We start with a review of basic concepts and constructions in the theory of Newton-Okounkov bodies. I will then talk about some related results and touch on some applications e.g. constructing toric degenerations and applications to symplectic geometry, and multiplicities of ideals.

Toric degenerations constructed in this context are directly related to the notion of a Khovanskii basis. This in turn is related to the key question of finite generation of the value semigroup. This approach provides a natural setup to generalize the notion of a SAGBI basis for a subalgebra of polynomials to an arbitrary algebra equipped with a full rank valuation. We will talk about some basic results on Khovanskii bases and the subduction algorithm.

Finally, I will discuss some recent results (joint with Chris Manon) about a connection between full rank valuations (appearing in the theory of Newton-Okounkov bodies) and rank one valuations (appearing in tropical geometry). More precisely, let A be a finitely generated positively graded domain. Then we make a correspondence between prime cones in tropical varieties of different ideals presenting A, and certain class of full rank valuations on A (which we call “good” valuations). Roughly speaking, a valuation v is good if its values semigroup is finitely generated and the subduction algorithm for v terminates.

Many well-known examples such as Plucker coordinates and coordinate rings of Grassmannians, as well as the Gelfand-Zetlin bases and coordinate rings of flag varieties fit into this general picture.