, I will provide an introduction and motivation behind the subject of weighted flag varieties, with particular emphasis on the first non-trivial case of weighted Grassmannians. I aim to show that how one may use them as ambient varieties to construct interesting classes of algebraic varieties (Fano, Calabi-Yau, etc), which are important from the point of view of the classification of algebraic varieties. As an application, I will present some new deformation families of smooth Fano 4-folds of index 1.