By Vincent Guedj

The objective of those lecture notes is to supply an advent to the speculation of advanced Monge–Ampère operators (definition, regularity concerns, geometric homes of ideas, approximation) on compact Kähler manifolds (with or with out boundary).
These operators are of primary use in different basic difficulties of advanced differential geometry (Kähler–Einstein equation, area of expertise of continuing scalar curvature metrics), complicated research and dynamics. the themes coated contain, the Dirichlet challenge (after Bedford–Taylor), Monge–Ampère foliations and laminated currents, polynomial hulls and Perron envelopes without analytic constitution, a self-contained presentation of Krylov regularity effects, a modernized facts of the Calabi–Yau theorem (after Yau and Kolodziej), an creation to countless dimensional riemannian geometry, geometric constructions on areas of Kähler metrics (after Mabuchi, Semmes and Donaldson), generalizations of the regularity thought of Caffarelli–Kohn–Nirenberg–Spruck (after Guan, Chen and Blocki) and Bergman approximation of geodesics (after Phong–Sturm and Berndtsson).

Each bankruptcy will be learn independently and relies on a chain of lectures by R. Berman, Z. Blocki, S. Boucksom, F. Delarue, R. Dujardin, B. Kolev and A. Zeriahi, dropped at non-experts. The ebook is hence addressed to any mathematician with a few curiosity in a single of the next fields, advanced differential geometry, advanced research, advanced dynamics, absolutely non-linear PDE's and stochastic analysis.

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