March 22,  2024

on-line on zoom

• 14:00-14:45 Bárbara Karolline de Lima Pereira  (Universidade de São Paulo) The Relative Bruce-Roberts Numbers of a Function on an Isolated Complete Intersection Singularity

  Abstract:
  In this talk I will give the formula for the relative Bruce-Roberts number of a function \(f\) with respect to an ICIS \((X,0)\), \(μ_{BR}^{-}(f,X)\). In this work we prove that \[μ_{BR}^{-}(f, X) = \mu(f^{-1}(0)\cap X, 0) + \mu(X, 0) - \tau(X, 0),\] where \(\mu\)  and \(\tau\) are the Milnor and Tjurina numbers, respectively, of the ICIS. We also consider the relative logarithmic characteristic variety, \(LC(X)^-\), and we show that \(LC(X)^-\) is Cohen-Macaulay. This is a joint work with J. J. Nuño-Ballesteros (Universitat de Valencia, SPAIN), B. Oréfice-Okamoto, (UFSCar, BRAZIL) and J.N. Tomazella, (UFSCar, BRAZIL).

•  14:55-15:40 Otoniel Nogueira da Silva (Universidade Federal da Paraíba) On Zariski multiplicity conjecture for quasihomogeneous surfaces with non-isolated singularities

  Abstract:
  In this talk, initially we will consider an \(\mathcal{A}\)-finite map germ \(f\) from \((\mathbb{C}^2,0)\) to \((\mathbb{C}^3,0)\). In this case, the double point curve \(D(f)\) plays a fundamental role in studying the topology of the image of \(f\). When \(f\) is quasihomogeneous and has corank \(1\) we present a characterization of the fold components of the double point curve \(D(f)\). As an application of this result, we will consider Zariski multiplicity question for a pair of germs of surfaces \((X_1,0)\) and \((X_2,0)\), in \((\mathbb{C}^3,0)\) with \(1\)-dimensional singular set \((\Sigma(X_i),0)\).

• 15:50-16:35 Maciej Denkowski (Uniwersytet Jagielloński) On Yomdin's version of a Lipschitz Implicit Function Theorem

  Abstract:
   In this talk I will present a complete proof of Yomdin's version of a general Lipschitz Implicit Function Theorem (LIFT). In his beautiful paper on the central set from 1981, Yosif Yomdin made use of this version assuming wrongly it to be a straighforward consequence of the Clarke Lipschitz Inverse Function Theorem. The latter gives only an apparently weaker version of LIFT. Yomdin's result has remained unproved for all these years. Which is more, a misinterpretation of an example due to Lev Birbrair and Dirk Siersma from the context of Lipschitz normal embeddings lead to the belief that Yomdin's LIFT is probably false. Several years has been spent chasing after a non-existent counter-example. Eventually, my first attempt to prove Yomdin's LIFT happened to be successful. Not only the proof is elementary, but it turned out Yomdin's LIFT is in fact equivalent to the 'weaker' LIFT. The proof consists in showing that Yomdin's condition implies the strong one known already to Clarke.