Torus actions and trinomials

In this talk we discuss algebraic torus actions of complexity one and relate them via Cox rings to affine varieties defined by trinomials. Our aim is to survey recent results on affine trinomial varieties and their automorphism groups.
We study the flexibility property for affine varieties, find all rational trinomial varieties, characterize rigid factorial trinomial hypersurfaces, describe the automorphism group of a rigid trinomial hypersurface, solve some Diophantine equations and characterize existence of certain polynomial curves on trinomial hypersurfaces.
An explicit description of primitive homogeneous locally nilpotent derivations on trimomial affine algebras allows to describe the automorphism group of a complete rational variety with a torus action of complexity one generalizing Demazure's description of the automorphism group of a complete toric variety.