Friday, October 30

on-line on zoom

• 9:45-10:35 Dmitry Kerner (Ben Gurion University) Which germs of vector fields admit 'nice' exponentials?
  

  Abstract: An everyday task in Singularities is to build a local coordinate change (e.g. analytic) with prescribed properties. The classical approach is to cook up a special vector field, and to take its time-one-flow. (A highly non-algebraic/not o-minimal operation!)
  In zero characteristic this 'time-one-flow' is defined as exp[vector field].
  In many cases the vector field can be chosen 'in a nice class', e.g. polynomials/algebraic power series/holonomic functions/higher Picard-Vessiot extensions/Denjoy-Carleman/ \(C^\infty\).
  When does the corresponding coordinate change belong to the same class?
  Our answer are:
  * 'Very rarely' in cases close to algebraic. (The criteria are of Liouville type.)
  * 'Very often' for normed rings.
  * Never in the \(C^\infty\) case.
  The talk is based on arXiv:2607.27279.

• 10:45-11:35 Ramzen Dzhafarov (Uniwersytet Jagielloński) On arc-continuity of subanalytic functions and the Łojasiewicz exponent
  

  Abstract: In a paper On the Gradient Inequality, S. Łojasiewicz and M.-A. Zurro give an example of a subanalytic function which is unbounded near zero, but continuous at this point on any analytic arc. However, their example turns out to be wrong, as we will show in this talk. Moreover, we prove that no such example exists in the subanalytic category, i.e. any subanalytic function that is arc-continuous at a point is continuous there. In addition, several new results concerning the Łojasiewicz exponent will be presented and discussed.