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Singularities and Artin Approximation
Singularités et approximation de Artin

January-June 2015

Artin Approximation concerns the solvability of algebraic equations in spaces of formal, convergent or algebraic power series. The classical version asserts that if a formal solution exists, then there also exists a convergent, hence analytic, and even algebraic solution which approximates the formal solution up to any given degree. As such, the theorem is instrumental for numerous constructions in algebraic geometry, commutative algebra and recursion theory in combinatorics. A series is Nash or algebraic if it is algebraic over the polynomials. Nash series can be codified by polynomial data deduced from the minimal polynomial by the normalization of the respective algebraic hypersurface. This makes them computable. The field has seen renewed activity through the recent research on Arc Spaces, Motivic Integration and Infinite Dimensional Geometry. Important questions remain still unanswered (nested subring case, composition problems, structure theorems for the solution sets) and shall be investigated during the program. Fruitful interchanges with the singularity theory, the combinatorics and the algebraic geometry groups are expected. The scientific program is to be complemented by an exhibition series of algebraic surfaces in the city of Marseille, based on the very successful « Imaginary » program designed by Hauser for the Mathematisches Forschungsinstitut Oberwolfach.
> Prof. Herwig HAUSER
Universität Wien
Gallery of Singularities
Local Project Leader 
> Dr. Guillaume Rond
I2M Marseille
Aix-Marseille Université

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Artin Approximation and Infinite Dimensional Geometry (1255)
Approximation de Artin et géométre dimensionnelle infinie

23-27 March 2015
Introduction to Artin Approximation and the Geometry of Power Series (1254)
Introduction à l’approximation de Artin et à la géométrie des espaces de séries formelles

26-30 January 2015
Approximation and Combinatorics (1310)
Approximation et combinatoire

20-24 April 2015
Nested Subring Conditions (1311)
Conditions d’emboîtements des sous-anneaux

11-15 May 2015
Algebraic Power Series (1325)
Séries algébriques

25 May – 5 June 2015
List of guests in residence during this semester
​Invités du semestre
Exhibitions, articles, interviews, etc.