1. Introduction and real algebraic geometry
   1. Real roots of univariate polynomials
   2. Eigenvalue techniques
   3. Real roots in the multivariate case

2. Optimization and real algebraic geometry
   1. Global optimization of polynomials and sums of squares
   2. Semidefinite programming
   3. Algebraic certificates and Positivstellens\"atze
   4. Constrained optimization

3. Tropical geometry
   1. Introduction to tropical geometry
   2. Algebraic techniques
   3. Amoebas, tropical geometry and deformations