Delve into the advanced mathematical concepts of optimal transportation and Monge Ampère type equations in this comprehensive lecture by G. De Philippis from Scuola Normale Superiore di Pisa, Italy. Explore key topics including the Merge problem, super differential, isoperimetric problem, parametric problem, probability measure, and eigenvalue concepts. Gain insights into the solution, definition, and proof of related theorems. Examine the intricacies of equality everywhere, constant speed maps, and variation in the context of these complex mathematical theories. This in-depth presentation is part of the School and Workshop on "Geometric Measure Theory and Optimal Transport" series, offering a rigorous exploration of cutting-edge mathematical principles.