Harald Helfgott's generals
May 19, 1999
Peter Sarnak (chairing), Zoltan Szabo, Arun Ram
Sosabbak itt a konnyek
S a fajdalmak is masok...
Complex Analysis
Riemann mapping theorem: both the proof with Koebe, Arzela-Ascoli & Co.
and Riemann's original proof (= with Dirichlet's problem; Ahlfors does this
more generally, for annuli); Green's functions.
A sequence of analytic functions on a compact set converges in L^2 -->
uniform convergence
(since I did not answer this immediately, they asked me whether it was true
for real functions (of course not!), then I used an annulus and Cauchy.)
Annuli; how many parameters? Why are two annuli with different ratios
not conformally equivalent?
Automorphisms of the unit disk.
Sarnak (with a murderous glance): No, not in terms of generators! As a
classical group!
Conformal mapping between upper half plane and unit disc.
Schwarz, Rouche...
Real:
Talk about Fourier analysis. What can you say about the Fourier transform
of something in L^1? (=where does it lie?) What about L^2? Schwarz? Isometry.
Poisson summation.
Cantor function.
Radon-Nikodym. what's a singular measure. What's an absolutely continuous
measure. Radon-Nikodym derivative: what can you say about it?
Give something that is singular with respect to the usual measure. I gave
delta, then the one from the Cantor set. Cantor function: derivative a.e zero.
What is the Hausdorff dimension of the Cantor set.
Algebra:
Jordan-Holder (finite groups). Give the proof in detail.
Structure theorem for modules/PID. Jordan, rational canonical forms.
What's the connection between the structure theorem for modules/PID and
conjugacy classes in GL(n,F)?
Structure of multiplicative group of a finite field.
Minimal polynomial and characteristic polynomial. Cayley-Hamilton.
Some commutative algebra.
Analytic # theory
Statement of Weyl's Inequality
Hardy-Littlewood method. How do you apply it to a generalization of Waring?
They made me work everything out for certain specific coefficients.
What is the singular series? How do you reduce it to the p-adic case? Why
do you need to reduce it? (To show that no term of the infinite product is
zero.) When is it not just necessary but sufficient? (Minkowski-Hasse.)
Hensel's lemma in several variables.
Consider this equation. (Much like Waring, but I do not remember exactly what;
it seems to me it was x_1^k+...+x_20^k plus a couple of quadratic terms with
and without non-1 coefficients.)
Take a quick look at it and tell us how many equations it should have. What
is the leading coefficient?
Prove that when you have a real quadratic field extension there is a nontrivial
unit.
Can you give a bound for the size of the smallest generator of Z/pZ?
Birch. I would have liked to give the proof, but they denied me the pleasure.
Extend zeta function. (To the right half of the plane? No, to the whole plane.
I sketched Davenport's proof, and was about to give the details, but Sarnak
said he knew a nicer proof, and we went on to the next equation.)
Three primes. Major, minor arcs. What do you need for each? Vinogradov's sieve:
sum over primes -> sum over sums over arithmetic progressions, get rid
of the short progressions.
(They would have liked me to work out the minor arcs in full. Vaughan's
treatment (divide into four pieces) is short but full of nice tricks which
are hard to remember. Thus I could only sketch it.)
Where in three primes do you need the large sieve? Uniformity of error term
in # of primes over arithmetic progressions for modulus <= (log N)^B.
Polya-Vinogradov. Bombieri. Siegel's theorem.
Bound \sum_{m=1}^N e^{nm\sqrt{2}} . No, do all of it: how close can \sqrt{2}
be to a/q? Sarnak did not much like my use of Roth's theorem, and asked me
to prove it (rhetorical question). Multiply \sqrt{2}-a/q by something.
Lie groups.
How do you go from representations of Lie groups to representations of
Lie algebras. Define adjoint. Show that any two maximal tori are conjugate.
Prove Brouwer's fixed point theorem (in all dimensions). Szabo seemed pleased
to see I had some idea of what the letter "H" means.
Tell me all you know about representations. (irreducibility... what are
weights....give the definition of highest weight.) Ram did not like my
definition of highest weight based on the geometry of the Weyl diagram.
(I gave a definition from which existence does not immediately follow;
by this point I was very tired, and could not give him the Lie-algebra-based
definition he wanted. I had read little besides Adams's lectures, where
algebras never appear. Not making sure that one is at ease with Lie algebras
is a very serious mistake if one has an algebraist on one's committee
(as nice a person as he may be).)
Weyl character formula. State all terms you have used. No, we do not want
the proof. Give some examples.
SU(2) and SO(3) so what can you say? SO(3) is not simply connected and
so SU(2) is the covering group. Prove that SO(3) is not simply connected.
The exam took 3 1/4 hours.
Notes:
(1) I had been told it was important to work out examples in preparation
for the exam. I had taken this to mean merely that I should be familiar with
certain useful pathologies (Cantor set, etc.). I was sadly mistaken. One
should take care to work out many of the exercises in the standard books,
especially the trivial ones (interesting problems can wait -- or at least
it is a lesser shame not to have solved them). Examples follow
from general statements in written exams, but not in oral exams,
or at least not before some noticeable delay and embarrassment.
Classical groups are especially important. It would be a good idea to work
out every single possible statement on Lie groups for them. One should store
all their supposedly obvious properties (incl. typical uses: automorphisms of
domains, quaternions inducing rotations & the sort) on the surface of the
brain cortex. After three hours of examination the interior will be gone.
There are also some examples which are considered part of basic literacy
(e.g. apply structure theorem for modules over PIDs to conjugacy classes
of matrices); one can find out about them by computing the intersection of
all examples in all books (or taking a course, or having common sense).
(2) You will not be allowed to present the proofs you like best.
(3) A tip for the algebra section: if you lisp, it is a good idea to
distinguish "Z" and "C" by adopting an approximation to the British name of
the former.