March 7,  2025

on-line on zoom

• 10:00 - 10:50 Dmitry Kerner (Ben-Gurion University of the Negev ) Deforming the weighted-homogeneous foliation of \((k^N,0)\), and trivializing families of semi-weighted homogeneous ICIS
  

  Abstract:  

  Denote by \(k\) the real/complex numbers. The essential tool in Geometry and Topology of weighted-homogeneous germs is the (weighted-homogeneous) action of the group \(k^*\). The group-orbits form the foliation of \(k^n\) into (real/complex) weighted-homogeneous arcs. The non-equal weights define a flag of coordinate planes on \(k^n\). The foliation is compatible with this flag.

  Deform such a weighted-homogeneous germ by 'higher order terms'. One would like to deform the foliation accordingly, preserving the flag-compatibility. This is not always possible.
  We identify the 'obstruction locus',  outside of which such a deformation does exist, and the family of deformed arcs possesses exceptionally nice properties. [In many case the obstruction locus is trivial.]

  As an application, take a weighted-homogeneous ICIS, and its deformation by higher order terms. This deformation is 'ambient-topologically-trivial', but usually not analytically-trivial. We construct a contact trivialization by a diffeomorphism that is:

  *  Nash off the origin, and its presentation in weighted-polar coordinates is globally Nash;
  *  with controlled Lipschitz/\(C^1\)-properties (depending on the weights);
  *  with many other nice properties.

• 11:00 - 11:50 Aftab Patel (Université de Rennes) An open problem in algebraic approximation of analytic singularities

  Abstract:

  In this talk I will describe a technique used to show the existence of algebraic approximations to analytic singularities over the real or complex numbers which share specified properties with the original such as, for example, the Hilbert-Samuel function. This methodology can described as follows: If we want to approximate a given analytic singularity by one that is defined by algebraic equations which preserves a certain property, we "encode" the targeted property in a polynomial system of equations, which has, by construction, an analytic solution in terms of the defining equations of the original analytic germ. Subsequently an application of a suitable version of Artin's approximation theorem gives us the required algebraic approximation. The primary motivation here is to construct equisingular algebraic approximations for various notions of equisingularity.

  I will briefly describe some results obtained using this method so far, before describing an approximation problem corresponding to a very strong property called topological equiresolvability. Here "equiresolvability" is in terms of Hironaka's Resolution of Singularities. I will show the main obstacle to using the above method for this problem and describe some approaches that may lead to progress on this problem. Some of the work mentioned in this talk was joint work with Janusz Adamus at the University of Western Ontario.