My name is Alessio Figalli, and I’m professor of Mathematics at Zurich. The optimal transport problem consists of finding the cheapest possible way Of transporting resources from one place to another. This problem comes form economics, but it turns out to have many impications in mathematics. In particular, one possible application is to study how the shape of crystals change under the action of exterior forces. The basic idea is that you can take The ideal shape of a crystal, the one let’s say equations predict But when you don’t take into account all the exterior forces acting on it then you can take the real shape of the crystal where forces are acting and you can try to transport the invisible particles that are composing these crystals from the ideal configuration to the real configuraiton and by understanding how these particles move you can understand a lot about the shape. In the same way you can use the principle to understand the evolution of clouds. The basic principle in this case is that If you look at the cloud in a set instance of time and then at the next instance of time particles have moved. And if you want to understand how the particles have moved again you can think there is an optimal transport behind it and by using this intuition you can actually get very good models. There is no real universal way to how you approach a problem Every problem has it’s own specifics. Now if the problem come’s from geometry, for instance when you study the shape of soap bubbles you have a lot of geometric intuition because you can see the geometric object, you can draw pictures, and let’s say by just looking around you and doing experiments and looking around you you can get a lot of ideas of what theorem could be true and so you have a clear path to follow The same thing is with problems that come from physics you will have physics intuition about what you want to do But, this being said, at some point you have to start using mathematics and using analysis which means you really have to work with formulas. And at that moment, it’s a completely different world You just hvae your pen, your piece of paper, and your imagination to try to make a geometric intuition a physic intuition appear in the formulas and so being able to prove to formulas that the equation your studying really model the phenomena’s you want to describe and that is really something which is difficult to explain but it’s beautiful it’s like art, very creative.