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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

The 2-Wasserstein Metric

• Geodesics and generalized geodesics; Santambrogio (202-207), Ambrosio, Gilgi, Savaré (158-160)
• Displacement convexity; Santambrogio (249-251,271-276); Villani (150-154)
• Asymptotic equivalence of $W_{2}$ and ${\dot {H}}^{-1}$; Santambrogio (209-211); Villani (233-235)
• Formal Riemannian Structure of the Wasserstein metric; Villani (245-247, 250-251); Ambrosio, Gigli, Savaré (189-191)

Numerical Methods for Optimal Transport

• Semidiscrete Optimal Transport (for $c(x,y)=|x-y|^{2}$); Santambrogio (242-248); Peyré Cuturi (85-89)
• Computing OT via Benamou-Brenier; Santambrogio (220-225); Peyré, Cuturi (102-108)
• Sliced Wasserstein Distance; Santambrogio (214-215); Peyré, Cuturi (166-169)
• Wasserstein Barycenters; Santambrogio (215-218); Peyré, Cuturi (138-144)

Applications of Optimal Transport

• Machine Learning Kolouri, et al, Optimal Mass Transport: Signal processing and machine-learning applications

Mathematical Foundations: Optimization

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

Mathematical Foundations: Differential Equations

• Gradient flows in metric spaces; 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)