Peter Humphries' generals.
10 May 2013, 1pm.
Special topics: analytic number theory and representation theory of compact Lie groups.
Committee: Peter Sarnak (chair), Charlie Fefferman, Nicolas Templier.
Algebra (all Sarnak):
Say you have a finite field. What's its cardinality? (Prime power -
must be prime characteristic to be a field, and must be a vector space
over F_p for some p.)
Prove that the unit group of a finite field is cyclic. (I of course
used the structure theorem for finite abelian groups. They made me
explain why x^m - 1 can only have at most m roots, which to me felt
like a tautology. A few hints about factorising out roots eventually
did the trick.)
You mentioned the structure theorem for finite abelian groups. What is
the module and what is the PID in this case? (The group, and Z.)
How do you prove the theorem? (Structure theorem for finitely
generated modules over a PID.)
How do you prove THAT? (Can't remember.) Seriously?! (Yep, forgot to
revise this. Oops!)
When are two matrices conjugate over a given field? (Is the field
algebraically closed?) Okay, let's say the field is algebraically
closed at first. (Jordan canonical form.) How do you prove it? What is
the module and what is the PID? (I fluffed around with this a bit at
first before getting it right.) What about over a field that isn't
algebraically closed? (Rational canonical form. I again had to state
the module and the PID.)
Why is Q[x] a PID? (Euclidean algorithm.) Explain it.
Give an example of a ring that is not a PID? (Z[sqrt{-5].) Why not?
(It's not even a UFD, as 6 factorises in two ways. We talked about
ideals and norms of rings of integers, in order to calculate the unit
group of Z[sqrt{-5].)
What are Sylow's theorems? Give an application. (I falsely claimed
that if G was a group of order n = mp with m < p, then G was abelian,
before correcting myself and stating rather that it was not a simple
group.)
Complex:
T: What is a Laurent series? (I started talking about where a Laurent
series converges, with some difficulty remembering the radii of
convergence.) How do you calculate the coefficients? (I differentiated
Cauchy's integral formula.)
T: Say f is holomorphic in the unit disc. How do you know its power
series converges inside the disc? (I looked at this for ages before
remembering to expand the denominator as a power series.)
F: Why does this work? (Estimate the tail, or use the dominated
convergence theorem.)
S: Say you have an entire function that is bounded. What can you say
about it? (Constant, by Liouville's theorem.) What if it's bounded by
a polynomial, say |z|^100? (Cauchy's integral formula shows that it
must be equal to c z^100 for some |c| = 1.)
S: Suppose now that you have a bounded function on a punctured
disc. What can you say? (Riemann's theorem on removable
singularities.) Proof? (Use a keyhole contour.) What if instead it may
not be bounded, but blows up like 1/sqrt{|z|}? (Can't happen - the
same method for removable singularities.) So what's the weakest
condition for a singularity at the origin to be removable? (Only need
o(1/|z|).)
S: When are two doubly-connected domains conformally equivalent? (I
stated that any doubly-connected domain is conformally equivalent to
an annulus.) Proof? (Don't know - it's not in Stein & Shakarchi!)
Really? (Yep, a notable omission. But I can prove that two annuli are
conformally equivalent iff the ratios of the radii are equal!) Please
do. (I gave a proof from Rudin that he hadn't seen before, by looking
at g(z) = 2(log(|f(z)|/r_2) - a log(|z|/r_1)) for a =
log(R_2/r_2)/log(R_1/r_1). Sarnak hadn't seen this before and thought
it was quite neat.)
S: Do you know a proof via Schwarz reflection? (Yes, and we went
through it.)
Real:
F: Define the Fourier transform. What is it useful for? (I couldn't
think of anything, so I said if you integrate by parts you can turn
differential operators in phase space into polynomials on frequency
space.)
F: What is the Poisson summation formula? Prove it. (I did.)
F: Why does the Fourier series of a smooth function convergs to the
function itself? (I started working through the proof but was quickly
stopped when he saw I knew how to do it.)
Sarnak decided he wanted to generalise this (uh-oh). How do you define
a (tempered) distribution? (Linear functional on Schwartz space.)
What's the topology? (No idea. Oops. They spent a while giving hints
as to what this should be, without success. Along the way I falsely
claimed that the space of smooth functions in the unit circle is a
Banach space, before digging myself out of that whole by quoting the
Stone-Weierstrass theorem.)
S: How do you define the Fourier transform of a tempered distribution?
What about in the case of the sum of Dirac delta functions, each
giving mass one at an integer? Is it a distribution? (Yes.) Now what
does Plancherel's theorem state about this sum applied to a Schwartz
function? (Gives back the Poisson summation formula.) Sarnak then told
me some rigidity results about this kind of thing.
Analytic number theory:
T: Define a Dirichlet character. What is a primitive character? (I
struggled to define it correctly. I probably should've mentioned the
word conductor at some point!) Why are Dirichlet characters useful?
(Orthogonality relations allow you to pick out elements of an
arithmetic progression.) What can you say about primes in arithmetic
progressions? (Stated Dirichlet's theorem.) Anything stronger?
(Dirichlet density is 1/phi(q).)
S: No, something else. (PNT in APs.) Which is stronger, that or the
Dirichlet density result? (The PNT version, of course.)
S: How do you show L(1,chi) doesn't vanish? (I explained why this is
true for complex characters, by multiplying together all the
L-functions of a fixed modulus.) What is this product of L-functions,
and why are the coefficients of the Dirichlet series nonnegative?
(Dedekind zeta function of a cyclotomic field.) Is this immediate?
(No, but I didn't know much about how to factorise Dedekind zeta
functions of abelian extensions into products of zeta functions and
Dirichlet L-functions. I said I could still prove that the
coefficients were nonnegative via their Euler products, but Sarnak was
unimpressed - he pointed out that the only recipe to cook up such
products of L-functions for which the resulting Dirichlet series has
nonnegative coefficients is via Dedekind zeta functions and their
like.)
S: What about when chi is real? (Multiply by zeta(s).) What is the
zeta function in this case? (Quadratic field. He made me write out
what it was.) How do you proceed from here? (Show that the Dirichlet
series has coefficients bounded below by 1 when n is a square, so the
series diverges at s = 1/2. Then Landau's lemma for the rest.) Sarnak
pointed out that this lemma was actually due to Pringsheim earlier,
but I countered with the fact that Pringsheim's version was only for
power series, not Dirichlet series, which Sarnak found amusing (and in
fact Vivanti published this result a year before Pringsheim).
S: What lower bound on L(1,chi) can you get from this (>> 1/sqrt{q},
though he didn't press for details.)
T: Do you know another way to show this? (I mentioned the analytic
class number formula, in full generality, though I didn't prove it.)
S: Can you get a better bound? (Landau-Siegel.) What's bad about this?
(Ineffective.) Why? (I went through the proof and discussed where the
ineffectivity arises. Once again, I had to mention that the product of
L-functions in this case is the Dedekind zeta function of a
biquadratic field.)
S: Let's talk about the circle method. (I started talking about three
primes, but then he changed tack.) Show that for every sufficiently
large N can be written as the sum of 20 cubes. (I talked about
Waring's problem.) How do you bound the minor arcs? (Hua's lemma and
Weyl's inequality.) What bounds do they give? How do you prove them?
(I mentioned Parseval for Hua's lemma and he was content. For Weyl, I
couldn't remember how the differencing method worked, so they talked
me through how to bound sum_{n < N} e(alpha n^2).) Why can you give a
good bound for an exponential sum where the polynomial inside is
linear? (He wanted me to say that it was a geometric series!)
They didn't ask about the major arcs at all.
Representation theory:
F: What are the irreducible representations of SO(n)? (I mentioned
harmonic polynomials.) Okay, let's assume that this representation is
irreducible. Does this give all representations? (I floundered for a
bit before mentioning the Weyl character formula, and that the theorem
of highest weights gives a way of indexing irreducible
representations.)
S: Let's step back for a second. Suppose that I just have some compact
topological group. How do I prove the existence of an irreducible
representation? (I started talking about the Peter-Weyl theorem, and
decomposing the left regular representation. This took a while, and
they were very quick to point out when a claim of mine was false
because I hadn't stated a necessary condition - e.g. why the
convolution operator I defined was actually compact and self-adjoint.)
S: Okay, all you've shown is that finite-dimensional representations
exist. How do you know that there are any finite-dimensional
irreducible ones? (Maschke's theorem - as soon as I said average over
the group, he was happy.)
S: You mentioned the Weyl character formula earlier on. What is it? (I
wrote it down. Sarnak didn't like that everything was in terms of Lie
algebras, so he made me define everything.) What IS a Lie algebra?
(This was embarrassing. I had a complete mental blank on how to define
it. After a few painful minutes, I mumbled something incomprehensible
about invariant vector fields at the identity, but did not stating the
crucial words ``tangent space''.)
S: How about we take a five minute break? (Fine with me! When Sarnak
and Fefferman left the room, Templier whispered the answer to me,
though thankfully by that point I had remembered the missing words!)
S: Back to Lie algebras. What are maximal tori? Cartan subalgebras?
(He never asked me prove that maximal tori are conjugate! I was very
surprised.) What are roots? Weights?
S: Let's get back to the original problem. Maybe make it a bit easier
- what are the irreducible representations of SO(3)? (I knew the
dimensions, the maximal torus, and the roots, but was stuggling to
work out the weights.)
S: How about some other group, of rank at least 2? (I talked about
SU(3): the maximal torus, Cartan subalgebra, simple and positive
roots.)
T: Can you draw the diagram associated to the roots? (I could not - I
forgot to practise this in my study for generals!)
They were unimpressed.
S: How about a simpler example still? (I looked at the representations
of SU(2) on the space of homogeneous polynomials, in particular the
action of the maximal torus.)
T: Is there a connection between SO(3) and SU(2)? (The latter is a
double cover of the former - I mentioned the adjoint representation
giving the cover.)
They decided to end here, perhaps to put me out of my misery. I left
the room and waited for what seemed like an eternity (but was more
likely three or four minutes) before Sarnak welcomed me back in and
told me that I had passed (somehow) but that there were some serious
things that I needed to improve on (which was painfully clear to me,
especially in the last 45 minutes). The exam lasted 2h45m.
Some general advice: there is a lot to study, but you will learn a lot
along the way. Enjoy it; you'll come out the other end much more
knowledgeable. Make sure you look at past exams, and try to actually
solve plenty of these questions yourself (I found this to be the most
productive way to study). Some examiners ask the same questions very
often, so make sure you know how to answer those ones in some
detail. In the weeks leading up to the exam, it's a good idea to test
your knowledge by doing mock generals (i.e. getting your friends to
ask you questions in front of a blackboard). This is especially useful
for more lengthy problems that you might be asked about, such as the
Riemann mapping theorem or Vinogradov's three primes theorem. For
representation theory of compact Lie groups in particular, it would be
worthwhile to completely memorise everything about SU(n), SO(2n),
Sp(n), and SO(2n+1) (i.e. their Lie algebras, maximal tori, Cartan
subalgebras, simple roots, positive roots, roots, Weyl chambers, Weyl
groups, Dynkin diagrams, and so on). I had most of this covered but
made the mistake of not knowing how to calculate the weights of these
groups, which my committee were not happy about.
Finally, don't worry about how screwed you might be. Everyone freezes
up and forgets things (in my case, some extremely obvious and simple
things). Your committee expect that and will give you some leeway, and
will try to help you dig yourself out any hole you might find yourself
in.
For what it's worth, here are the books that I used to study:
For complex analysis, Sarnak recommended Ahlfors, though I couldn't
get a hold of a copy so instead I studied Stein & Shakarchi (volume II
of their Princeton Lectures in Analysis). Its only faults are missing
details about conformal mappings for multiply-connected domains (which
Ahlfors does cover). Both don't really have anything about the
uniformisation theorem, which is something worth knowing (especially
if Sarnak is on your committee).
For real, I used the third volume of Stein & Shakarchi. It has pretty
much everything you need except some functional analysis (in
particular L^p spaces), which is covered at the beginning of the
fourth volume. Don't bother with the first volume - although it covers
Fourier analysis, it's all in terms of Riemann integration, which is
pretty silly.
For algebra, I used both Dummit and Foote's ``Abstract Algebra'' and
Michael Artin's ``Algebra'', though I can't say I was a huge fan of
either.
For number theory, I used Montgomery and Vaughan's ``Multiplicative
Number Theory I. Classical Theory'' and Davenport's ``Multiplicative
Number Theory'' for multiplicative number theory - both books are
written similarly (Montgomery revised the later editions of the
latter), with the latter being briefer and less comprehensive, though
the proofs were often neater. For additive number theory, I liked
Nathanson's ``Additive Number Theory'', though it covers a lot of
extra stuff and can be a bit lengthy. For the proof of the three
primes theorem, I actually preferred Montgomery's lecture notes (see
http://www-personal.umich.edu/~hlm/math775/handouts.html).
For representation theory, Sarnak recommended Adams' ``Lectures on Lie
Groups'', which I found pretty heavy going - I much preferred the Lie
algebra approach. Hall's ``Lie Groups, Lie Algebras, and
Representations'' is a good introduction but has too many proofs
ommitted. I mostly ended up using Sepanski's ``Compact Lie Groups''
and Fegan's ``Introduction to Compact Groups''. Note that if you want
to read about noncompact groups (e.g. anything about SL_n(R)) then
you're going to have to look elsewhere.