January 19,  2024

on-line on zoom

• 10:00 - 10:50 Dmitry Kerner (Ben-Gurion University) Artin approximation. The ordinary, the inverse, the left-right, and on quivers

  Abstract:
  On many occasions we have to resolve systems of equations, F(x,y)=0. The only hope (in most cases) is to prove: "The solution y(x) exists and is analytic/Nash/...". Even this is often difficult, as one can only resolve the equations order-by-order. But the resulting power series might be non-analytic.
  (Artin approximation) Any formal solution y(x) of the (analytic/Nash) system F(x,y)=0 is approximated by (analytic/Nash) solutions. This result is highly useful when studying singularities of sets and maps. (Deformations, unfoldings, stability, determinacy.)
  (The inverse question) Suppose several (analytic/Nash) power series satisfy a formal relation, \(F(y_1(x),..,y_n(x))=0\). Is this relation approximated by analytic/Nash relations? The answer is 'Yes' in the Nash case and 'No' in the analytic case. More precisely, in the analytic case this ``Inverse Artin approximation" holds for maps of finite singularity type. The left-right equivalence of map-germs asks for the left-right Artin approximation. 
  After a brief introduction I will present some new results. The inverse and left-right Artin approximations hold for maps of ``weakly-finite singularity type". Moreover, these versions of approximation appear to be particular cases of the general "Artin approximation problem on quivers".

• 11:00-11:50 Jean-Baptiste Campesato (Universite d'Angers) Motivic, logarithmic, and topological Milnor fibrations

  Abstract:
  We compare the topological Milnor fibration and the motivic Milnor fibre of a regular complex function with only normal crossing singularities by introducing their common extension: the complete Milnor fibration for which we give two equivalent constructions. The first one extends the classical Kato-Nakayama log-space, and the second one, more geometric, is based on a the real oriented version of the deformation to the normal cone. In particular, we recover the topological Milnor fibration by quotienting the motivic Milnor fibration with suitable powers of (0,+∞). Conversely, we also show that the stratified topological Milnor fibration determines the classical motivic Milnor fibre. (joint work with Goulwen Fichou and Adam Parusiński)