Paul Pollack
October 18, 2004
Committee: Sarnak (chair), Fefferman, Ellenberg
Special Topics: Algebraic Number Theory, Analytic Number Theory
BUFFY: It was just too much to deal with. It was like nothing made
sense anymore. The things that I thought I understood were gone. I just
felt... so alone.
GILES: Was that the math or the verbal?
BUFFY: Mostly the math.
-- Buffy Summers on her SATs
Sarnak explained that they usually start with the core topics and I was
asked which of those I wanted to begin with. I said algebra.
[NOTES: I have listed the core topics and the primary questioner(s) in that
topic. But there were frequent interjections from others. I have tried to
list the questions in the order they were asked but have not always succeeded.]
ALGEBRA [Ellenberg]
What are the conjugacy classes in GL_2(C)? [After overcoming some initial
terror, I managed to squeak out the words "Jordan canonical form." But they
wanted details, which I botched a bit.]
One of my mistakes led to a discussion of the minimal polynomial of a linear
transformation. I gave the definition then said it could be described by
decomposing the vector space V as a direct sum of C[x] modules C[x]/(a_i),
with each a_i dividing a_{i+1}, and then taking the largest of the a's.
State the Fundamental Theorem of Finitely Generated Modules over a PID.
[M = R^r + direct sum of R/(a_i).] Suppose D is a domain; does the conclusion
of the theorem imply that D is a PID? I wasn't quite sure where to go with
this, so we started with Z[x] and then showed that any nonprincipal ideal was
a Z[x]-module that gave a counterexample to the "theorem" there. I had to
define "torsion module" and "torsion free" somewhere in here.
Which (complex) n by n matrices are exponentials? [All invertible matrices;
I had seen this before on S.J. Miller's generals account.] For what other
fields does this proof work? [This puzzled me and I never really gave
a satisfactory answer.]
Several questions on GL_2(F_q). How many elements does this have? How would
you construct representations? [I didn't really know.] What can you say about
the one dimensional representations? We [read:I] were [was] stuck here for
a while -- there was a discussion of simplicity of related groups (where I
had to be told the relevant facts), and somewhere along the way I was asked
to characterize (ha) the group characters of F_q* and to prove F_q* was cyclic.
Sarnak asked me if I had done many examples when I was studying representation
theory and I confessed I hadn't. (see ADVICE below)
REAL [Fefferman]
Write down the Poisson summation formula (in one variable). How do you prove it?
In the sketch of the proof I mentioned that e^{-pi*x^2} was its own Fourier
transform. Sarnak asked if there were a lot of functions with this property
or whether this was some sort of miracle. I didn't know. I was prompted to
consider what happens when you apply the Fourier transfer twice, and then
further prompted to think about the Fourier transform having order 4 and
then to take an average (so to consider f + f-hat + f-hat-hat + f-hat-hat-hat).
What can you say about the Fourier transform of e^{-|x|^3}? I said that
e^{-|x|^3} was an L^1 function and stated the standard properties of the
Fourier transform of any L^1 function. But they wanted more, e.g., is in
the Schwarz class? Why not? Along the way they asked why is e^{-|x|^4} easier
[it is in the Schwarz class, and the Fourier transform maps the Schwarz class
into itself].
Can a function and its Fourier transform both have compact support?
COMPLEX [Sarnak, Fefferman]
Take the Fourier transform of e^{-|x|^3} that you wrote down before. Now
consider it as a function of a complex variable. Is it an entire function? [Yes.]
What is the "order" of an entire function? What can you say about the order
of the entire function above? I had serious trouble estimating the integral
and received substantial hints. But eventually we got that its order was <= 3/2.
Sarnak then asked what the zeros were, then explained he was just kidding.
What is special about entire functions of finite order? [Weierstass factorization;
they can be written as e^(poly) * product over zeros.] What is the degree
of poly in the example we just worked out? [At most 1.]
As a serious footnote to the above joke, Sarnak asked whether this entire
function *had* any zeros. Yes, otherwise it would be e^poly, and in fact
it must have infinitely many else it's e^(poly1) * poly2, and "surely"
we could rule out those cases. Can you use a similar argument to prove
the Riemann zeta function has infinitely many zeros?
Draw two annuli. Are they always conformally equivalent? [No. The ratio
R_2/R_1 must equal r_2/r_1.] Prove it.
Can you map the doubly connected region you drew in the course of the previous
proof onto some annulus? I confessed ignorance and we moved on. Or more
accurately we took a 5 minute break before starting the special topics.
---
I was asked whether I wanted to do ANALYTIC or ALGEBRAIC number theory first.
I chose analytic.
ANALYTIC NUMBER THEORY [Sarnak]
Write down the sum Sum(e(alpha*n^2), |n| <= N). How would you estimate it?
What is the end result? [i.e., what does Weyl's inequality end up giving
in this case.] Why are these sorts of estimates useful? [Application to
Waring's problem.]
Three Primes Theorem: What is the analogous exponential sum you need to
estimate in this case. [f(theta) = sum(e(theta*p) log p, p<=n).] How is
this carried out? [Reexpress the von-Mangoldt function Lambda(m) using
"Vaughan's identity," multiply by e(theta*m) and sum.] Where do you need
a bilinear forms estimate?
What information about primes do you need for estimating f(theta)? I said
you use the Siegel-Walfisz form of the PNT for arithmetic progressions
and was asked to state that theorem.
Why doesn't the proof of three primes work for two primes? [I described
the role Parseval plays in the three primes proof. Sarnak pointed out
that for the two prime case a natural approach would be to try to estimate
the L^1 norm of f. I didn't know how to attack such an estimate, but wasn't
expected to.]
Somewhere in here I had discussed rational approximations a/q of a real
number alpha to within 1/q^2. Jordan asked whether the exponent 2 could
be improved in general. [No; if alpha=sqrt(2) or any other quadratic
irrational we always have |alpha-a/q| >= c(alpha)/q^2.] Can you connect
this with the continued fraction expansion somehow?
Why do the Dirichlet L-functions have no zero on Re(s) = 1? [I said multiply
them all together, get essentially the Dedekind zeta function of Q(zeta_m),
which has a pole at 1.] Sarnak wanted to know what lower bound, if any,
you could coax out of this proof but I didn't know.
State the class number formula for an imaginary quadratic field. What lower
bound on L(1,chi) does this give you? What is Siegel's theorem in this
subject? [L(1,chi) >>_{\epsilon} q^{-epsilon} for any real primitive
character chi mod q.] What is HORRIBLE about this? [The ineffectivity.]
Where does ineffectivity enter the proof? Do you know the statements of
any other results in number theory that are ineffective? I wasn't sure
so Ellenberg asked me to state "Faltings' Theorem."
ALGEBRAIC NUMBER THEORY [Ellenberg]
It seems Jordan had prepared some questions in advance centering around Q(zeta_7).
What is the ring of integers of this field? How would you prove it?
Which rational primes ramify? (First, what does it mean that a prime ramifies?)
How can you determine this? [Discriminant]
Ok, 7 is the only ramified prime. Can you be more specific about how it factors.
[Totally ramifies and is a power of the principal ideal (1-zeta_7).]
How do the primes other than 7 factor (i.e., give e, f and g explicitly in this
case). [e=1, f= order of p mod 7, n= 6/f.]
What is the structure of the units? How do you produce these fundamental units?
[an application of Minkowski's theorem.]
Can you name any units in Z[zeta_7] other than the roots of unity?
[I said (1-zeta^r)/(1-zeta^s) with r and s coprime to 7.]
What is the Chebotareff density theorem? I stated it and was told to define
the Artin symbol and to define what I meant by "density" in the statement.
What does Chebotareff say in the case of Q(zeta_7)? [Primes are uniformly
distributed with respect to Dirichlet density in congruence classes mod 7.]
Is there a prescribed value of the Artin symbol which guarantees that a prime
splits completely? [yes] What if "splits completely" is replaced by "is inert"? [no]
Are there imaginary quadratic fields with class number 1? [Yes, Q(i).]
Ok, are there infinitely many? [No, the last is -163. The finiteness
of the list comes from Siegel's theorem and the class number formula.]
What is the regulator of a number field? In the quadratic field case
what is the connection between the regulator and the elementary theory
of the Pell equation? Why does a "large" fundamental unit suggest a
"small" class number and vice versa; I was asked to quantify this
[using the class number formula and Siegel's theorem as above].
somewhere in here was this exchange:
[Sarnak]: Do you know what an Artin L-series is?
[Me]: No.
They did not seem to mind my ignorance here.
---
The opening Buffy quote notwithstanding, the professors were actually
quite kind and were willing to assist me whenever I was having trouble,
which was often (only some of which is described above; e.g., I left out
the part where I was asked to write down the identity matrix and wrote
down [[1 0][1 1]]). There were no harsh words save for amusing critiques
of my notation (e.g., using gamma as an integration variable or writing
Z_p for the integers mod p...) and my strange looking thetas. The exam
lasted somewhere between 2 1/2 and 3 hours (?). I was surprised there were
no questions about class field theory proper, but perhaps there would
have been had the exam gone longer; Sarnak was eager to end it after
2 1/2 hours and asked Jordan to try to wrap things up.
IMPORTANT ADVICE: do exercises! I failed to follow this advice when prepping
for my core topics and think that I would have had a more pleasant time if
I had taken it.
I would like to conclude this account by thanking my friends and family
for their support during the panicky weeks prior to generals.
"I'm talking about mathematics--hard, brutal, extreme. I'm talking about
pushing your mind beyond the limits to understand what no one else can
because they're afraid to risk it all, to lose their freaking worthless
minds in the push to know." -- James S. Harris