Nonlinear dispersive PDE are characterized by the dispersion of the underlying linear flow, meaning that, in the absence of a boundary, solutions tend to spread out in space as they evolve in time. Equations of this type emerge as asymptotic models in a plethora of physical phenomena, such as quantum mechanics, plasma physics, nonlinear optics, oceanography, and general relativity.

The most fundamental question when studying any PDE is that of well-posedness, that is, whether solutions exist, are unique, and depend continuously on the initial data. Completing the well-posedness picture of the most famous dispersive equations has been a major goal in the field. One such equation is the derivative nonlinear Schr\"odinger equation (DNLS), which arises as a model in magnetohydrodynamics. This model is known to be completely integrable and $L^2$-critical with respect to scaling. However, until recently not much was known regarding the well-posendess of the equation below $H^{\frac 1 2}$. I will discuss why the existing methods failed to solve this problem and recent progress towards closing this gap, culminating in our proof of sharp well-posendess in the critical space $L^2$ in joint work with Benjamin Harrop-Griffiths, Rowan Killip, and Monica Visan.