Boris Bukh's generals, 2pm, 30 May 2006
Special topics: analytic number theory, probabilistic combinatorics
Committee: Sudakov (chair), Sarnak, Kohn
I was offered to choose the one of the general areas to
start with.
Questions were asked by:
[Sd] - Sudakov
[Sr] - Sarnak
[K] - Kohn
If not committee member who asked a question is
not indicated, it means I do not remember.
Complex analysis:
- [K] State Cauchy's theorem.
- Classify singularities.
- Where does Laurent series converge?
- [Sr] Does it converge uniformly?
Where does it converge uniformly?
- [Sr] Can a meromorphic function have infinitely
many poles in a compact set?
(I said no, but blanked on the proof, and was given
a hint to prove this triviality).
- [Sr] Entire function satisfies |f(z)|0
what can you say about O/(prime ideal in O)
(Finite.)
- [Sr] Anything else?
(No.)
- How much do you know about representation theory?
(I said little. They did not pursue it, but
Sarnak recommended to learn it well).
- Talk about finite fields, their extensions, and corresponding
Galois groups.
- What does the multiplicative group of a finite field look like?
Prove it.
- What is Jordan canonical form?
- When/how can we solve matrix equation e^A=B?
It was 3:30pm and the committee decided to take
a tea break. So, we went to the Common room for 5mins
to get some tea. After the break the exam resumed with
probabilistic combinatorics.
- [Sd] Markov and Chebyshev inequalities. Chebyshev
for non-negative variable. (At this point
Sarnak started discussing with Sudakov what
the correct names for these inequalities are).
- [Sd] Define a martingale.
(I managed not to answer it right away.)
- [Sr] (Interjecting) Can you state some
martingale central limit theorems?
(No.)
- [Sd] State Azuma's inequality.
- [Sd] Apply Azuma to isoperimetric inequality for
hypercube.
- [Sr] Do you know Poisson's inequality?
(No.)
- [Sd] Define entropy.
- [Sd] What are its properties?
- [Sr] (Interjecting) Why entropy? What is so special
about it?
(Me: it measures information, etc)
- [Sr] Is there some theorem that uniquely characterizes
entropy as function that possesses some properties.
(I remembered reading such a theorem long ago, so
I said yes, but could not state it).
- [Sd] Prove that sum_{i<=pn} binom(n,i)<=2^{nH(p)}
- [Sd] State Shearer's generalization of subadditivity
property of entropy
(It took me a while to recognize the name because
I did not know how "Shearer" is pronounced)
- [Sr] (Eagerly) You show us construction of graphs
of high girth and high chromatic number. Or we
fail you.
- [Sd] Yes, or we fail you.
(I did it.)
Analytic number theory (all asked by Sarnak):
- Prove that zeta(s) has infinitely many zeros.
- No, not trivial ones.
- No, without counting them.
(He wanted "zero-free"<->"equals to the product
of an exponential and a polynomial")
- Now, count them.
- What kind of function is zeta(2+it) as a function
of t?
("Almost periodic" satisfied him)
- Is zeta(1/2+it) almost periodic?
(No, it is unbounded)
- Give an upper bound on sum chi(n)/n
for non-principal character chi.
(I gave some weak bounds, he explained
then how to get strong bounds).
- Give a lower bound on sum chi(n)/n.
(Siegel)
- Why is zeta(s) free of zeros on the line 1+it?
- Why doesn't Vinogradov's method work for two primes?
- Estimate from below L^1 norm of sum_{k<=M} e(xk^2)
(with a hint).
The exam lasted 2hrs50min. There were probably more
questions than written above, but I do not remember them.
Advice: Do not overprepare. The generals are merely an
exam. On the exam, take your time to define everything
you write down, it will save you from mistakes, and
give you time to think about the answer while you write.
Once you done writing, you will have mere seconds to
answer the question.