January 24,  2025

on-line on zoom

• 14:00 - 14:50 Hellen Santana (Universidade Federal de São Carlos) Brasselet number: a generalization of the local Euler obstruction
  

  Abstract:  Let \(f,g:(X,0)\to(\mathbb{C},0)\) be germs of analytic functions defined over a complex analytic space \(X\). The Brasselet number of a function \(f\) describes numerically the topology of its generalized Milnor fibre. We present formulas to compare the Brasselet numbers of \(f\) in \(X\) and of the restriction of \(f\) to \(X\cap\{g=0\}\), in the case where \(g\) has a one-dimensional stratified critical set and \(f\) has an arbitrary critical set. If, additionally, \(f\) has isolated singularity at the origin, we compute the Brasselet number of \(g\) in \(X\) and compare it with the Brasselet number of \(f\) in \(X\). As a consequence, we obtain formulas to compute the local Euler obstruction of \(X\) and of \(X\cap\{g=0\}\) at the origin, comparing these numbers with local invariants associated to \(f\) and \(g\).

• 15:00 - 15:45 Thaís Maria Dalbelo (Universidade Federal de São Carlos) Stratified Morse critical points and locally tame singularities

  Abstract: The generalization of the Morse theory presented by Goresky and MacPherson is a landmark that divided completely the topological and geometrical study of singular spaces. Let \(X \subset \mathbb{C}^n\) be the germ at \(0\) of a complete intersection variety and \(f, g\) non-constant polynomial functions on \(X\). If the germs \(X\), \(X \cap f^{-1}(0)\) and \(X\cap f^{-1}(0) \cap g^{-1}(0)\) are non-degenerate, locally tame, complete intersection varieties, we prove that the difference of the Brasselet numbers,  \({\rm B}_{f,X}(0)\) and \({\rm B}_{f,X\cap g^{-1}(0)}(0)\), is related with the number of Morse critical points {on the regular part of the Milnor fiber} of \(f\) appearing in a morsefication of \(g\), even in the case where \(g\) has a critical locus with arbitrary dimension. This result connects topological and geometric properties and allows us to determine some interesting formulae, mainly in terms of the combinatorial information from Newton polyhedra. 
  
  This project was supported (in part) by grant 2019/21181-0, São Paulo Research Foundation (FAPESP).
  
  

  This is a joint work with Hellen Santana