Taking the Temperature of Mathematical Problems: What Makes Some Hard?

Take a glass of water at room temperature and cool it by a few degrees centigrade. Consider the changes in the water. Although the average speed of the water molecules is a little lower now than before, you still have a refreshing glass of water ready for drinking. Continue steadily cooling the water by a few degrees at a time, and depending on how hot or cold you are, the water becomes more or less refreshing-but it is still a glass of drinkable water. Now suppose the water is at 1∞ centigrade and you cool it an additional couple of degrees. If you are very hot, the resulting "water" might still be very refreshing; however, it will not be drinkable. Its physical state has changed from a liquid to a solid. At the other end of the thermometer, water at 98∞ centigrade is still liquid water, but when heated just a couple of degrees it becomes a gas.

The study of matter as it changes from one state to another involves the physics of phase transitions; in the examples above, from liquid to solid, or from liquid to gas. Recently, the concept of a phase transition has been usefully applied to problems in mathematics - some very hard problems in mathematics. The result of this analysis has been a better understanding of exactly what makes some of the hardest problems so very hard.