# Lectures

This gives the approximate coverage of the syllabus, lecture by lecture. Topics/lectures in green are not yet covered.

###### Chapter 1: Introductory lecture

1. Critical points, folds and cusps.

2. Some applications in geometry

###### Chapter 2: Diffeomorphisms

3. Diffeomorphisms, changes of coordinates and inverse function theorem.

4. Proof of inverse function theorem by homotopy method.

5. Immersions and submersions

6. Submanifolds and parametrizations.

7. Linearly adapted coordinates and Lyapounov-Schmidt reduction; germs.

###### Chapter 3: Some algebra

8. Ring of germs of smooth functions; its maximal ideal, Hadamard's lemma.

9. Newton Diagram, finite codimension ideals, cobasis.

10. Modules, Nakayama's Lemma ...

###### Chapter 4: Right Equivalence

10. Right equivalence,

11. Jacobian ideals/modules. Codimension of critical points.

12. Morse Lemma. Splitting Lemma ...

13. Splitting Lemma; corank of critical points.

###### Chapter 5: Finite determinacy

14. $$\mathcal{R}$$-trivial families and $$\mathcal{R}$$-tangent space; vector fields along $$f$$

15. Finite determinacy theorem + Examples.

16. Refinements of Theorem

17. Proof of theorem.

###### Chapter 6: Classification

17. Classification of critical points of functions

18. Classification ctd -- corank 1 critical points and classification of binary cubics.

19. Classification for corank 2 critical points

###### Chapter 7: Unfoldings of function germs

20. Families, equivalence,

21. Unfoldings

22. Versality theorem for $$\mathcal{R}$$-equivalence ...

###### Chapter 8: Singularities of Maps

23. Bifurcation problems - what are they?

24. Contact equivalence. Tangent spaces.

25. Classification of corank 1 singularities (the $$A_k$$ family) and corank 2.

###### Chapter 9: Unfoldings & versality

26. Families of maps as unfoldings. Equivalent and versal unfoldings.

27. Versality theorem (no proof) and examples.

JM