Bilipschitz geometry and surface singularities resolution

• Grant number: 15/08026-4
• Support Opportunities: Scholarships in Brazil - Post-Doctoral
• Start date: September 01, 2015
• End date: May 31, 2018
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Maria Aparecida Soares Ruas
• Grantee: Helge Pedersen
• Host Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil
• Associated research grant: 14/00304-2 - Singularities of differentiable mappings: theory and applications, AP.TEM

Abstract

One of the most important tools in singularity theory is the notion of a resolution of the singularity. If (X,0) is the germ of a singular variety, then a resolution of (X,0) is a smooth germ (Y,E) together with a proper map p from (Y,E) onto (X,0), such that p restricts to an isomorphism (Y-E)to (X-0)$. There are several ways to obtain a resolution. The classical way is through repeated applications of normalized blow-ups. This was used by Zariskiand later by Hironaka to prove the existence of resolutions in some cases. Another way, due to Spivakovsky, to obtain resolutions for complex surfaces, is through repeated applications of normalized Nash modifications. It is conjectured that these two processes are dual to each other. Recently the subject of bilipschitz geometry of complex singularities has seen a great deal of research. It is the study of the bilipschitz class of the outer metric, which is the restriction of an ambient euclidian metric, and the inner metric, which is the metric induced by the Riemannian metric on the smooth part. The goal of this project is to use bilipschitz geometry to explain the conjectured duality between resolution through normalized blow-ups and resolutions through normalized Nash modification. The results will be applied to study the bilipschitz geometry of determinantal surfaces singularities. (AU)

Scientific publications

(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)

KERNER, DMITRY; PEDERSEN, HELGE MOLLER; RUAS, MARIA A. S.. Lipschitz normal embeddings in the space of matrices. MATHEMATISCHE ZEITSCHRIFT, v. 290, n. 1-2, p. 485-507, OCT 2018. (15/08026-4, 14/00304-2)