\( \def\Fix{\mathrm{Fix}} \def\D{\mathsf{D}} \def\C{\mathsf{C}} \def\D{\mathsf{D}} \def\GL{\mathsf{GL}} \def\OO{\mathsf{O}} \def\SO{\mathsf{SO}} \def\RR{\mathbb{R}} \def\ZZ{\mathbb{Z}} \def\Aut{\mathrm{Aut}} \def\xx{\mathbf{x}} \)
jam - wiki

Feedback on the 2018 exam:

You can see the exam here

Question 1

(a) Most obtained 4/4 for this, though a few lost a mark for not giving the orders of the group (I'm sure they knew, but I can't give marks if the answer isn't there!).

(b) Some confusion (again - like last year, see Feedback2017). Symmetries of \(f\) must satisfy \(g\cdot f = f\): allowing reversal of sign doesn't mean \(s\cdot f(x)=-f(x)\) is a symmetry! For example \((T_1,-1)\cdot f\) is the function \[ ((T_1,-1)\cdot f)(x) = -f(T_1^{-1}(x)) = -f(x-1)\] and looking at the graph we see that \(-f(x-1)=f(x)\) so it's a symmetry. [Here \(T_1\) is the translation \(x\mapsto x+1\).]

Some missed \(x\mapsto -x\) with \(s=-1\) (equivalently \(R_\pi\) in the \(\RR^2\) notation which I did accept, even though it wasn't the notation suggested in the question), and 2 marks were lost there. Full marks were obtained if you got all the generators. A few got the transformations, but lost marks for not checking them all on the given function.

Question 2

(a) Almost all did this well, including the statement of the orbit-stabilizer theorem. One or two gave some extra points in the orbits, such as \((\sqrt2,\,0)\), perhaps confusing \(D_4\) with \(D_8\), but there was no explanation, so I couldn't give many marks there. A few unfortunately confused the orbit-stabilizer theorem with the Burnside orbit-counting theorem we saw in the coursework, so lost the 3 marks for that part of the question.

(b) Overall a challenging question. Only a very few candidates got 20 or more (out of 24).

(i) [3 marks] Many confused effective with free. The action here fails to be effective because \(R_\pi\cdot f=f\) for all \(f\in X\).

(ii) [5 marks] This asked for Fix(\(D_4,X\)). Some did this well: there are two functions \(x^2+y^2\) and \(-x^2-y^2\). Some included \(x^2-y^2\) but that function is not fixed by \(R_{\pi/2}\) (in fact, \(R_{\pi/2}\cdot(x^2-y^2)=-x^2+y^2\)). Some listed which functions are fixed by which elements, but unfortunately didn't go that extra step and put the information together to find the 2 functions fixed by all elements of the group.

(iii) [8 marks] Very few remembered what the Burnside type of an action is (see Chapter 1). Some had the right general idea and got partial credit.

Question 3

(a) This was done well by almost everyone. A few failed to find the Seitz symbol for \((A|v)^{-1}\) and lost 1 or 2 marks depending on how far they did get. One or two students wrote \[(A|v)(B|u) = AB+Au+v\] which is adding matrices and vectors! You needed to include the vector \(x\) in this.

(b) (i) [6 marks] Done well. Almost everyone got the diagram of the lattice correct.

(ii) [6 marks] Almost all correctly identified vectors \(a,b\) that generate the lattice (more than one answer was possible here). Most proved \(L=\mathbb{Z}\{a,b\}\), although a few only proved 1 inclusion (sometimes twice!) and lost 2 marks for that. A few gave an argument including the words 'it is clear that', but that is not a `careful' proof, so marks were lost there. The lattice is a square lattice, which only a few got (1 mark for that bit).

(iii) [6 marks] About half of candidates got this right. Some said `a reflection and a translation' but that's not the definition. The main point is that the translation should be parallel to the line of reflection. If you didn't understand that it was unlikely you could do the rest of the question (although some did manage). A few tried using \(r_{\pi/3}\) but that doesn't preserve the lattice, even after a translation.

Question 4

(a) Using invariance of fixed point spaces requires a careful statement of results used - lost 2 marks if that was not done. In the course, we had two chapters on the importance of fixed point spaces, but only a few saw their relevance to this question (disappointing!).

There was also a confusion between `invariance under evolution of the system' and `invariant functions' (the latter being irrelevant here) lost all marks for that part of the question.

(b) The question only asks about equilibria on the 2 subspaces the \(z\)-axis and the \(x\)-\(y\)-plane, which makes the calculation very easy and a very easy 5 marks. Some tried to solve the full equations, which was possible but lengthy and not a good idea to do in an exam.

(c) The initial point is fixed by \(B\), and hence the entire solution will also be fixed by \(B\) (that is, Fix(\(B\)) is an invariant subspace (invariant under the evolution). A few students put \(x=y\) in the equations, but that subspace is not invariant (under the evolution - it would be if \(x\longleftrightarrow y\) were a symmetry, but it isn't!).

Scaling

Since there were quite a few poor marks (mostly due to the challenges of Question 2) the marks are being scaled up a little.