Quantum mechanics and semi-definite optimization
We begin with a very light introduction to mathematical formalism of quantum mechanics. Then we will study entanglement and play the CHSH game which leads to a design for an experiment which "proves" the quantum nature of our universe. We will see an algebraization of the game which is stated in the form of positive functionals on monoid *-algebras. These in turn can be approximated using semi-definite optimization and indeed, a finite-dimensional relaxation of the problem is enough to mathematically confirm the superiority of quantum strategy.
Traditionally quantum mechanics is described using complex numbers. I will sketch a very modern development in the field and show how a similar program can be used to confirm that complex vector spaces are indeed needed for modelling the real world.