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Lipschitz Geometry of tame singular sets and mappings.

Grant number: 15/12667-5
Support type:Scholarships in Brazil - Post-Doctorate
Effective date (Start): February 01, 2016
Effective date (End): May 31, 2019
Field of knowledge:Physical Sciences and Mathematics - Mathematics - Geometry and Topology
Principal researcher:Maria Aparecida Soares Ruas
Grantee:Saurabh Trivedi
Home 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

O-minimal structures originate in Model theory. They can be seen as a generalisation of semialgebraic and subanalytic geometry. They also provide an excellent framework for developing Grothendieck's `topologie modérée' (tame topology) proposed in his `Esquisse d'un Programme' in 1984 and provide a platform to study most general singular sets on which geometry can be performed. The project is to study geometry of singular sets in O-minimal structures in two parts. In the first part we propose to study Lipschitz geometry of singular sets of log-exp o-minimal structures. In the second part we propose to study bi-Lipschitz classification of map germs. Part I. The most common examples of o-minimal structures are the semialgebraic sets and globally subanalytic sets. They are also polynomially bounded structures, i.e. every for every function f : R \to R definable in this structure there exist an a>0 and an integer r > 0 such that |f(x)| \leq x^r for all x > a. Wilkie proved that the extension of R by exponential function is model-complete, and van den Dries and Miller showed that the extension of polynomially bounded structures by exponential and logarithm functions is o-minimal. Sets definable in such structure are called log-exp sets. There is a preparation theorem for log-exp sets due to Lion and Rolin. Thus, it is natural to ask whether log-exp sets admit local Lipschitz contractibility and the cohomology restrictions as in the case of globally subanalytic sets. Part II. Two map-germs are called equivalent if they look 'same' up to diffeomorphisms. One can also ask if two map-germs are equivalent up to bi-Lipschitz homeomorphisms. Henry and Parusinski proved that while bi-Lipschitz equivalence of real and complex analytic set does not admit moduli, it is not true for function germs. Through this project we propose to study bi-Lipschitz A-equivalence of functions and mappings. We would like to study invariants for bi-Lipschitz A-classification of holomorphic function (extending Henry-Parusinski result) and for map-germs, in low dimensions, for instance (2,2) and (2,3).

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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)
TRIVEDI, SAURABH. Cohomology of flat currents on definable pseudomanifolds. Journal of Mathematical Analysis and Applications, v. 468, n. 2, p. 1098-1107, . (15/12667-5)
NGUYEN, NHAN; TRIVEDI, SAURABH. Transversality of smooth definable maps in O-minimal structures. MATHEMATICAL PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY, v. 168, n. 3, p. 519-533, . (15/12667-5, 16/14330-0)
SOARES RUAS, MARIA APARECIDA; TRIVED, SAURABH. BI-LIPSCHITZ GEOMETRY OF CONTACT ORBITS IN THE BOUNDARY OF THE NICE DIMENSIONS. ASIAN JOURNAL OF MATHEMATICS, v. 23, n. 6, p. 953-968, . (14/00304-2, 15/12667-5)

Please report errors in scientific publications list by writing to: cdi@fapesp.br.