DESCRIPTION
The programme is organized around four classes covering the main techniques and results in the field. The afternoon will be dedicated to problem sessions and work in small groups.
The planned contents are: (1) Basic commutative algebra techniques (completion, henselization, Weierstrass division, Zariski Main Theorem), rank theorem in infinite dimensional geometry. (Herwig Hauser) (2) Classical approximation theorems, counterexamples to various generalizations, nested approximation theorem. (Guillaume Rond) (3) General Néron Desingularization theorem and Artin Approximation, étale and smooth morphisms. (Dorin Popescu) (4) Algebraic power series, generating series, algebraic power series in positive characteristic, automata, applications to number theory. (Boris Adamczewski) The participants will be informed in advance of the contents of the classes and preparatory reading material will be distributed. They should thus arrive with a good knowledge of the basic techniques. The goal of the school is to deepen and broaden this knowledge and to propose attractive research topics so as motivate them to do genuine research in the field. In addition to this one week school, we propose to give a one semester course on commutative algebra and power series techniques for the more advanced students of the Marseille area. 
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