Seminar Activity Project: A Primer on Optimal Mass Transportation and the Supremal Monge Problem (English)
Published:
Featuring a high cortisol graph for solving Monge problem in the \(L^\infty\) case.
Download the notes directly from here [version 20260508].
Presentation slide can be downloaded here.
Submitted abstract for the presentation can be downloaded here. I suggest the reader to check lists of UNIPD master students’ seminars in the following link.
Recording of the presentation can be accessed through this link.
This seminar explores foundational concepts and central existence results in optimal transport theory. We transition from the intuitive but restrictive Monge problem to the Kantorovich relaxation, leveraging direct methods of calculus of variations to guarantee the existence of optimal plans. By constructing the dual problem and applying tools from convex analysis—specifically \(c\)-transforms and \(c\)-cyclical monotonicity—we establish strong duality. This primal-dual framework allows us to demonstrate the existence and uniqueness of optimal maps for strictly convex costs, culminating in Brenier's Theorem in \(\mathbb{R}^d\). Finally, we resolve the supremal (\(L^\infty\)) Monge optimal transport problem, rigorously proving that a secondary variational problem selects a unique optimal solution induced by a transport map.
Within this project, I’m thankful for the mentorship and support from:
- Prof. Elio Marconi for his thorough mentorship and patience in guiding me through these “baby steps”.
- Prof. Marco Alessandro Cirant for his recommendation on contacting Prof. Elio for this project.
- My MAPPA Colleagues + Jacopo + (ALGANT Colleagues!) for your support and acceptance. Thank you as well for the epic photograph!
- Prof. Alessio Figalli for being willing to give your signature to my project. Thank you as well for your short advice. I will do what I love and I hope someday you will check this page! It is an honor for a Fields Medalist to sign this project, especially since his work is into optimal transport as well!
Below here are the archives of miscellaneous items. It will be updated as time goes on!
- Update 2026/04/25: First public version of the manuscript. Feedbacks are welcome!
- Update 2026/04/29: Added motivation to the Monge Problem in page 1 with an illustration (now 67 pages!), added bold on new definitions, fixed slight grammar error, added a remark for another notation of pushforward measure after we defined it, optimized filed size by compressing the scanned preface page, slight overhaul on abstract.
- Update 2026/05/02: Added an additional remark suggesting the interpretation of the consideration of dual problem in the notes, slight adjustment on abstract, and uploaded a downloadable abstract for the seminar in this site.
- Update 2026/05/05: Fixed errors within the assumption for weak and weak* topologies on finite measure space where weak topology can simply be defined on general metric (even topological) spaces and weak* can be defined with an additional assumption of locally compact. I made the assumption too strong as I put separability as well. Removed the assumption of separability for now-named Riesz-Markov-Kakutani Representation Theorem (Theorem 2.1.5). Added a remark on Definition 2.1.6 when X is compact, we can instead view M(X) as the dual of C(X) to have coherency with subsequent contents, added sharper assumption for Weierstrass Method (Theorem 2.1.2) which should be sequentially compact instead. Additionally, I adjusted some non-fatal typos on some theorems and adjusted the positioning of Figure 2.1 to be more convenient.
- Update 2026/05/08: Thanks to discussion with Prof. Elio during my presentation on May 6th, added contents on Remark 3.0.6 regarding uniqueness of solution to the Supremal Monge Problem and emphasized the statement of Theorem 2.3.3 of h need to be strictly conved on the whole space instead of just the compact domain to assure Liploc of h.
Proof of Alessio Figalli signing this report. Credits: Jacopo Lisciandra.
Special thanks to: 
Can’t believe the best girl this season (Spring 2026) is actually a guy. Thank you as well to Akasaki since your ending theme for the anime is stuck in my head the whole time I’m writing this report.