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

Questions I've been asked recently

Answers in blue

General

Do we need to know how to prove, say, the principle of symmetric criticality?

No. I don't ask for proofs of theorems like that. I ask proofs where they test your understanding of the material, and are basically straightforward. For example last year one question was to prove that stabilizers are subgroups (Qu 2). And another asked for a proof that certain elements were generators of a given lattice (Qu 3).

Can we use the \(g\cdot x\) notation, rather than the \(\rho(g)x\) notation in the exam?

Yes certainly. (In fact it's the one I usually use, but which you use is up to you.)

We didn't do much on the 3D groups - do we need to know that material?

Not in any detail. You could be asked a question about the symmetry of the tetrahedron or cube, but you'd be expected to work it out, not remember any results about it.

Is it the same to say that a map f is G-equivariant and that f has symmetry G?

Yes - they are the same thing.

Do we need to know the conditions for the 5 types of lattice.

Yes!

Lecture notes

In the notes, in Example 6.5, for \(H=D_3\), how do we know that \(\theta(R_{2\pi/3})=0\)?

I presume you mean part of the final bullet point. What are the possible homomorphisms of \(\D_3\) to \(S^1\). The answer there is a bit concise, so let me elaborate:
As always, \(\ker\theta\) must be a normal subgroup, and the only normal subgroups of \(\D_3\) are the trivial group, the cyclic group \(\C_3\), and the whole group \(\D_3\). Now the kernel can't be the trivial group, as if it were then the image of \(\theta\) would be isomorphic to \(\D_3\), but \(S^1\) is Abelian so that's not possible. If the kernel is all of \(\D_3\) then the \(\theta\) is trivial (i.e., \(\theta(g)=0,\;\forall g\in\D_3\)), which we've already dealt with. Finally the only other possibility is \(\ker\theta=\C_3\), and that's the one that's stated: \(\theta(R_{2\pi/3})=0\) and \(\theta(r_0)=1/2\) (and \(\theta\) of all 3 reflections = 1/2).

Notation: How is \(\ZZ_3= \left<(123)\right>\)? Am i wrong to say that \(\ZZ_3=\{0,1,2\}\) ?

No, you’re not wrong! But I (and others) get lazy, and denote any cyclic group by \(\ZZ_n\). To be more careful, the subgroup \(\left<(123)\right>\) of \(S_3\) should be denoted \(A_3\). But they are isomorphic (\(A_3\) and \(\ZZ_3\), that is). However, for \(n>3\) the subgroup \(A_n\) is not cyclic, so how would you denote the subgroup generated by \((1\;2\;3\;\dots\;n)\) say? Probably \(\ZZ_n\) is as good a notation as any. [In an exam (or any other written work), you should write something like, "Let XXX denote the subgroup of \(S_n\) generated by ... ", then it's clear (whatever you decide to use for XXX.]

Exams

This year (2019) there are 5 questions instead of 4. What's the difference?

The only difference is that the final question has been split into two parts - one on ODEs and one on periodic orbits

Please can you upload the solutions to last year's exam?

As a matter of principle I don't give out solutions to exams. The problem with doing so is that I find students learn exam solutions rather than learning to understand the material. But make sure you read the Feedback to last year's exam. I should also point out that it is useful to compare your solutions with those other students obtain, and discuss them critically: you can learn a lot from each other in that way.

Are the proofs of Appendix B examinable?

No - nor are those of Appendix A.

Which proofs are examinable? (Eg, Theorem 2.9 on the classificationof subgroups of O(2))

As you can see from past papers, I do not expect you to reproduce lengthy proofs (such as that of Theorem 2.9). However, shorter proofs may be examined.

2017 Exam

Last part of 2(b)iv): I have found the stabilizer for \(\D_6\) and \(\D_6\times \ZZ_2\) however I don’t understand how to show it’s the graph of a homomorphism from \(D_6\) to \(\ZZ_2\), or what revelevance the kernel has with this.

Each element of \(\D_6\) either maps \(S_0\) to itself or to its complement. And the homomorphism tells you which it is. That means the stabilizer is the graph of a function from \(\D_6\) to \(\ZZ_2\); why is it a homomorphism? you can either check explicitly, or use the fact that it's a homomorphism iff the graph is a subgroup - see Problem A3.5 in the appendix. The kernel is the stabilizer for the \(\D_6\) action.