The Peskin problem models the dynamics of a closed elastic string immersed in an incompressible 2D stokes fluid. This set of equations was proposed as a simplified model to study blood flow through heart valves. The immersed boundary formulation of this problem has proven very useful in particular giving rise to the widely used immersed boundary method in numerical analysis. Proving the existence and uniqueness of smooth solutions is vitally useful for this system in particular to guarantee that numerical methods based upon different formulations of the problem all converge to the same solution. In this project ``Critical local well-posedness for the fully nonlinear Peskin problem'', which is a joint work with Stephen Cameron, we consider the case with equal viscosities but with a fully non-linear tension law. This situation has been called the fully nonlinear Peskin problem. In this case we prove local wellposedness for arbitrary initial data in the scaling critical Besov space $\dot{B}^{3/2}_{2,1}(\mathbb{T}; \mathbb{R}^2)$. We additionally prove the high order smoothing effects for the solution. To prove this result we derive a new formulation of the equation that describes the parametrization of the string, and we crucially utilize a new cancellation structure.