Title: Tropical geometry and amoebas in matrix groups

Abstract: We start with the basic and remarkable notions of amoeba and tropical variety of a subvariety Y in the algebraic torus $(\mathbb{C} \setminus \{0\})^n$. We will demonstrate how these notions lead us to finding a minimal compactification of Y (usually referred to as "tropical compactification"). In the course of this we will introduce the notion of a toric variety as well. Next, I will discuss recent results about extending these notions from the algebraic torus to other matrix groups such as $GL(n, \mathbb{C})$. Some interesting linear algebra, such as singular value decomposition and Smith normal form, pops up. For the most part, I assume only basic background from algebra and geometry and the talk should be understandable to a general math crowd. There will be a nonzero number of pictures!