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.

Want to write about something that's not listed here? Email me!

The Optimal Transport Problem

Unless otherwise specified, all topics are for general cost functions c(x,y).

• Monge Problem; Villani (3-4, 6-9), Santambrogio (xiv-xvii,1-9)
• Kantorovich Problem; Villani (1-3, 6-9), Santambrogio (xv-xvii,1-9)
• Kantorovich Dual Problem (for general costs); Villani (17-21), Santambrogio (9-16)
• Kantorovich Dual Problem (for $c(x,y)=d(x,y)$ where $d$ is a metric); Villani (34)
• Kantorovich Dual Problem (for $c(x,y)=d(x,y)^{2}$ where $d$ is a metric); Santambrogio (16-18)
• Optimal Transport and the Monge Ampère equation; Santambrogio (xvi, 54-57)

Numerical Methods for Optimal Transport

• Discrete Optimal Transport; Villani (5), Santambrogio (235-237), Peyré Cuturi (7-12)
• Auction Algorithm; Santambrogio (238-240); Peyré Cuturi (37-39, 52-56)
• Entropic Regularization; Santambrogio (240-241); Peyré Cuturi (57-62)
• Sinkhorn's Algorithm; Peyré Cuturi (62-73)
• Semidiscrete Optimal Transport (for $c(x,y)=|x-y|^{2}$); Santambrogio (242-248); Peyré Cuturi (85-89)

Applications of Optimal Transport

• Machine Learning Kolouri, et al, Optimal Mass Transport: Signal processing and machine-learning applications
• Economic Matching Problems; Santambrogio (44-48) Galichon, A survey of some recent applications of optimal transport methods to econometrics

I'm hoping graduate students in other areas of science may write an article about how they are using optimal transport :). If you would like me to suggest references, let me know!

Mathematical Foundations: Functional Analysis

• The dual of $C_{o}(X)$ vs. $C_{b}(X)$; Villani (39-43); Santambrogio (4); Rudin Real and Complex Analysis (127-132)
• Convergence of Measures and Metrizability (click on link for references and explanation)

Mathematical Foundations: Optimization

• Fenchel-Rockafellar and Linear Programming; Brezis (15-17); Rockafellar, Variational Analysis (505-507)