  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.
