Ruixiang Zhang's generals (Jan 24, 2013, 1:00 PM - 4:00 PM)
Special Topics: Harmonic analysis, Analytic Number Theory
Committee: P. Sarnak (chair), C. Fefferman, V. Lie
There might be more questions but they were likely to be standard ones.
Questions were asked by:
[S] - Sarnak
[F] -Fefferman
[L] - Lie
If I don't label a question in this way, it means I don't remember who asked
it.
Several days before the exam we agreed to move the starting time from 1:30 PM
to 1:00 PM,since there would be a seminar starting at 4:30 PM.
They first asked what order I would like. I said I would like to do algebra,
which I was most worried about, first. Sarnak sad that I shouldn't do that and
asked Fefferman to start the Real part.
Real Analysis[Part 1]
[F]Define the Fourier transform.
(I defined the Fourier transform of an L1 function and was interrupted when I
wanted to extend the definition)
[F] What is the Fourier transform of an L1 function? Prove it.
(A continuous function that vanishes at infinity)
[F]What else can you say about it?
Is the Fourier transform surjective from L1 to C_0 (space of continuous
functions that vanish at infinity)?
(I blanked with the first question and was asked the second by someone I don't
remember. I had a hard time with the second one. Intuitively the answer should
definitely be no but I didnot know a proof. First I tried to modify the
example of the Fourier transform of p.v. 1/x but failed. I defined the Fourier
transform on the tempered distributions and computed the Fourier transform of
p.v. 1/xand talked a bit about the Hilbert transform somewhere and was asked
to compute standard things like the Fourier transform of 1/(1+x^2). But I
could not use this function to give a counterexample either. After I abandoned
this idea Sarnak suggested I do some functional analysis. I mentioned the open
mapping and said that it suffices to show the inverse is unbounded to obtain a
contradiction. But again I blanked before they let me off the hook. Later I
realized that by superposition of wave packets (just like what I did with the
Hausdorff - Young below) one could easily show that the inverse of this map is
unbounded. I also learned that on other books like Fourier Analysis on
Euclidean Spaces by Stein there are other solutions.)
[S] What is a tempered distribution? Is e^x sin (e^x) a tempered distribution?
(Yes. This was a question somewhere in the above discussion. I first said that
if there was no sin (e^x) then it's not. Then I found that e^x sin (e^x) is
just the derivative of - cos (e^x)!)
[L] What is the Hausdorff - Young inequality?
(The Fourier transform is bounded from Lp to Lp' if 1<=p<=2.)
[L]Can it hold for p>2?
(I said it couldn't but blanked for the reason. They then made me realize that
a superposition of space and frequency translations of a fixed wave packet
would work.)
We then moved to Algebra and Sarnak said that perhaps we should come back
later for some "standard" questions like measurability, but I never got such
questions! They did come back to Real later but the topic was again about the
Fourier transform.
Algebra
Sarnak asked what I learned and started with linear algebra.
[S]Talk about canonical forms of square matrices over C.
(The Jordan canonical form.)
[S]For which square matrix B does the equation e^A = B have a solution?
(Nondegenerate ones. I used the Jordan canonical form and the power series of
log.)
[S]What canonical forms do we have on other fields?
(Rational canonical form. I also mentioned that these canonical forms are
consequences of a more general theorem on structure of finitely generated
modules over a PID.)
What is the PID here and what is the module?
(Standard answer.)
[S]Can you give an example of a UFD that isn't a PID?
(k[x, y]. Sarnak then asked what makes k[x] a PID. I explained the algorithm
but forgot the name. He said 'long division'!)
[S]What is the structure of finitely generated Abelian groups?
(Stated it.)
What can the order of a finite field be?
(p^n since it is a vector space over some Fp.)
[S]What is the multiplication group of Fp*? Why?
(Cyclic. I said we could use the structure theorem of finite generated Abelian
groups and did the proof for F5* as an example.)
[S]What is the algebraic closure of Fp?
(The union of F_(p^n).)
[S]You mentioned Galois extensions. What is the Galois group of F_(p^n)/Fp?
(Cyclic group of order n. I explained the Frobenius automorphism.)
[S]Do you know representation theory?
(I said I mainly know basic theory of finite groups and compact groups but did
not prepare much on the latter.Sarnak said we would only do finite groups.)
[S]Talk about complete reducibility.
(Maschke's theorem.)
[S]Talk about more properties about representation theory (on complex vector
spaces) of finite groups.
(I mentioned Schur's lemma. Then said that if there are m inequivalent
irreducible representations in total and they have dimensions d1 , ... , dm
respectively, then the sum of dj^2 is |G| and m = # of conjugacy classes.)
[S] Explain the latter fact.
(Standard argument.)
Talk about representations of D4.
(I draweda picture of a square and showed all irreducible representations.)
[S] Do 1-dimension representations (of any finite group) factor through some
standard subgroups?
(It will be trivial on the commutator subgroup.)
Real Analysis [Part 2]
[L]Do you know the uncertainty principle?
(I explained it in three different ways. First, the Fourier transform of a
compact supported function cannot have compact support. Second I stated a
lower bound of the product of two integrals concerning f and its Fourier
transform (which I think should be the "standard" inequality when we talk
about the uncertainty principle). Finally I explained if a function is
concentrated in a cube then its Fourier transform is "mainly" concentrated in
its dual cube, which is a useful heuristic I learned in the subject of
restriction.)
[L&F] If a function f on R satisfies that the integrals of |x^k f/k!| over R
are uniformly bounded independent of k, can its Fourier transform have compact
support?
(No.The initial question did not have a k! on the denominator.Fefferman
claimed that such f must have compact support and added the k!.After a long
time I realized that in this problem we also have analyticity for the Fourier
transform.)
Complex Analysis
[S] Some question I did not remember about certain analytic functions on the
unit disk. I used the maximum principle.
[S] On an annulus with inner radius 1 and outer radius 2, what can you say
about an continuous function that is analytic inside and s.t. |f(z)| <= 1 when
|z| = 1 and |f(z)| <= 2 when |z| = 2?
(Standard argument using the corresponding harmonic measure.)
[S] What is the harmonic function that is 1 on the inner circle and is 2 on
the outer circle?
(I said an exponential function maps a strip into it and on the strip it is
the function Re(z). Sarnak forbade me to do this and asked me to write down
the function directly. An embarrassing moment came because I was so nervous
and blanked! I completely forgot that it is the fundamental solution. Panicked,
I solved the corresponding ODE and realized that it should be a linear
function of log|z|. I laughed because I mentioned the exp at the beginning but
forgot the log!)
[S] What about the semi-disk? Assume I am 1 on the diameter and 2 on the half-
circle.
(Some linear function of arg((z+1)/(z-1)).)
[S] Talk about the Riemann mapping theorem. What can you do if the domain is
doubly connected?
(Map it to an annulus.)
[S] When can two annuli be conformal equivalent? Prove it.
(Iff they are similar. I used the reflection principle.)
[S] What can you do for multi-connected domains? Can you map them conformally
into circles with circular holes missing?
(I said that I can map them into annuli with slits missing but have no idea
for the second question. Sarnak said that it was a much deeper theorem shown
by Koebe in the first rigorous proof of Riemann mapping theorem. He then said
"sadly, no textbook mentions this".)
We took a break and started the special topics.
Harmonic Analysis
They first asked what I learned. I said maximal functions, singular integrals,
H^1 and BMO, Littlewood – Paley, pseudo differential operators, oscillatory
integrals, restriction and Kakeya.
[L] Talk about Littlewood – Paley theory and talk about the proof of some key
theorems.
(I talked about a vector – valued singular integral inequality and talked
about the Hormander multiplier theorem. They then wanted me to come up with a
method to prove the main theorem in L – P theory without using the vector –
valued singular integral. I didn’t know what to do and they gave me a hint:
consider the sum of a_jS_j f. I came up with the random argument (Khinchin’s
inequality). But again I didn’t know what to do next. Fefferman said that
since we didn’t do standard things such as singular integrals, we could do
those first.)
[F] Talk about singular integrals.
(I talked about Calderon – Zygmund theory. Then Fefferman asked me what could
I say about the kernel associated with the operator \sum a_jS_j. This time I
realized that it is a singular integral!)
[S] Talk about the decay rate of the Fourier transform of the standard measure
of the unit circle on the plane.
(Decay like |?|^{-1/2}, using the stationary phase.)
[S] What about an ellipse?
(I said I can do a linear transform to transform it to the circle and they
laughed.)
[S] So what is the common property that makes their Fourier transform decay?
(Nonzero curvature.)
Analytic Number Theory
[S] Prove the zeta function doesn’t vanish on the line Re(s)=1.
(Standard argument.I also gave a zero free region and was asked to explain it.)
[S] Explain the proof of Vinogradov’s theorem. Why can’t you prove Goldbach in
this way?
(Because the method is to estimate the integral of S(\alpha)^3 and we know the
L^2 norm and the L^infty norm on the minor arc well. We could not prove
Goldbach by estimatingS(\alpha)^2 in this way.)
[S] Give a lower bound of ||\sum_{n=1}^N \Lambda(n)e^{\alpha n}||_{L^1 (T)}.
(Use the L^4 norm.)
[S] Prove that ||\sum_{n=1}^N e^{\alpha n^2}||_{L^1 (T)} has a lower bound
that is almost N^{1/2}.
(By noticing the L^4 norm is almost the same size as the L^2 norm.)
The exam took us exactly three hours. They were really nice and gave me many
hints.
I want to thank all my friends who helped me prepare for my generals.
Good luck for everyone!