Mathematics / research
Optimal transport and displacement geometry
Relate mass rearrangement to convex duality, geometric evolution, statistical distances, and computational approximation.
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Module 01 research
Optimal transport and displacement geometry
Relate mass rearrangement to convex duality, geometric evolution, statistical distances, and computational approximation.
- 01
Monge and Kantorovich: maps versus couplings
Formulate mass rearrangement with deterministic maps or joint distributions and understand why allowing mass splitting changes feasibility.
36 min - 02
Couplings, tightness, and existence
Prove the direct-method existence argument for transport plans and see why lower semicontinuity of the cost matters.
38 min - 03
Kantorovich duality and the c-transform
Certify an optimal transport value with feasible potentials and construct the best companion potential using the cost transform.
38 min - 04
Brenier maps and convex potentials
Identify when quadratic optimal transport is a unique gradient map and separate that structural theorem from smooth PDE regularity.
40 min - 05
Wasserstein metrics and weak convergence
Track the moment condition that upgrades weak convergence and distinguish a transport metric from its powered objective.
38 min - 06
Displacement interpolation and Wasserstein geodesics
Build constant-speed distributional paths from optimal couplings and contrast moving particles with mixing endpoint weights.
39 min - 07
Continuity equations and the Benamou–Brenier formula
Express quadratic transport as minimum kinetic action while stating the weak continuity and integrability conditions needed for measure-valued paths.
41 min - 08
Wasserstein gradient flows and JKO time steps
Derive a variational implicit transport step, separate an exact quadratic toy from entropy-driven diffusion, and state a valid convergence setting.
43 min - 09
Entropic regularization and Sinkhorn scaling
Fix the entropy sign convention, derive positive matrix scaling, and distinguish solver convergence from regularization bias.
42 min - 10
Sample complexity and transport statistics
Connect empirical Wasserstein error to moments and dimension without treating every transport estimate as a root-n statistic.
42 min