Tropical geometry studies a piecewise linear combinatorial shadow of classical algebraic geometry. In this talk I will explain what little we currently know about the tropical geometry of algebraic groups. Different avenues of progress towards an answer to this question lead to new insights both within and beyond tropical geometry, in particular in the context of vector bundles and the geometry of buildings. This talk will touch upon joint work with Luca Battistella, Desmond Coles, Andreas Gross, Inder Kaur, Kevin Kühn, Arne Kuhrs, Annette Werner, Alejandro Vargas, and Dmitry Zakharov.

Quiver Grassmannians are projective varieties parametrising subrepresentations of quiver representations. Their geometry is an interesting object of study, due to the fact that many geometric properties can be studied via the representation theory of quivers. In this talk, we construct a special quiver with relations and consider two classes of quiver Grassmannians for this quiver. For an appropriate choice of dimension vector for this quiver, we provide an isomorphism between the corresponding quiver Grassmannians and certain Bott-Samelson resolutions of type A Schubert varieties. Furthermore, for smooth type A Schubert varieties, we identify a suitable dimension vector such that the corresponding quiver Grassmannian is isomorphic to the Schubert variety.