My primary PhD thesis work is on optimal fluid mixing in collaboration with my advisor Prof. Charles Doering. This project investigates the fundamental mixing limitations of the advection-diffusion equation. This single equation describes how a scalar $\theta(\mathbf{x},t)$ such as chemical composition, dye concentration, temperature, or salinity is moved by an incompressible flow $\mathbf{u}(\mathbf{x},t)$ with molecular diffusion coefficient $\kappa$. Mixing is important to industry and science with many open questions. We ask "given mild constraints on the flow $\mathbf{u}$, what is the best mixing rate?" Given these conditions on $\mathbf{u}$, what does this tell us about the transport properties of $\theta$? In particular, we are interested in efficiently mix or homogenize $\theta$ over a domain. To address this problem, we measure 'mixed-ness' of $\theta$ by the $H^{-1}$ Sobolev mix-norm that quantifies mixing with or without diffusion, and utilize techniques from optimal control theory. We have studied optimal mixing within a shell model [1]. A shell model is a reduced model describing the spectral dynamics of the full advection-diffusion equation using a system of ordinary differential equations. Recently, we have explored optimal mixing within the advection-diffusion equation [2] with and without diffusion. The video below shows the optimal stirring stategies for the case of pure advection compared to that of advection and diffusion with a fixed enstrophy constraint in both cases.