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Aubry-Mather Theory through Optimal Transportation

Voronine, Filip (2023) Aubry-Mather Theory through Optimal Transportation. Bachelor's Thesis, Mathematics.

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Optimal transport theory is the study of efficient movement and allocation of resources. The field can be traced back to the eighteenth century where Gaspard Monge aimed to understand how to move mass from the ground to construct fortifications in the most efficient manner. This problem is intimately related to the field of linear programming through the Kantorovich problem which extends the problem considered by Monge to include mass splitting. We characterize the solutions of the Kantorovich problem using the theoretical framework developed by Rüschendorf which generalizes tools from convex analysis using the cost function that is inherent to the optimal transport problem. Under suitable assumptions, this characterization provides sufficient conditions for the existence of a solution to the Monge problem in the Euclidean setting. Aubry-Mather theory can also be studied through optimal transport problems where the cost function is given by the minimal action between two points and both initial and final measures are the same.

Item Type: Thesis (Bachelor's Thesis)
Supervisor name: Seri, M. and Sterk, A.E.
Degree programme: Mathematics
Thesis type: Bachelor's Thesis
Language: English
Date Deposited: 02 Aug 2023 08:36
Last Modified: 09 Aug 2023 10:37

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