DIMA DOLGOPYAT'S GENERALS.
Committee: Sarnak, Sinai (chair), Trotter.
Subjects: Stochastic processes & Dynamical systems.
Duration: 1:30.
All questions were asked by Sarnak unless otherwise noted.
COMPLEX ANALYSIS.
Here I always forgot to mention some important conditions in the
formulations of theorems, so a half of questions was 'Give a
counter-example to what you just said.'
- State Riemann mapping theorem.
- Which simply-connected domains are not equivalent to the unit disc,
how to prove it.
- Weil lemma about elliptic operator and diffusion processes
(This was probably stochastic processes question since no other
question qualify.)
- (S) Talk about isometries of Lobachevsky plane. (At this point
Sarnak seemed to be unhappy that somebody also asked question, so he
said that it was not complex analysis question and they discussed this
for 5 minutes.)
- Talk about conformal isomorphisms of compact Riemann surfaces.
Which of them have an infinite group.
- (S) Weierstrass rho-function, differential equation for it.
- Talk about field of meromorphic functions on compact surfaces.
Are Mer(S^2) isomorphic to Mer(T^2)? Now this problems seems not very
difficult but on the exam I couldn't solve it so they went to the next
topic.
REAL ANALYSIS.
- (T) Does the series f(x)=\sum 1/((2^n) sqrt(x-a_n)) where a_n runs over
rationals on [0,1] converge at some point.
- State the monotone convergence theorem.
- (S) That can you say about f(x,s)=\sum 1/(x p_n+q_n)^s where
a_n=p_n/q_n are as above? (It has analytic continuation to complex s-plane by Cauchy method.) Talk about Ruelle-Perron-Frobenius operator. (This may be
dynamical systems question.)
- (S) Give the asymptotics of the n-th moment of the standard normal
distribution. (I used Laplace method to calculate the answer.)
- What is the name of the formula you just proved? (After a small
reflection I said it was Stirling's formula.) Do you know it for
complex values of parameter. (I had to recall them that I didn't
know it for reals, because they seemed to forget this.)
- Talk about multi-dimensional Laplace method.
ALGEBRA.
- (T) Give an example of UFD which is not Euclidean ring. At least
do you know any example of ring? (I gave polynomials of several
variables.) Do you know that this is not a principal ideal domain?
Have you ever heard that every Euclidean ring is a principal ideal
domain.
- (S) State structure theorem for finitely-generated abelian group.
- Structure of the multiplicative group of a finite field.
- State and prove Mashke theorem. How to prove that
any finite-dimensional representation of a compact Lie group is
equivalent to unitary one?
DYNAMICAL SYSTEMS.
Sinai said that maybe it was time he should ask a dynamical system
question, but Sarnak said he had a lot of question in this topic.
- Can you formulate the content of KAM-theory in one word (no, Sarnak
mentioned that he could do that.)
- What does the abbreviation KAM mean.
- State Moser twist theorem. What is geometrical meaning of twist
condition. Can you deduce this theorem from the implicit function
theorem? (no) Give some arguments why it is not possible.
- How many numbers satisfy the Diophantine condition you mentioned
recently? After that Sarnak asked several simple questions about
Diophantine approximations which I couldn't answer. I therefore
said that I was nervous and that I wanted to think a little bit
without interruption, but they went on.
- Is it a written test or a oral test? (Oral.) Are you supposed
to think or to speak? (This I couldn't answer either.)
- Isn't a time for tea. (This was probably the question for Sinai
because he was the chair. After this questions they shake my hand
and we went to tea-room.)
General comments: They seemed to be in harry. A lot of Sarnak questions
was of the kind of 'Can you prove this statement IN ONE LINE?',
'Can you formulate this theorem in ONE WORD?', 'What is the main idea
in this theory?' and so on. Usually immediately after asking the question they
started to tell solution, but sometimes they began solution before
formulating the question so I had to guess what the problem was. Also,
they seemed to pay much attention not to whether I knew answer or not but
to whether I answered immediately or not.
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