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Topological invariants of stable maps and classification of singularities


The main subject of the project is to study the global classification and topology of stable maps. This problem needs of the study of different areas of mathematics, specially of geometry and topology and a combination of these with algebraic methods and complex analysis. The two main pillars for this goal are the classification of singularities and the study of topological invariants. The main objectives of the project are the continuity of the study of Vassiliev invariants started during my thesis, with applications to the global study of generic mappings and the development of new methods for the classification of multigerms. For this goal, the collaboration with professor J. N. Tomazella e B. O. Okamoto from UFSCar is going to be crucial. Joining our knowledge about classification of singularities can produce a deeper knowledge of the area. Continuing the big project of recovering geometry of codimension 2 objects through their projections to Euclidean spaces is also a main objective. It would be very positive to work with researchers in the area of Differential Geometry such as G. A. Lobos Villagra to take advantage of their knowledge in that area to join it to mine on singularities. Another main objective is the development of a theory of versality for 2-parameter families of Lagrangian maps with applications to gravitational lensing, for example. We will try to use the work on Lagrangian maps done in my thesis to study 2-dimensional sections of caustics in space-time. Besides continuing the research I have already worked on, I propose to study a new area in Singularity Theory. The theory of Thom polynomials based on Chern characteristic classes seems to be a new string theory for Singularity Theory. The idea is to study these techniques and develop them together with the topology group of UFSCar (E. L. dos Santos and D. Vendrúscolo amongs others).In general, the proposal wants to establish a collaboration in order to consolidate a group of Singularity Theory in UFSCar. (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)
NISHIMURA, T.; OSET SINHA, R.; RUAS, M. A. S.; WIK ATIQUE, R. Liftable vector fields over corank one multigerms. MATHEMATISCHE ANNALEN, v. 366, n. 1-2, p. 573-611, OCT 2016. Web of Science Citations: 1.
ICHIKI, S.; NISHIMURA, T.; SINHA, R. OSET; RUAS, M. A. S. Generalized distance-squared mappings of the plane into the plane. ADVANCES IN GEOMETRY, v. 16, n. 2, p. 189-198, APR 2016. Web of Science Citations: 7.
SINHA, R. OSET; RUAS, M. A. S.; ATIQUE, R. WIK. ON THE SIMPLICITY OF MULTIGERMS. MATHEMATICA SCANDINAVICA, v. 119, n. 2, p. 197-222, 2016. Web of Science Citations: 1.

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