Mathematics

• Degree theory
• Sard's Theorem
• The Brouwer fixed point theorem with applications
• The Schauder fixed-point theorem with applications
• Critical points
• Differential calculus and critical points; constrained critical points
• Minimization problems
• Linear eigenvalues and their variational characterization
• Ekeland's variational principle
• The Palais-Smale condition
• Min-Max methods
• Linking and Mountain-Pass  theorems
• Multiplicity via symmetry: the Lusternik-Schnirelamann index
• Multiplicity via category
• Applications to elliptic PDE's

Further applications to semilinear nonlinear differential equations will be presented, depending on students' interests and time availability.