Optimal Transport: Heat Equation and Metric Spaces

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Conditional Expectation

Covers conditional expectation, Fubini's theorem, and their applications in probability theory.

Convergence and Completeness

Covers convergence, completeness, and properties of metric spaces and Banach spaces.

Harmonic Forms: Main Theorem

Explores harmonic forms on Riemann surfaces and the uniqueness of solutions to harmonic equations.

Properties of Complete Spaces

Covers the properties of complete spaces, including completeness, expectations, embeddings, subsets, norms, Holder's inequality, and uniform integrability.

Joint Equidistribution of CM Points

Covers the joint equidistribution of CM points and the ergodic decomposition theorem in compact abelian groups.

Convergence in Law: Theorem and Proof

Explores convergence in law for random variables, including Kolmogorov's theorem and proofs based on probability lemmas.

Proofs: Logic, Mathematics & Algorithms

Explores proof concepts, techniques, and applications in logic, mathematics, and algorithms.

Fundamental Groups

Explores fundamental groups, homotopy classes, and coverings in connected manifolds.

Lipschitz Maps and Compact Domains

Covers Lipschitz maps, compact domains, changing variables, counting problems, and probability of ideals.

Numerical Analysis: Implicit Schemes

Covers implicit schemes in numerical analysis for solving partial differential equations.