Topology and Singularity Theory
Team and Interlocutors
P. Schweitzer, M. A. Ruas e seus colaboradores
Differential and algebraic topology was one of the fields of mathematics that had the greatest development during the twentieth century. In Brazil, its development occurred principally in the second half of the century, not only as an independent area but also in relation to dynamical systems, differential geometry, and holomorphic structures. At present, topology continues to be a very active field in Brazil with various directions: 3 and 4-dimensional manifolds, foliations, singularity theory, topological group actions, and applications to other fields of mathematics.
All of these subfields are active in Brazil. P. Schweitzer has been working with foliations for almost 30 years with groups at PUC-Rio and UFF (S. Firmo, P. H. Gusmão, S. Druck), including various projects in dimensions 3 and 4. Now there is a beginning of work on laminations, which promise to provide much information about the global topology of 3-manifolds. Geometrical structures in 4-manifolds is another active area (N. Goussevski (UFMG), F. Fang (UFF), K. Resende (UNICAMP), W. Marar (USP São Carlos).
In Rio, several mathematicians use foliation theory as a tool (R. Sá Earp, L. J. Díaz, and R. Ruggiero of PUC-Rio in differential geometry and dynamical systems) and there is a strong group studying holomorphic foliations at IMPA (C. Camacho, P. Sad, A. Lins Neto). Thus it is clear that topology is not only an active field of research in Brazil, with many international contacts, but also one that interacts fruitfully with several other fields of Brazilian mathematics.
Singularity Theory draws on ideas and techniques from algebraic and analytic geometry and algebraic and differential topology, to study the geometry and topology of spaces and mappings defined by polynomials or analytic equations with singularities. It is applied to problems in differential geometry, dynamical systems and the physical and life sciences. The main directions of the research of M. A. Ruas are: a) geometry and classification of singularities; b) applications of singularity theory to extrinsic geometry. They aim to develop the following projects: 1. Topology of real analytic varieties; 2. Equisingularity of families of analytic mappings; 3. Generic properties of submanifolds of euclidean spaces; 4. Differential equations of the geometry, and singularities of vector fields; 5. Classification of singularities: topological invariants.