Yi Ni
May 4th, 2004
Committee: David Gabai(chair), Andrei Okounkov, Elias Stein
Length: one hour and 35 minutes 
Special topics: Algebraic Topology, Functional Analysis


Okounkov was 5 minutes late, so Stein asked me some questions about
myself before Okounkov came. I guess he wanted to make me relax.
Dave asked what topic I would like to begin with, I said real analysis. 


Real and Functional Analysis (Stein, a few by Okounkov)

Cantor set? What special properties does it have?
 - Turns out that he wanted the Hausdorff dimension of Cantor set.
I described the meaning of such a "fractional dimension", but didn't
know the definition of Hausdorff dimension.

What special function is related to this set? How to define it?
What property does it have?
 - I said that it is constant almost everywhere, and looks like
an infinite stair. He seemed not satisfied, so he asked the more explicit

Suppose we have an increasing function, whose derivative is 0 everywhere,
must it be constant? 

What's the correct version of this theorem?

Definition of absolutely continuous function?

Lipschitz condition? Is it absolutely continuous? What if
|f(x)-f(y)|<C|x-y|^{\alpha} for 0<\alpha<1?
 - I said no, then Stein wanted an example. I wrote x^{\alpha} on the
board and erased it quickly, but Stein asked me whether x^{1/2} was A.C.
I said no, (actually I was thinking that it was not Lipschitz,) then
Stein helped me to correct it. I didn't know the example he wanted,
but he didn't care.

Do you know Hilbert space? (Okounkov said it was functional analysis,
Stein confirmed that he did turn to FA.)

spectral theorem in any form you like. Properties of spectral measure.
What's good for a normal operator?
 - I mentioned the polar decomposition, but he wanted a simpler one.
I couldn't answer, then Stein said it was the sum of two symmetric operators.
I was very surprised and said I didn't know that, but Okounkov seemed more
surprised that I didn't know this fact. Immediately I realized that Stein
meant the sum of a symmetric operator plus i times another one. 

What's the property of the spectrum? (Okounkov: Can it be empty?) How to
get the measure? (I didn't know.) Example of a symmetric operator? 
 - I wrote down the integration operator, but Stein wanted a concrete
example. Then I turned to laplacian. I said I didn't like this example
because it was unbounded. But Stein liked it very much! I felt that
I had chosen a bad example. After the exam, I realized that it was really
a tragedy for me to mention laplacian. Because it led me to the formidable
PDE and harmonic analysis, which I know nothing but he knows everything!

Let's consider the laplacian on the unit disk, what's the spectrum?
(I need to define laplacian on a suitable space first. Okounkov thought
that Stein had wanted the laplacian on the unit circle, Stein said that
was a better example.)
 - I told them some general facts, but Okounkov wanted the eigenvalues
and Stein wanted the eigenfunctions. Then I mentioned the Green function,
which Stein then asked for a definition. After some hints I got that
an eigenfunction should be the product of a radial part and an angular
part. Okounkov also asked if the spectrum contains 0, and if the first
eigenvalue is simple.

Can you give me an interesting space in FA other than Hilbert space?
 - Banach space, but Stein want a concrete example, so I said L^p.

Do you know a special theorem about L^p? 
 - I mentioned a lot but he was not satisfied, then he said the theorem
which was useful in PDE. I was almost faint, but still managed to state
the Sobolev embedding theorem. Then he asked for a proof. I used
the method in Gilbarg-Trudinger which involves integration inequality.
He seemed not familiar with this method, but still agreed with me.
I guess he wanted the method using Fourier transform.

The last question: what's your favorite theorem in FA?
 - I stated Banach-Steinhaus. (After the exam I realized that I only
stated a version for linear functionals, but they didn't point out.)
Proof? What is Baire theorem?

Then Stein sat back and said he had finished. I was confused that he
didn't continue with complex analysis.


Algebraic Topology (Gabai & Okounkov)

(G) What's your favorite theorem in AT? (Okounkov laughed and said
I could just mention Brouwer. So I did, but then changed to Lefschetz.)

(O) What's the index of a fix point?

(G) relation with vector field?

(O) trace in this case?
 - I had a lot of trouble at this time because I didn't know his purpose.
It turns out that he wanted me to answer "it's the Euler characteristic!"
Then he asked for a proof, I said because they had the same definition...
After some hints he lost his patience:"Do you know homotopy?" (He wanted
me to say that the homeomorphism obtained from the vector field is homotopic
to the identity.)

(O) Do you know Lefschetz for an algebraic variety over a finite field?
(I didn't know, then he explained Frobenius to G & S.)

(G) can a non-orientable 3-manifold be a homology sphere? rational homology
3-sphere? Why the Euler number of a 3-manifold is 0?
 - I said we could use Z/2 coefficient for non-orientable manifolds.
This was not what he was after, he wanted me to state the Poincar\'e duality!

(G) Homology of a simply connected closed 3-manifold? \pi_2? Proof?

(G) Orientation reversing homeomorphism of a 3-manifold? (It must have
a fixed point.)

(G) Can a genus 3 surface cover a genus 2 surface?

(O) Branched cover, genus 2 over genus 3?
 - I used cohomology product to get contradiction. He seemed not so satisfied,
I guess he wanted Riemann-Hurwitz. 

(G) \pi_3(S^2)?

(G) Counter example for Lefschetz if not triangulable? (There were some, but
I didn't know. Dave said Bing had a simple construction.)

Now Dave fetched a piece of newspaer, containing an advertisement with a picture.
He let me show the picture to Stein, then he asked me to describe it. I said
it was a packing of hexagons on the 2-sphere, which was impossible. Dave said that
there were some pieces other than hexagon. So I found some pentagons on it.
The picture was by no means accurate because there were so few pentagons.
(Yes, it is a soccer. I then knew that he wanted me to count the pentagons
and hexagons. I should have known the answer because it was once a problem
of chemistry in the college enrollment exam in China.) I wrote down the equations
and tried to solve them, but Dave asked me to guess. I could only guess there were
12 pentagons. Then Dave let me think it after the exam. (The number he wanted
of hexagons is 20 because it is related to the icosahedron. But I still think
his question was not well-defined.) The other two examiners were very interested
in this question. Stein said he never asked question from the newspaper, Okounkov
even began to do the computation himself.


Complex Analysis (Stein, few by Okounkov)

Relation of the growth of an entire function with the distribution of its zeros?
 - I didn't remember the result, so Stein turned to basic ones.

zeros of an entire function? Weierstrass factorization? If a function has finite
order, what happens?
 - It has finite genus. I also mentioned that we could estimate the growth by
integrating log|f|, he then asked the properties of log|f|. He didn't get what
he wanted.

Which function has zeros at the negative integers? Product form and integration
form of Gamma function? How to deduce the poles of Gamma function from the
integration form? (satified once I wrote down the functional equation.)
How to estimate the behavior of Gamma function at infinity? 
 - It turns out he wanted the Stirling formula. I didn't remember the constant
in Stirling, so Okounkov asked me to deduce it from the integration form,
I didn't know how to begin, we moved to another question.

Formula of comformal map from the upper half plane to a polygon.


Algebra (Okounkov, few by Stein)

\pi_1 \pi_2 \pi_3 of a compact Lie group? (G & S were all surprised. Stein said
that Lie group was not algebra, but analysis. But I didn't complain because
I knew the answer...)

If a lie group has no abelian factor, can you prove that the universal cover is
compact? (I couldn't. Then he asked whether I knew how to deduce \pi_1 from the
diagram of representations. I didn't, he was disappointed.)

Hilbert's Nullstellensatz?

How many doubly periodic functions are there, which have only one pole of order n
at the origin? 
 - I thought he was asking Riemann-Roch, but I couldn't believe that, because he
was supposed to test me algebra. So I only mentioned they could be represented
by Weierstrass Pe function, but he immediately asked for the Riemann-Roch theorem.
I asked:"Is it algebra?" but still wrote down the formula. Then he helped me to get
the answer from RR. Now Stein asked me to deduce the differential equation of
Pe function from this result.

What is a C^*-algebra? classification of commutative C^*-algebra?


Now he finished, I guess the last two questions were not algebra, but supplement
to Analysis. 

They asked me to go out of the office. As soon as I went out, they bursted out laughing.
I don't know why:(   A few seconds
later they came out and said congratulations to me.


Comments:
All professors are nice. It doesn't matter even if you don't know about
a few questions. Stein gave me a lot of hints. After the exam he told me
that I did well, but I guess he merely meant that I could follow his hints...
Okounkov is really tough, because his interest is so broad. In my case, I think
he already decided to let me pass when he began algebra, because he hardly asked
any algebra question seriously. And it's ok even if I couldn't answer his questions
about algebraic geometry orrepresentation theory. But if your field is AG
or representation, you may have big trouble if you can't answer.

It's very important for you to understand the purpose of their questions.
They seldom ask question directly. You may know the theory well, but you need
to give them what they want, sometimes a rather trivial fact.