In this two-part talk I will first introduce the Vlasov-Maxwell system of equations describing the evolution of a hot plasma. I will then survey recent results on (in)stability theory of steady-state solutions. This will lead us to the following question: is it possible to calculate the spectrum of a Schrodinger operator as the limit of finite dimensional approximations? Questions of this type will be the topic of the second part, in which I will introduce a new notion of computational complexity and a number — the Solvability Complexity Index — measuring how “difficult” a given problem is. The first part includes joint work woth Tom Holding and the second part is joint work with Anders Hansen, Olavi Nevalinna and Markus Seidel.