Alfonso Sorrentino's Generals Exam
Fine Hall, Office 706 (Prof. Mather's)
05/03/2004
10.00 am - 11.30 am
Committee:
[M]: Prof. John N. Mather (Chair)
[C]: Prof. Sun-Yung Alice Chang
[K]: Prof. János Kollár
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Algebra:
[K]: What do you Know about Galois theory? (I gave the basic definitions
and stated some results and the Main theorem)
[K]: Compute the Galois Group of x^3-2? (I computed the splitting field of
this polynomial and showed that the extension has degree 6. Hence the group
must be S_3)
[K]: Consider a non-constant function between two compact Riemann Surfaces.
How is it related to Galois theory? I confessed that I had no clue and we
spent some time talking about Riemann surfaces, and the Riemann surface
associated to y^2=x^3-1. I don't know what he wanted me to answer, but
after a while, he said that I was on the right track and that was enough,
since this was beyond what I was expected to know.
With my great surprise, Kollár said that he was fine with Algebra (just
these few questions? Algebra was one of my greatest worry!) and we moved
to the other topics.
Complex Analysis:
[C]: What is an analytic function? (I gave the definition of holomorphic
and analytic functions, and justified they were equivalent in the complex case)
[C]: Are these two notions still equivalent in R? (No) What is the main
difference with the complex case? (I talked about Cauchy-Riemann equations
and how this kind of symmetry is related to the regularity properties). Is a
C^{\infty}(R) function analytic? (No, the example I gave was exp(-1/x^2),
that is not analytic in 0, but it's C^{\infty}(R))
[C]: Do you Know Schwartz's lemma? (I stated and proved it). Can you eliminate
the condition f(0)=0 ? (yes, using Moebius trasformations).
[C]: How do you find a conformal map, that transforms a quarter of circle
into half circle? (I suggested f(z)=z^2). Can you find a Moebius transformation?
(I first mapped all the circle into the real axis, in such a way to send a
quarter of circle into the positive semiaxis, and then mapped this one back
into the half circle)
Real Analysis;
[M]: Can you find a Meager subset of the real line with full measure?
(I considered the classical example of the "fat rationals", i.e. I took the
complement of the usual residual of measure zero, intersections of union of
balls centered at rationals of decreasing radii)
[M]: Where does such a kind of set appear in KAM theory? (Diophantine frequencies)
[M]: Find an example of sequence of functions that converges pointwise, but
not in L^1. (I considered a sequence of functions in [0,1], whose graph is
a triangle of constant area 1, but whose support tends to 0)
[C]: Under which conditions does "pointwise" imply "L^1" ? (I stated
Lebesgue theorem, i.e. dominated convergence thm). Any other easier condition?
(uniform convergence). How can you check if a sequence of functions converges
uniformly? ( I said "using the definition" and I also stated some results,
such as Dini's theorem for a sequence of monotone decreasing continuous functions)
[M]: What can you say about a family of equibounded and equicontinuous functions?
(I stated Ascoli-Arzela' theorem. I also said that equicontinuity can be
weakened, asking local equicontinuity, and how the claim changes in this case.
Moreover I highlighted that in the complex case, equiboundness and analyticity
are enough, since equicontinuity comes from Cauchy estimates)
[M]: Do you know anything about Spectral theory or Compact operators? (I chose
to talk about Compact operators, giving the definition of a compact operator
and stating Rellich theorem... opening the way to PDE)
Elliptic PDE (I):
[C]: Rellich theorem and Sobolev immersion theorem (just to link to the
previous answer). What can we say about the extremal case q=p*? Can you find
a variational equation that this kind of function satisfies? (We worked on it
for a while; actually I received several hints and suggestions in order to
figure out an answer).
[C]: Why does Sobolev immersion theorem play an important role in the proof
of De Giorgi-Nash-Moser theorem? (I described Moser's iteration method and
how we choose the exponents in order to make it converge)
I think there were other questions in this first section of PDE, but I can't
remember them right now. Actually the discussion above was not so "plain":
we get into different topics and there were several other questions.
Now it was Hamiltonian dynamical systems' time:
Hamiltonian Dynamical systems:
[M]: Tell us what you know about KAM theory (I started talking about
integrable and nearly integrable systems, the geometry of the phase space in
the integrable case, how it changes under a small perturbation and pointed
out some problems related to the stability and instability of the solutions.
I stated Kolmogorov theorem in the analytical case and discussed why the same
technique doesn't work in the differentiable setting and how this problem can
be avoided, using a smoothing operator. I also talked about diophantine
frequencies and the non-degeneracy conditions for the integrable hamiltonian)
[M]: What can you say about the convergence of the iterative scheme? How do
you choose the radii of the domains?
Mather said that he was fine, and asked if there were further questions.
Alice Chang said she wanted to ask something more on PDE.
Elliptic PDE (II):
[C]: What do you know about local regularity results for a solution of an
elliptic operator? (I talked about Hoelder regularity results, stating
perturbation and non-perturbation ones. Namely: Schauder estimates, relation
between local growth of integrals and Hoelder regularity, Local boundness and
De Giorgi-Nash-Moser theorem. This was an articulated discussion and we came
in a lot of other topics, such as Moser's Harnack inequality, W^{2,p} estimates,
Calderoni-Zygmund inequality, John-Nirenberg lemma etc...)
At this point I was completely exhausted and it seemed to me that I had been
there for an eternity, while it was just 1h and half. Since noone had any more
questions, I was asked to step out the office for a while. After 1 minute
(the longest ever!), they came out, shaking my hand and congratulating!
All of them created a nice atmosphere and the exam was really much easier
than what I had thought (altought it was everything but relaxing!). I think
they mostly want to test your abilities, rather than your knowledge; so they
spend a lot of time on topics you seem to be less confident with. But at
the same time they are aware you cannot know everything, so don't be ashamed
to say that you are not very familiar with something...
Good luck to everyone!