Toric degenerations: a bridge between representation theory, tropical geometry and cluster algebras

In this thesis we study toric degenerations of projective varieties. We compare different constructions to understand how and why they are related. In focus are toric degenerations obtained from representation theory, tropical geometry or cluster algebras. Often those rely on valuations and the theory of Newton-Okounkov bodies. Toric degenerations can be seen as a combinatorial shadow of the original objects. The goal is therefore to understand why the different theories are so closely related, by understanding the toric degenerations they yield first. We choose Grassmannians, flag varieties and Schubert varieties as starting point as here many different constructions are applicable. One of our main results shows how toric degenerations obtained using full-rank valuations, independent of how these are constructed, can (under certain conditions) be realized using tropical geometry.