April 4,  2025

on-line on zoom

Meeting ID:    970 1800 8217
Passcode:       726900

• 10:00 - 10:50 María Pe Pereira (Universidad Complutense de Madrid) A theory to study metric degenerations: Moderately Discontinuous Algebraic Topology

  Abstract:   In the same way algebraic topology gives a language to talk about properties of topological spaces up to homeomorphism (more precisely, up to homotopy), we give a theory to talk about metric degenerations where the metric and dynamical information matters. In the works [1] and [2] we developed a first version where we impose subnanalytic hypothesis (which is roughly speaking as asking the spaces and mappings to be triangulable). In this talk I will explain the more general framework of continuous families of metric spaces we are working on. 

   

  [1] with J. Fernández de Bobadilla, S. Heinze, E. Sampaio,  Moderately Discontinuous Homology,   Communications on Pure and Applied Mathematics, volume 75, Issue 10 p. 2123-2200 https://doi.org/10.1002/cpa.22013   Also available in arXiv:1910.12552v3

  [2] with J. Fernández de Bobadilla and  S. Heinze. Moderately Discontinuous Homotopy. International Mathematics Research Notices, Volume 2022, Issue 23, December 2022, Pages 18346–18400,  https://doi.org/10.1093/imrn/rnab225 Available in ArXiv:2007.01538.

• 11:00 - 11:50 Maciej Denkowski (Uniwersytet Jagielloński) Lipschitz Normally Embedded Hölder Triangles in \(\mathbb{R}^4\)

   

  Abstract: Bi-Lipschitz classification of semi-algebraic (or definable in some polynomially bounded o-minimal strucutres) surface germs in \(\mathbb{R}^n\) was made possible due to the finiteness theorems of Mostowski, Parusiński and Valette. It can be considered from three different points of view: the ambient one, the outer one and the inner one. Only the latter is done (L. Birbrair 1999) and led to the introduction of the so called Hoelder triangles and horns which are the basic building blocks for surfaces. The outer classification problem is the one we will concentrate on. Since the Lipschitz version of the Whitney Embedding Theorem of L. Birbrair, A. Fernandes and Z. Jelonek (2021), it has been narrowed down to the cases \(n=2,3\). Recently, the case \(n=3\) has been almost completely solved by L. Birbrair and D. Lopes Medeiros, but the methods do not apply in \(\mathbb{R}^4\) due to the presence of knots. As proved by L. Birbrair,  M. Brandenbursky and A. Gabrielov in 2023, the case n=4 is strictly related to Knot Theory. In the present talk we will introduce the notion of microknots and present a Universality Theorem concerning Lipschitz normally embedded Hoelder triangles in \(\mathbb{R}^4\). This result will allow us to produce a counter-example to a conjecture of Birbrair and Gabrielov: in \(\mathbb{R}^4\) there exist two Lipschitz normally embedded surface germs with isolated singularity that are outer bi-Lipschitz equivalent, ambient topologically equivalent but not ambient bi-Lipschitz equivalent. We will also announce a classification result for \(n=4\) we are working on with L. Birbrair.  (Joint work with Lev Birbrair, Davi Lopes Medeiros, Jose Edson Sampaio)