Topological degrees and applications
It is well-known that mathematics is the language of science, that gives toolsto model real-world phenoma. Most of those are intrinsically very non-linear, and the study of the mathematical models they lead to necessitates to assessthe existence of solutions to non-linear equations.The Brouwer’s and Leray-Schauder’s topological degrees are extremely pow erful tools to ensure the existence of solution to non-linear equations. They are based on advanced topological and functional analysis techniques and have a number of exciting consequences.The purpose of this project is twofold. First, we will cover the topological and functional analysis theories needed to understand the construction and usage of the topological degrees, both in finite dimension (Brouwer’s degree) and in infinite dimensions (Leray-Schauder’s degree). Then, we will study the degrees themselves, their construction as well as some of their powerful applications in various domains of mathematics (topology of R n, geometry, differential equations, etc.).