In this course, we will cover three main topics:

• The degree in finite dimensions ( Brouwer degree) and applications (fixed point theorems, periodic ODEs)
• The degree in infinite dimensions (Leray-Schauder degree) and applications (fixed point theorems, periodic ODEs, existence of solutions to ODEs, topological applications.)
• Introduction to bifurcation theory and applications (results on completely cont inuous operators, nonlinear eigenvalue results, the bifurcation theorem of Krasnoselskii and then its extension to a global bifurcation theorem of Rabinowitz)