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**Abstract:** Assume that we have to move a load of coal from source pile A to destination pile B, both of whose shape we know, and whose capacities are identical. It is known that moving a unit mass of sand from source location x to destination y requires an effort of c(x,y). The problem of optimal transport deals with performing this task in the most efficient possible way, that is we ask for a transport "plan", u(dx,dy), which tells us how much coal should be moved from x to y. Over the past two decades, this problem has received a large amount attention, with applications ranging from finance to meteorology.

In this talk, we will review some basic ideas about optimal transport, and see one application in metrizing the space of probability distributions. Further, we will see how such a metrization can be used to derive fixed points of distribution valued mappings (mappings which take one probability distribution to another).

I will only assume basic familiarity with linear programming for the purpose of this talk.