Why are bubbles round? Why do honeycombs have hexagonal chambers? Why are salt crystals cubic? 
The answers all rely on one principle: energy minimization. For each of these problems there 
is an associated energy functional whose critical points can be described geometrically. 
These critical points, coming from a constrained minimization problem, often characterize 
the equality state in a certain "geometric" inequality (e.g., the isoperimetric inequality, 
which relates perimeter to volume). We will discuss how to use tools from the calculus of 
variations to show when minimizers exist and how their geometric information can be deduced.