Weighted PBW degenerations and Vinberg's polytopes

We study algebraic, combinatorial and geometric aspects of weighted PBW-type degenerations of flag varieties in type A. These degenerations are labeled by degree functions lying in an explicitly defined polyhedral cone. Varying the degree function in the cone, we recover the classical flag variety, its abelian PBW degeneration and toric degeneration corresponding to Vinberg's polytopes. We also identify the cone of degree functions with a maximal cone in the tropical flag variety. The talk is based on the joint work with Xin Fang, Ghislain Fourier and Igor Makhlin.