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 Lame system of linear Elasticity.

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 consider a collocation based quadrature method for obtaining stable approximations for the solution of ill-posed integral equations.

We consider a fourth order traveling wave equation associated to the Suspension Bridge Problem (SBP). This equations are modeled by the traveling wave behavior on the Narrows Tacoma and the Golden Gate bridge. We prove existence of homoclinic solutions when the wave speed is small. We will also discuss the associated fourth order Liouville theorem to the problem and possible link with the De Giorgi's conjecture. This is an attempt to prove the McKenna-Walter conjecture which is open for the last two decades.

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).

I will talk about my results on Modica type estimates for parabolic reaction diffusion equationsobtained in a series of joint work with Prof. Nicola Garofalo. I will describe the connection ofthese estimates to a geometric conjecture of De Giorgi. I will also show how one can obtaincertain rigidity/symmetry type results as a consequence of these estimates.

The Moser-Trudinger embedding has been generalized by Adimurthi and Sandeep to a weighted version. We prove that the supremum is attained, generalizing a well-known result by Flucher, who has proved the case \beta= 0