Ian Tobasco - University of Illinois at Chicago, Mathematics We consider the related questions of how to (1) optimally transport heat across a fluid layer and (2) optimally cool an internally heated disc. In either setup, the problem is to select an optimal divergence-free velocity field maximizing the rate of thermal transport. A natural constraint is on the power expended to move the fluid, but other constraints are possible as well. This talk will discuss a connection between such optimal heat transport problems and a family of problems from mathematical materials science known as "energy-driven pattern formation". The result is a multi-scale branching ansatz attaining nearly optimal transport of heat in either setup. The analysis is rather general and various extensions will be discussed. This is joint work with Charlie Doering (U Michigan).