Scholarship 24/13488-6 - Singularidades, Teoria dos nós - BV FAPESP
Advanced search
Start date
Betweenand

New Frontiers in Singularity Theory and Bi-Lipschitz Geometry of Semialgebraic Set Germs.

Grant number: 24/13488-6
Support Opportunities:Scholarships in Brazil - Post-Doctoral
Start date until: September 01, 2024
End date until: February 28, 2026
Field of knowledge:Physical Sciences and Mathematics - Mathematics - Geometry and Topology
Principal Investigator:Maria Aparecida Soares Ruas
Grantee:Davi Lopes Alves de Medeiros
Host Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil
Associated research grant:19/21181-0 - New frontiers in Singularity Theory, AP.TEM

Abstract

One of the most interesting problems in geometry is the existence and classification of extensions of mappings, whether they are homeomorphisms or diffeomorphisms. A classic result is Alexander's Trick, which says that two homeomorphisms in Bn that are isotopic on the boundary are isotopic. Such isotopies generally do not preserve differential structures, and interesting examples of this are the exotic spheres studied by Milnor, Kervaire, Brieskorn and others.An "intermediate" variant of this problem is related to Lipschitz geometry, whose most famous result is Kirszbraun's theorem, which states that any Lipschitz map defined on a subset of a Hilbert space extends to the whole space. However, such maps are generally not bi-Lipschitz, i.e. a Lipschitz map whose inverse is Lipschitz. Understanding how to classify and extend bi-Lipschitz maps is fundamental to understand the geometry of analytic and subanalytic sets. In fact, the studies of Maria Ruas, Alexandre Fernandes, Walter Neumann, Juan Jose Ballesteros, Edson Sampaio and many others show that topological and differentiable structures are closely related to their Lipschitz structures.In the case of germs of subanalytic sets of dimension 2, the bi-Lipschitz classification under inner metric was elucidated by Lev Birbrair in 1999. As for the bi-Lipschitz extension problem, Birbrair, Gabrielov and Brandenbursky proved that, even though two germs in R4 are bi-Lipschitz equivalent by Euclidean metric and such map could be extended to a single homeomorphism class in R4, the possible bi-Lipschitz equivalence classes of this extension are infinite! This discovery unveiled the metric knot theory, a theory that differs from the usual one because metric obstructions can appear in the Lipschitz classification, even though the links of these surfaces do not have knots.In the postdoctoral fellow's PhD thesis, Davi Lopes Medeiros showed that if two germs of normally embedded semialgebraic surfaces in R3, with isolated singularity, are bi-Lipschitz equivalent by the Euclidean metric, then such map can be extended to all the ambient space germ. This result is pioneering, as it establishes a bridge between Alexander's trick and Kirszbraun's theorem in a specific case, and in this thesis, the author develops techniques for analyzing and extending bi-Lipschitz maps. One of the innovative concepts established there is the concept of a synchronized triangle, a parameterized surface germ that induces a rigidity in the ambient space. Medeiros proved that every normally embedded surface germ can be decomposed into synchronized triangles, and from this he carried out an analysis that resulted in the aforementioned bi-Lipschitz extension theorem.In every innovative theory, one question is answered and many others arise, and with the metric knot theory is no different. To what extent can these techniques be generalized? Can we construct synchronized spaces of dimension greater than 2? Can we decompose analytic and subanalytic sets normally embedded in Rn, with n greater than or equal to 4? What happens to the bi-Lipschitz extension when the surface has no isolated singularity? And if the surface is not normally embedded, what additional obstructions will we have? It was to answer these and other questions, and to establish new frontiers in singularity theory, that Medeiros decided to take part in this research project. The supervision of Maria Ruas, who has a vast and brilliant career in singularitiy theory, will be of paramount importance, as her great experience, combined with the innovation brought by the work of the postdoc fellow, will result in a collaboration with the potential for wonderful fruits and remarkable mathematical advances, which will greatly enhance Brazil in the scientific community.

News published in Agência FAPESP Newsletter about the scholarship:
More itemsLess items
Articles published in other media outlets ( ):
More itemsLess items
VEICULO: TITULO (DATA)
VEICULO: TITULO (DATA)

Please report errors in scientific publications list using this form.