Start date
Betweenand

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).