Grant number: 12/01139-0 Support type: Scholarships in Brazil - Post-Doctorate Effective date (Start): June 01, 2012 Effective date (End): March 31, 2013 Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology Principal Investigator: Maria Aparecida Soares Ruas Grantee: Aurélio Menegon Neto Home Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil Associated research grant: 08/54222-6 - Singularities, geometry and differential equations, AP.TEM Abstract This project proposes the study of the topology of real and complex varieties with non-isolated singularity, through the study of the topology of the Milnor fibre and its degeneration to the singular variety.In my doctoral thesis, we extend the concepts of the Lê Polyhedra to non-isolated singularities, describing the degeneration of the Milnor fibre of a line singularity (given by a holomorphic germ $f: (\Cn,0) \to (C,0)$ with singular set a smooth curve) to the singular fibre. We also describe the degeneration of the boundary of the Milnor fibre to the link of certain classes of real and complex singularities. In this project, we shall continue the application of the concepts of the Lê Polyhedra to the study of more general classes of non-isolated singularities. Moreover, we shall study the homotopy of the Milnor fibre of a line singularity by means of its projection on the singular set of f.On the other hand, we also show in the thesis that the boundary of the Milnor fibre of a non-isolated singularity given by a real analytic map-germ of the type $f \bar{g}: (\C3,0) \to (\C,0)$, with an isolated critical value, is a Waldhausen manifold, that is, a graph manifold. Now, in this project, we also propose the construction of an algorithm to calculate the corresponding graph.