Semester on 'Artin Approximation and Singularity Theory'

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source: Centre International de Rencontres Mathématiques 2015年6月1日
Jean-Morlet Chair - Research Talks - Hauser/Rond
Semester on 'Artin Approximation and Singularity Theory'
January - June 2015
General themes
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.

Herwig Hauser : Commutative algebra for Artin approximation - Part 1 1:23:55
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In this series of four lectures we develop the necessary background from commutative algebra to study solution sets of algebraic equations in power series rings. A good comprehension of the geometry of such sets should then yield in particular a "geometric" proof of the Artin approximation theorem.
Recording during the thematic meeting: «Introduction to Artin Approximation and the Geometry of Power Series» the January 26, 2015 at the Centre International de Rencontres Mathématiques (Marseille, France)
Film maker: Guillaume Hennenfent
Herwig Hauser : Commutative algebra for Artin approximation - Part 2 1:28:25
Herwig Hauser : Commutative algebra for Artin approximation - Part 3 1:29:59
Jan Draisma: Stabilisation in algebraic geometry 1:02:47
Jack Hall: Tannaka duality and formal glueings 59:45
Matthias Aschenbrenner: The algebra and model theory of transseries 1:06:29
Raf Cluckers: Pfaffian functions: real and non-archimedean, and an application to... 48:59