Djordje Milicevic - General Examination
Committee: Andrew Wiles (Chair), Peter Sarnak, Charles Fefferman
Advanced Topics: Analytic Number Theory, Algebraic Number Theory
Coordinates: 602, May 20, 2002. Exam lasted 2 hours and 20 minutes
(1.30-3.50).
m: I'm yours.
S: Why don't we start with real?
F: Write Poisson summation formula. In dimension 1 of course. Prove it.
Why does the Fourier series of a nice function converge to it? (yes, they
ask you to prove tautologies sometimes. and these can be puzzling. I first
proposed orthogonal complement zero but was unable to complete this
triviality. Then I said OK let's approximate interval, which is to say
that sum(sin(nx)/n)=sgn(x), then I turned this into an integral, used R-L
lemma, and would you believe I was lost once again. Finally made it.)
m: I'm already humiliated enough.
F: Everyone can be humiliated, it's just that it is you at the board (how
comforting). Well, I'm satisfied with the real. (Sure he was).
S: Let's move to complex. Why don't you write down the general Poisson
summation. "How many lattices are there in R^2?" (He wanted the
fundamental region for SL_2(Z). Notice how nicely focused the questions
tend to be.) Now generalize to n dimensions. To help me, he asked: "How
would you integrate over lattices?" We did this. Onward: Riemann mapping
theorem (he actually preferred my expert hand waving plus the argument
outline to the formal details). What group do the automorphisms of the
unit disk form? Hint: I am asking you the question. (Aha, SL_2(R)).
Conformal equivalence of annuli. Conformal equivalence of simply connected
compact Riemann surfaces. (I added the Abel-Jacobi theorem. I was then
informed that 'Abel' should be pronounced as in original, whereas
'Abelian' may be Americanized. Curiously enough, they didn't protest my
Americanization of 'Jacobi'). Zero-free entire functions of finite order
(=e^poly-s. I also mentioned order/genus theorem.)
S (to W): Hey, you're supposed to be the chair. Why don't you ask him some
algebra?
W: Multiplicative group of a finite field is cyclic. Compute the Galois
group of x^5-2. Brauer theorem.
S: Describe and define induction. Frobenius reciprocity. How do you
compute dimensions of irreducible representations in induction. Maschke
theorem. (S insisted that I prove it, which was a futile pain as I hadn't
reviewed group representations. Embarrassment again.) [It's possible W
asked some more algebra, but my memory goes only about this far.]
They offered to take a break now, which I declined, so we forced forward
to Algebraic.
W: Describe units in quadratic fields. How do you compute fundamental
units (cont'd fractions). How do you solve x^2+7=y^3? (I said first I'd
like to check that Q(sqrt(-7)) is PID, but he threatened to change 7 to a
large number, so we agreed to temporarily ignore pid-ness and be happy
with class number (=1!) being prime to 3.) Finiteness of class group +
Minkowski bound. Hilbert class field(s) (I described the relation between
HFs offhand, to his ambivalence). Why does every ground ideal become
principal (Ver:G/G'->G'/G'' is trivial. Of course they don't expect you to
actually prove this group-theoretic beast.) Why does every principal prime
decompose completely. Class field tower problem. Ray class fields.
Kronecker-Weber. Artin map (I described the global reciprocity with this.)
When can you make it explicit? (I offered Lubin-Tate theory for local
extensions, but all he wanted was-) Describe it explicitly in cyclotomic
extensions.
S: Let's move continously. (to Analytic.)
m: We should move to Professor Fefferman (who was sitting in the middle)
if we want to move continuously. (My remark was graciously ignored. In
what follows the order of questions is probably not as it was in the real
life, where it started with something number-fieldish. Later he changed to
core analytic by saying "now let's go for the real thing".)
S: Zeta not zero on the sigma=1 line (I noted that a modified argument
works for Dirichlet L f-ns.), oh well does it work for Dedekind zeta
(clearly yes), la Valle-Poussin and Vinogradov's mean value method
zero-free regions. L(1,chi)<>0. (I gave an elementary argument from
Murty's book, then offered zeta of cyclotomic field and class number
formula. He liked the first two -- ) [Oh, is that formula for zeta of
cyclotomic extension really automatic? He knew it isn't so we did that as
well.] ( -- and noted that the first one actually gives 1/sqrt(q) bound,
but nevertheless opted for the third. I gave the general class no
formula.) Give an explicit lower bound from this (log(q)/sqrt(q)). Improve
- aka Siegel theorem with proof. Why are coefficients positive (this is
zeta of a biquadratic field; or elementarily), discuss ineffectivity. Why
don't you finally define Artin L-series. Then talk a while about them,
zeta_L/zeta_K formula and such. Would you expect Dedekind zeta to have
multiple zeroes? (I said yes, given that Artin's have zeroes at all, which
provoked some amusement and-) Now prove that Dedekind zeta has non-trivial
zeroes at all. (Now I was amused.) Hint: is it of finite order? (Then,
after quickly dismissing any thought of partial summation, I said sure as
each completed partial zeta is the Mellin transform of an integral of the
corresponding theta over a compact fundamental domain in norm-one-surface.
Then it can't be zero-free for it would be e^poly). How could you force a
double zero at 1/2? (S then hinted at how to do this.) Why doesn't three
primes work for two primes.
Then S asked the final amusement question about L^1 norm of an exponential
sum over primes actually being large, which clearly remained unanswered,
as was the expectation. At this point three out of four were happy.
m: Could we perhaps do some more real?
S (with smile): Don't worry. If we decide to fail you because of the real,
we'll first ask you some more of it.
Then I was asked to leave the room. A minute later they congratulated me.
In the end they were very satisfied, or at least so did they act. Despite
some obvious pitfalls my performance was mostly even up to my
satisfaction. Committee was generally nice, and Wiles even seemed to have
prepared his questions in advance.
The following is a short, manifestly non-comprehensive list of books I
actually enjoyed reading during the pre-exam hell, most of which
generally, and undeservedly, don't seem to be in wider use at Fine.
Real - Folland for general stuff, Dym&McKean for Fourier analysis.
Algebra - Bell&Alperin (some of it is obviously fatally obscure though)
Algebraic NT - Neukirch's Algebraische Zahlentheorie (there is an English
translation). Most definitely worthwhile reading, give every word whatever
time it takes.
Analytic NT - Murty's introductory problem book for fun. Vaughan's book on
Hardy-Littlewood method. Do the exercises in the latter.
I am eternally thankful to Akshay, Andy and Harald, for careful
realization of the wonderful and most useful mock exam, and praising me
when comforts were actually deserved. Also to other supportive few.
Some advice is in place to the poor yet-have-to-pass colleagues.
1. Do not overstudy for the elementary topics. You'll end up forgetting
all the nonsense (cf. above). 2. Do not ignore this database. 3. You have
all your life to impress the mathematical community; generals aren't
exactly the best opportunity. Don't even try. And keep this rule above
your work desk. 4. Take a mock exam. 5. Leave absolutely nothing for the
last few days. Remember, "last 10% of the work take another 90% of the
time". 6. Finally, generals, just as math anyway, are not about knowing
everything. Relax.