Singularities: Algebra, Geometry, Topology and Applications

1. Singularities and invariants. Map germs, action of Mather groups A, R, L, C and K local forms, codimension and stability, simple map germs. Invariants: Milnor number, cusps and double fold points.
Lecturer: Marcelo José Saia.

2. Computacional methods in singularity theory: Introduction to Singular, multiple point spaces in the source, multiple point spaces in the target via Fitting ideals, Mond-Pellikaan algorithm for constructing a presentation. computing topological invariants. 
Lecturer: Aldicio José Miranda.

3. Lipschitz Geometry of Singularities: Classification of singularities up to bi-Lipschitz homeomorphisms, Lipschitz classification of germs of complex curves for the outer metric, classification of inner Lipschitz geometry of surface singularities. Lipschitz invariants. Recent results on Lipschitz normal embeddings of complex surface singularities. Examples.
Lecturers: Alexandre Fernandes and Anne Pichon

4. Topology of Singularities. Function Germs, Milnor Fibers, topology of the Milnor fiber, germs of isolated singularity hypersurfaces, Milnor Number. 
Lecturer: José Luis Cisneros Molina

5. Flat Curves: Flat Curves and analytical invariants, flat curve representation, parameterization, topological and analytical equivalence. topological Invariants: Milnor number, semigroup of values, topological classification, analytical invariants: Tjurina number, set of differential values. analytical classification. 
Lecturer: Marcelo Escudeiro Hernandes.

Referências:
1. Curves and Singularities A geometrical introduction to singularity theory. J. W. Bruce and P. J. Giblin. Cambridge University Press. 1984, 1982.
2. Bi-Lipschitz geometry of weighted homogeneous surface singularities. L.Birbrair, A. Fernandes; W. Neumann, Math. Ann. 342 (2008), no. 1, 139–144
3. Bi-Lipschitz geometry of complex surface singularities. L.Birbrair, A. Fernandes; W. Neumann, Geom. Dedicata 139 (2009), 259–267.
4. Singular Points of smooth mappings. C. G. Gibson. Pitman Research Notes in Mathematics 25. London.
5. A. Hefez, M. E. Hernandes. Computational methods in the local theory of curves. Publicações Matemáticas do IMPA. [IMPA Mathematical Publications] 23o Colóquio Brasileiro de Matemática. [23rd Brazilian Mathematics Colloquium] Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 2001. viii+115 pp. ISBN: 85-2440172-9.
6. A. Hefez. Irreducible plane curve singularities. Real and complex singularities, 1-120, Lecture Notes in Pure and Appl. Math., 232, Dekker, New York, 2003.
7. J. Milnor, Singular points of complex hypersurfaces. Annals of Mathematics Studies, No. 61 Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo 1968 iii+122 pp.
8. Neumann, Walter D.; Pichon, Anne Lipschitz geometry of complex curves. J. Singul. 10 (2014), 225–234.