Ryan Peckner's Generals Committee: Sarnak (chair), Skinner, Nelson
Special Topics: Algebraic number theory, Functional analysis
May 12, 2011, 1:30 PM - 4:30 PM
REAL ANALYSIS
[N]: Suppose you have a sequence of functions on the unit interval
converging in L^1. What can you say about pointwise convergence?
Much to my humiliation I completely flubbed this one. He was even kind
enough to suggest thinking about a subsequence, but I was too
flustered to do anything but spew some nonsense about G_deltas. He
moved on quickly.
[N]: Consider the set of irrational numbers in the unit interval. What
can you say about closed subsets thereof?
I said that since the irrationals have measure 1, inner regularity of
Lebesgue measure implies that there are closed subsets with measure
arbitrarily close to 1.
[N]: Consider the Fourier transform, except with exp(-itx^2) in the
integral instead of the usual expression. What can you say about this
function (which he amusingly dubbed the "Furious Transform")?
I showed that it belongs to C_0(R) a la usual Riemann-Lebesgue, by
showing that the "Furious transform" of a Schwartz function is
Schwartz. This prompted Sarnak to have me replace x^2 in the exp with
an arbitrary smooth function g(x) whose first derivative is bounded
below near 0. I multiplied the integrand by g'(x)/g'(x) and integrated
by parts.
COMPLEX ANALYSIS
[Sa]: State the Riemann mapping theorem. What was Riemann's original
(flawed) proof?
I talked vaguely about Green's functions, harmonic conjugates, etc. I
hadn't studied them in detail but fortunately knew enough to squeak
by.
[Sa]: Suppose you have a closed semicircle in the plane, and an
analytic function on it which is bounded by 1 on the circular arc and
2 on the straight line below. Can you give a better bound than 2 for
points inside the region?
This led to a discussion of the Hadamard three circle
theorem. Ultimately I realized that I had to construct a harmonic
function on the semicircle which is constant on the arc and on the
line, but I had no idea how to do this. Sarnak told me that the answer
is the angle subtended by the point and the radial segments.
ALGEBRA
[Sk]: How can you tell when two matrices over a field are conjugate?
Routine discussion of rational canonical form, modules over a PID,
etc.
[Sa]: For what matrices B can you solve exp(A) = B?
[Sa]: Tell me about representations of finite groups.
I basically said everything I know (which isn't much): Maschke's
theorem (with proof), Artin-Wedderburn, representations of D_8.
ALGEBRAIC NUMBER THEORY
[Sk]: Consider the field Q(sqrt(5)). What can you tell me about the
behavior of 7 in the ring of integers?
I reduced the minimal polynomial of (1 + sqrt(5))/2 (mod 7), figured
out there were no zeros, hence 7 is inert.
[Sa]: You got lucky because 7 is small. What if we take a much larger
prime?
I screwed up a bit in stating quadratic reciprocity and was
corrected. Sarnak asked me to prove it using anything I wanted. I gave
the cyclotomic fields proof, which Sarnak liked.
[Sk]: You said the Galois group (of Q[zeta_q], q a prime) is
(Z/qZ)*. What element of this group does the Frobenius of p define (p
another prime)?
[Sk]: A famous theorem of Dirichlet...
He was of course asking about the one on arithmetic
progressions. Sarnak then asked if I knew Tchebotarev density. I
stated it and got about three-quarters of the way through the class
field theory proof before they stopped me.
[Sa]: Give a one-line proof that infinitely many primes split
completely in an arbitrary extension of number fields.
The Dedekind zeta function has a pole at 1.
FUNCTIONAL ANALYSIS
[N]: Suppose we have a bounded operator on a Banach space. What is its
resolvent set? Why is it unbounded? Why isn't it the entire plane?
[N]: What kind of continuity conditions should one impose on a
one-parameter group of unitary operators? What else can you say about
it?
I said that strong and weak continuity are equivalent, and stated
Stone's theorem. This led to a discussion of the spectral theorem for
unbounded self-adjoint operators.
[N]: What property do all unbounded self-adjoint operators have?
(They're closed.) What does essentially self-adjoint mean? Give an
example of a symmetric operator that isn't essentially self-adjoint.
I gave id/dx acting on C_0^1(R_>0) inside L^2(R_>0).
[N]: Take the Laplacian acting on L^2(R^n). What multiplication
operator is it unitarily equivalent to?
It took me an embarassing amount of time to realize I should use the
Fourier transform to get (x_1)^2 + (x_2)^2 + ... + (x_n)^2.
[Sa]: Let's talk about unitary representations of locally compact
groups.
I started talking about Pontryagin duality but was cut off.
[Sa]: Take a compact Lie group. How do you exhibit an irreducible
finite-dimensional representation?
I said Peter-Weyl but he wasn't satisfied.
[Sa]: No, I want you to show me an actual representation.
I started with the regular representation of G on L^2(G). They agreed
that it was completely reducible but wanted to know why the summands
were finite-dimensional. I said that they're the eigenspaces of a
compact operator, which I wrote down (convolution). They asked me what
the name for such operators is in general. I said "integral operator"
and they weren't amused. After thirty excruciating seconds of not
knowing what they wanted I finally blurted out "Hilbert-Schmidt", much
to their relief (and mine).
I was sent into the hallway to await my fate. A minute later Sarnak
came out and congratulated me.
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Some words of advice:
You will probably freak out on numerous occasions while preparing for
this exam. That is natural. However, try to bear in mind that the vast
majority of students who take generals pass them, and that the
professors truly are on your side in this. No one wants or expects you
to fail.
Don't obsess over details of proofs of minor theorems and lemmas. Your
committee won't expect you to know everything (not even close). That
said, you should have a good understanding of the important elements
of the proofs of the major theorems in your special topics. The big
picture matters much more than technicalities.
Finally, you will probably find yourself feeling much better during
the exam than you had for several weeks (or months) preceding it. My
committee was very friendly and helpful, and didn't seem to mind even
when I made some really ghastly mistakes. Relax. You're going to be
fine.