In this expository talk, we give a gentle introduction to the theory of Ginzburg Landau vortices. We will mostly be talking about the two dimensional theory, because in this context complex variables methods are quite useful. This development by Bethuel Brezis and Helein paved way to the theory of weak Jacobians which proved crucial for the problem in higher dimensions. Time permitting, we will briefly describe this and some related time dependent problems.

We shall review some recent results concerning the local stabilization, around unstable steady states, of fluid flows and of fluid structure systems. In all these problems the fluid flows will be described by the incompressible Navier- Stokes equations. The control is either a control acting at the boundary of the fluid domain, or a control acting in the structure equation. We consider models in which the structure is located at the boundary of the fluid domain and described by either a damped beam equation in 2D or a plate equation in 3D. Another fluid structure model, that we consider, consists in coupling the incompressible Navier-Stokes equations with the Lam ?e system of linear Elasticity

In this lecture we will first revisit Hodge theory from the point of view of the Heat equation. In particular, we will see how to prove the Hodge theorem by assuming the existence of the Heat Kernel. We will then look at the asymptotic expansion of the Heat Kernel and see how it leads to the signature theorem. In particular we will see how both the Hodge theorem and the signature theorem are special cases of the general statement "Analytical Index = Topological Index" (which is basically the statement of the Atiyah Singer Index Theorem).