Theory of singularities and catastrophes

All lectures: All lectures at the Academy of Intelligence on the discipline: theory of singularities and catastrophes. These fascinating lectures provide us with the opportunity to explore the deep principles and phenomena underlying our knowledge and our desire to understand the world.
1. Introduction: the essence and applications of singularity and catastrophe theory. Thom's seven elementary catastrophes
2. 1.1. Hadamard's lemma and its corollaries
3. 1.2. Critical points of functions, the Morse lemma. Classification of singular points of smooth functions and maps
4. 1.3. Stability of nondegenerate critical points
5. The division theorem and its corollaries. 2.1. Multiplicity of a function of one variable
6. 2.2. The Weierstrass division theorem as a fundamental result of singularity theory: its essence and purpose
7. 2.3. Newton-Puiseux series
8. 3.1. Germs in singularity theory
9. 3.1. Germs in singularity theory
10. 3.2. An algebraic digression (algebras, rings, ideals)
11. 3.3. Jets and flat functions, versal deformations, stable singularities and sufficient jets. Jet calculus and transversality theorems. Stratified sets
12. 4. Critical points of functions: 4.1. Corank and codimension of a degeneracy
13. 4.2. Critical points of corank 1 and 2
14. 5.1. Formal power series
15. 5.2. Smooth functions with special properties
16. 5.3. Three lemmas on smooth functions for the study of singularities of «continuous» objects
17. 6. Singularities of plane curves
18. 6.1. Normal forms of germs of curves
19. Examples and construction of plane curves
20. 7. Multiplicity of germs of functions and maps. Multiplicity of functions Rn → Rn and Rn → R
21. 7.3. The Malgrange preparation theorem
22. 8. Left-right equivalence, contact equivalence
23. 8. The philosophy of general position
24. 9. Singularities of maps R2 → R2: the fold and the cusp
25. 9.2. Singularities of codimension 3
26. 10. Singularities of maps Rn → Rn
27. 11. Singularities of maps R2 → R3
28. 12. Proof of the division theorem
29. 13. Implicit differential equations
30. 14. Envelopes of families of solutions of implicit differential equations. Catalan's paradox
31. The topology of the nonsingular fiber near a singular point of a complex analytic function. Milnor's theorem. Vanishing cycles, Dynkin diagrams
32. Bifurcation diagrams: swallowtails and caustics. Generalized braid groups. Local monodromy of complete-intersection singularities. Picard-Lefschetz formulas
33. Ramification of integrals and the Gauss-Manin connection. The global monodromy group
34. Applications of Picard-Lefschetz theory: Landau singularities, ramification of the volume function, sharpness of shock fronts
35. Thom polynomials, non-removable singularities, and characteristic classes. The Maslov index and its generalizations. The universal complex of singularities and Kazarian's theory
36. The topology of discriminants and of the space of nonsingular maps. Smale-Hirsch-Gromov type theorems. Invariants of knots and of nonsingular plane curves
37. The local algebra of a singularity. Reduction of function singularities to normal form. The Milnor number, modality. The Newton diagram. Filtrations in the space of germs. Quasi-homogeneous functions and maps
Complain