General topology: topological spaces, continuous functions, metric spaces, compactness, separation axioms. Introduction to algebraic topology: fundamental group and covering spaces, classification of surfaces. Introduction to differentiable topology: regular values, Sard's theorem, degree theory, notions of Morse theory. Notions of computational geometry and geometric mechanics. 



•    Lima, Elon Lages. Metric Spaces - IMPA; 
•    Rudim, Walter. Principles of mathematical analysis;
•    Introduction to Differential Topology. 


•    DI PRISCO, C. A. (1997) An introduction to set theory; 
•    Lima, Elon Lages. Fundamental Group and Covering Spaces. Euclides Project Collection - IMPA; 
•    Yukio Matsumoto. An Introduction to Morse Theory (Translations of Mathematical Monographs, Vol. 208) (9780821810224); 
•    Lima, Elon Lages. Differential Varieties; 
•    Milnor, John. Topology from the differentable.