New Frontiers in Singularity Theory and Bi-Lipschitz Geometry of Semialgebraic Set...
Singularities of differentiable mappings: theory and applications
| Grant number: | 15/12667-5 |
| Support Opportunities: | Scholarships in Brazil - Post-Doctoral |
| Start date: | February 01, 2016 |
| End date: | May 31, 2019 |
| Field of knowledge: | Physical Sciences and Mathematics - Mathematics - Geometry and Topology |
| Principal Investigator: | Maria Aparecida Soares Ruas |
| Grantee: | Saurabh Trivedi |
| 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 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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