 That was a bit of math. In fact, lecture 4 is the math heaviest lecture of the entire class. But it's important. The reason I took you through it is that understanding these concepts will now enable us to study applications in protein structure on a much deeper level, and you will be able to make predictions about what happens, how fast does it happen, and why does it happen. And that will prove very valuable. But to aid you in reviewing the stuff today, I have prepared the usual set of study questions. You will see here that three quarters of these are focused on understanding energy landscapes, partition functions, Boltzmann, stability, instability, kinetics, hydrophobic effect, etc. And then at the very end, we have these handful of applications to protein structure that we're going to come back to. But I suspect that you might find it easier to interpret the math if you think about concrete applications already now. Next lecture, we're going to be looking much more at concrete proteins. But until then, enjoy the statistical mechanics.