Interview at CIRM: Herwig Hauser
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Herwig Hauser (Chair) and Guillaume Rond (Local Project Leader) held a Jean Morlet semester at CIRM from mid January to mid July 2015. Their scientific programme focused on 'Artin Approximation and Singularity Theory'. 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 were investigated during the program. Fruitful interchanges with the singularity theory, the combinatorics and the algebraic geometry groups took place. The scientific program was complemented by an exhibition of algebraic surfaces in the city of Marseille, based on the very successful "Imaginary" program designed by Hauser for the Mathematisches Forschungsinstitut Oberwolfach. CIRM - Chaire Jean-Morlet 2015 - Aix-Marseille Université
