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In 1781 Gaspard Monge (1746–1818), published his Mémoire sur la théorie des déblais at des remblais , where he discussed the following, very simple, problem.
I would like to thank all the people that, during the three years of my doctoral studies, supported me both from the academic and the personal point of view. In particular, I thank my supervisor, Prof. Sergio Caracciolo, for his countless advices and his constant encouragement. I would like to thank also Prof. Giorgio Parisi, for the opportunity he gave me to collaborate with him on some aspects of the main topic of this thesis, and Prof. Luigi Ambrosio, that gave me precious hints and introduced me to the theory of optimal transport. Finally, I had many stimulating conversations with Elena Boniolo, Dr. Davide Fichera and Dr. Andrea Sportiello both on the matching problem and on the algorithmic aspects of its solution.
I thank Prof. Piergiulio Tempesta, Prof. Rosario Antonio Leo and Prof. Mario Leo for many interesting and pleasant discussions. I want to thank here also Prof. Giuseppe Mussardo, for his kind hospitality at SISSA in Trieste for two long periods, Prof. Andrea Gambassi and Prof. Markus Müller, for their lectures on disordered systems and stochastic processes.
The present document is a significantly enlarged version of my Ph.D. thesis. It contains more details about my work as Ph.D. student and some additional paragraphs for the sake of completeness and clarity. This revision process was financially supported by the John Templeton Foundation.
I am grateful to all the people that have never stopped to be on my side during this long period, starting from my family, and Alice in particular. Thanks to all my friends, Francesca, Giorgio, Marco, Pasquale, Ivan, Marta, Silvia, Alessia, Angela, Alessandro, Chiara, Cinzia, Claudia, Giovanni, Ottavia, Riccardo, and many others.
Finally, thank you, Elena, for everything you taught me, every day.
May 2016, Rio de Janeiro, Brazil
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