Abstract: In mathematics, the concept of a fiber bundle was first introduced in a formal way by H. Seifert during the first International Topological Congress which took place in Moscow 1935. During that time it was realized that fiber bundles could provide important topological invariants, thereby making them an important tool in algebraic topology and in the classification and construction of manifolds. The main representatives of that school are H. Seifert, H. Hopf, H. Whitney, J. Serre, J. Milnor and others. In physics, fiber bundles were first utilized by P. Dirac (1931) in his study of magnetic monopoles, by means of a mathematical structure that was later realized to be equivalent to the sphere bundles constructed by Hopf. Almost twenty years later, C. N. Yang and R. Mills realized that principal fiber bundles can provide the framework necessary for generalizations of electromagnetism to higher gauge theories. This framework is now called Yang-Mills theory and it is the foundation of modern particle physics. During the 70’s and 80’s, the progress made in the study of fiber bundles both in mathematics and physics culminated to the seminal works of M. Atiyah and R. Bott who provided a powerful link between gauge theory and algebraic geometry. This work then led to the works of S. Donaldson and K. Uhlenbeck, who revolutionized the study of four-dimensional manifolds.

The aim of this set of lectures is to provide the basics of the geometry of fiber bundles, hoping that one can then dive into the foundational results showcased previously. We are nonetheless going to focus on vector bundles and principal fiber bundles.

• In the first lecture we will introduce the notion of a vector bundle, the category of vector bundles over a smooth manifold and review some basic schemes of classification. We will also aim to briefly discuss how vector bundles naturally lead to algebraic K-theory as well as provide some idea of the proof of Adams’ theorem concerning the parallelizability of spheres.
• In the second lecture we will introduce the concept of a principal fiber bundle and build our way towards understanding the basics of Yang-Mills theory. After reviewing some basic constructions and theorems relating principal fiber bundles to vector bundles we will discuss some of Uhlenbeck’s results concerning the solutions of Yang-Mills equations and mention Donaldson’s work on the geometry of four-manifolds
• In the third lecture we will discuss characteristic classes and further applications in algebraic topology