The Dressian, the tropical Grassmannian and their positive parts

The Dressian and the tropical Grassmannian parametrize abstract and realizable tropical linear spaces, which are defined using the notion of valuated matroid introduced by Dress and Wenzel. In the last twenty years there has been an extensive study of their beautiful geometric and combinatorial structures, with a recent focus on their positive parts motivated by problems in mathematical physics. In this talk we will introduce these spaces and their main properties.