Welcome to the Roadmap!

This is a list of materials I recommend for learning a number of topics in Mathematics. Each list is classified under three levels of depth and/or complexity; namely,

• Introductory materials: aimed at those who have just finished the degree in Mathematics.
• Further reading: graduate-level texts I consider to be more accessible.
• Mastering the concepts: advanced materials, including bleeding-edge results.

Each level has a series of requirements labelled. It is not necessary to have a deep understanding of the subject, but it is recommended to have some notions and references in hand. Think MathOverflow, not Wikipèdia!

Singularity Theory

Introductory materials

Requirements: differential topology, group theory.

• Oset Sinha, R.; Wik Atique, R.; New techniques for classification of multigerms (2018)
• Milnor, J.; Stasheef, J.D.; Characteristic Classes (1962) (Appendix A)
• Milnor, J.; Singular Points of Complex Hypersurfaces (1968)
• [Gibson, C. G.; Singular Points of Smooth Mappings (1979)] https://www.amazon.com/Singular-Mappings-Chapman-Research-Mathematics/dp/0273084100

Further reading

Requirements: differential topology, group theory.

• Golubitski, M.; Guiillemin, V.; Stable Mappings and their Singularities (1973)
• Siersm, D.; A bouquet theorem for the Milnor fibre (1995)
• Milnor, J; Stasheef, J.D.; Characteristic Classes (1962)
• Hatcher, A.; Algebraic topology (2002)

Mastering the concepts

Requirements: differential topology, commutative algebra.

• Mond, D.; Nuño-Ballesteros, J.J.; Singularities of Mappings (2020)

Frontal topology and geometry

Introductory materials

Requirements: differential topology

• Chern, S.S.; Chen, W.H.; Lam, K.S.; Lecutres on Differential Geometry (2000) (Chapter 2)
• Geiges, H.; An Introduction to Contact Topology (2008)