New article ideas

Below, you can find a list of new article ideas and suggested references. (Feel free to incorporate additional references! Please list all references you use at the bottom of your article.) If you choose to write about one of these ideas, remove it from the list below and create a new link on the main page.

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The Optimal Transport Problem

• Optimal transport in one dimension; Villani (73-78); Santambrogio (59-67)
• 1-Wasserstein metric, duality, and measures with unequal mass; Piccoli, Rossi, and Tournus A Wasserstein norm for signed measures, with application to nonlocal transport equation with source term
• Regularity of optimal transport maps and the Monge-Ampére equation; Figalli-Glaudo (108-110); Santambrogio (54-57),[1] (make sure to link to the existing wiki article on Optimal Transport and the Monge Ampère equation)

Variants of Optimal Transport Problems

• Wasserstein barycenters and applications in image processing; Santambrogio (215-218); Peyré-Cuturi (138-148)
• Multi-marginal optimal transport and density functional theory (perhaps discussing applications in economics or density functional theory); Figalli-Glaudo(105-106); Santambriogio (48-51)
• Entropic optimal transport and the Schrödinger bridge problem [2][3]
• Martingale optimal transport and mathematical finance; Santambrogio (51-53); [4]

The 2-Wasserstein Metric

• Benamou-Brenier dynamic characterization of W2; Santambrogio (187-198); Villani (238-249) (make sure to cite existing wiki article on Geodesics and generalized geodesics)
• Displacement convexity; Santambrogio (249-251,271-276); Villani (150-154) (make sure to cite existing wiki article on Geodesics and generalized geodesics)
• 2-Wasserstein metric and Ricci curvature; Figalli-Glaudo (112-114); [5]

Numerical Methods for Optimal Transport

• Computing OT via Benamou-Brenier; Santambrogio (220-225); Peyré, Cuturi (102-108)
• Wasserstein Barycenters; Santambrogio (215-218); Peyré, Cuturi (138-144)

Mathematical Foundations:

• The Moreau-Yosida regularization; Bauschke-Combette Ch 12.; Santambrogio (6); Ambrosio-Gigli-Savare (59-61); please include an example with a picture :)
• The continuity equation; Chorin-Marsden (1-11); Santambrogio (123-126); Ambrosio-Brué-Semola (183-189)
• Gradient flows in metric spaces; Figalli-Glaudo (107-108); Santambrogio, 'OT for Applied Mathematicians' (285-290); Santambrogio, 'Euclidean, Metric, and Wasserstein GFs' (90-107; don't need to cover all topics, just what interests you)
• Gradient flows in Hilbert spaces; Ambrosio-Brué-Semola (109-124)

Applications:

• Wasserstein Generative Adversarial Networks [6], [7]
• Optimal transport methods in economics; see introduction of book by Galichon (I have a copy you can borrow) and [8]
• 2 layer neural networks as Wasserstein gradient flows [9]
• Quantization and Lloyd's algorithm [10], [11], [12]