\( \def\kK{\mathcal{K}} \def\RR{\mathbb{R}} \)

Comments on 2014/15 exam for MATH45051

See here for the exam paper.

Question 1

(a) was done mostly ok. (b) The relevant equivalence for bifurcations is \(\kK\)-equivalence (most got that right). However some didn't know how to write a map: the map \(\phi\) inducing \(F\) from the versal unfolding \(G\) should be \(\phi:\RR\to \RR^2\) (not the other way), and then \(F=\phi^*G\) (not the other way!). But in particular, one can't write \(\phi(x,y) = (a^2+b^2, ab)\) for example (it doesn't make sense - what is \(\phi(1,2)\) equal to?). You should write say, \((x,y)=\phi(a,b)=(a^2+b^2, ab)\). Part (c) was not answered correctly at all.

Question 2

This was mostly done well, except that no-one explained why the \(f\) in part (b) was 'precisely' 3-determined. The finite determinacy theorem only shows that it is 3-determined, but not that it isn't 2-determined. (The reason here is easy: the 2-jet of \(f\) is zero, so can't be equivalent to \(f\).

Question 3

(a) Mostly done well (although no-one mentioned that it was a graph, which I expected everyone to do!) One person made the mistake of treating \(F\) as a function of 5 variables, rather than a 3-parameter family of functions of 2 variables: this meant that their catastrophe set was obtained by putting all 5 partial derivatives equal to 0 - quite a different answer, and it meant their subsequent answers to this question made little sense)!

(b) Fine for the hyperbolae, but only one person recognized the equation of a cone (\(x^2+y^2=z^2\) is the equation of a circular cone - now you know!).

(c) Some ignored showing \((0,0,a)\not\in\Delta_F\), but this just consisted in showing that for such parameters the critical points are all non-degenerate and would have given a couple of easy marks.

Question 4

(a) Some made the basic error of showing one set was contained in the other, but not the reverse inclusion.

(b) Almost all noticed correctly that the diffeomorphism required was \(\phi(x,y) = (y,x)\). One tried \(\phi(x,y) = (x^{2/3},\,y^{3/2})\), but this is not differentiable, and not even defined for \(y<0\), so it's certainly not a diffeo.

(c) One or two tried finding a cobasis for \(Jf\) instead of \(T\kK_e\cdot f\).

Question 5

Very few attemped this question, and even fewer did (b). Part (a) most mostly done well by those who attempted it.