This thesis by Steven Avery from Harvey Mudd College explores the development and exploration of noncommutative field theory, starting from a basic background and delving into recent and significant results in classical noncommutative field theory. The background section presents a coherent presentation of mathematical and physical interpretations of differential geometry, which is not commonly found in the literature. The thesis covers several interesting examples that have emerged from recent research in the field. The introduction highlights the importance of noncommutativity in both mathematics and physics, discussing its applications in quantum mechanics, string theory, and condensed matter physics. It also addresses the challenges and motivations for studying noncommutative geometry, including its connection to quantum gravity and the discovery that string theory leads to noncommutative geometry in certain limits. The thesis is structured into several chapters, each focusing on different aspects of noncommutative geometry and field theory. Chapter 2 provides the necessary mathematical and physical background, including tensor fields, manifolds, and the metric tensor. Chapter 3 introduces noncommutative geometry in string theory, while Chapter 4 defines noncommutative field theories and discusses the Seiberg-Witten map, which maps noncommutative field theories to commutative ones. Chapter 5 explores the Seiberg-Witten map in more detail, showing how it can introduce non-flat curvature corresponding to gravitation. Chapter 6 presents a generalization of the Seiberg-Witten map that is a symmetry of noncommutative field theories, leading to a conserved current in the space of noncommutative planes. The thesis concludes with a discussion of the implications and future directions of noncommutative field theory, emphasizing its potential contributions to understanding the fundamental structure of the universe.This thesis by Steven Avery from Harvey Mudd College explores the development and exploration of noncommutative field theory, starting from a basic background and delving into recent and significant results in classical noncommutative field theory. The background section presents a coherent presentation of mathematical and physical interpretations of differential geometry, which is not commonly found in the literature. The thesis covers several interesting examples that have emerged from recent research in the field. The introduction highlights the importance of noncommutativity in both mathematics and physics, discussing its applications in quantum mechanics, string theory, and condensed matter physics. It also addresses the challenges and motivations for studying noncommutative geometry, including its connection to quantum gravity and the discovery that string theory leads to noncommutative geometry in certain limits. The thesis is structured into several chapters, each focusing on different aspects of noncommutative geometry and field theory. Chapter 2 provides the necessary mathematical and physical background, including tensor fields, manifolds, and the metric tensor. Chapter 3 introduces noncommutative geometry in string theory, while Chapter 4 defines noncommutative field theories and discusses the Seiberg-Witten map, which maps noncommutative field theories to commutative ones. Chapter 5 explores the Seiberg-Witten map in more detail, showing how it can introduce non-flat curvature corresponding to gravitation. Chapter 6 presents a generalization of the Seiberg-Witten map that is a symmetry of noncommutative field theories, leading to a conserved current in the space of noncommutative planes. The thesis concludes with a discussion of the implications and future directions of noncommutative field theory, emphasizing its potential contributions to understanding the fundamental structure of the universe.