Why do liquids spread on some surfaces, but not on others?

Why do liquids spread on some surfaces, but not on others? When you spill your coffee, why does it dry to form a ring-shaped stain? How fast does a cooled liquid solidify? Why do some liquids and also more complex soft matter systems self-assemble to form simple regular crystalline structures, but others arrange themselves to form complex structures such as quasicrystals? And why does the liquid state even exist?

In this lecture, I will show how mathematics can give a rich and deep understanding of the liquid state. Making sense of the correlated dance of the molecules in just a drop of liquid on a surface is intellectually satisfying and the lessons learned can even be useful to manufacturers of paints, lubricants and many others.