Title: On the Decision Boundaries of Neural Networks. A Tropical Geometry Perspective. Abstract: This work tackles the problem of characterizing and understanding the decision boundaries of neural networks with piecewise linear non-linearity activations. We use tropical geometry, a new development in the area of algebraic geometry, to characterize the decision boundaries of a simple network of the form (Affine, ReLU, Affine). Our main finding is that the decision boundaries are a subset of a tropical hypersurface, which is intimately related to a polytope formed by the convex hull of two zonotopes. The generators of these zonotopes are functions of the network parameters. This geometric characterization provides new perspectives to three tasks. Specifically, we propose a new tropical perspective to the lottery ticket hypothesis, where we view the effect of different initializations on the tropical geometric representation of a network's decision boundaries. Moreover, we propose new tropical based optimization problems that directly influence the decision boundaries of the network for the tasks of network pruning (removing network parameters not contributing to the tropical geometric representation of the decision boundaries) and the generation of adversarial attacks.