Optimal Transport: Heat Equation and Metric Spaces

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Explores the existence and uniqueness of solutions for differential equations through local Lipschitz continuity and the Cauchy-Lipschitz theorem.

Covers manifolds, charts, compatibility, and submanifolds with smooth analytic equations.

Explores the heat equation, equilibrium equations, heat flux, and harmonic functions in heat distribution.

Explores VC dimension, gradient descent, convex sets, and Lipschitz functions in convex optimization.

Explores equivalence in vector spaces, covering conditions for statements to be considered equivalent and properties of algebraic bases.

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Explores regularity in higher dimensions for scalar problems and discusses weak sense solutions and interior regularity.

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