This work starts with the intention of using mathematics to understand the intriguing vulnerability observed by ~\citet{szegedy2013} within artificial neural networks. Along the way, we will develop some novel tools with applications far outside of just the adversarial domain. We will do this while developing a rigorous mathematical framework to examine this problem. Our goal is to build out theory which can support increasingly sophisticated conjecture about adversarial attacks with a particular focus on the so called ``Dimpled Manifold Hypothesis'' by ~\citet{shamir2021dimpled}. Chapter one will cover the history and architecture of neural network architectures. Chapter two is focused on the background of adversarial vulnerability. Starting from the seminal paper by ~\citet{szegedy2013} we will develop the theory of adversarial perturbation and attack. Chapter three will build a theory of persistence that is related to Ricci Curvature, which can be used to measure properties of decision boundaries. We will use this foundation to make a conjecture relating adversarial attacks. Chapters four and five represent a sudden and wonderful digression that examines an intriguing related body of theory for spatial analysis of neural networks as approximations of kernel machines and becomes a novel theory for representing neural networks with bilinear maps. These heavily mathematical chapters will set up a framework and begin exploring applications of what may become a very important theoretical foundation for analyzing neural network learning with spatial and geometric information. We will conclude by setting up our new methods to address the conjecture from chapter 3 in continuing research.