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)|<e^(|z|^100)
       what can you say about it?
       ( sum 1/|rho|^(100+eps) over zeros rho converges)
- [Sr] Why?
       (Weierstrass factorization; started proving it)
- [Sr] More specifically?
       (Jensen's formula, stated it, forgot factor of
        2pi, and was corrected)
- What are harmonic functions? What is their relation
  to holomorphic functions?
- [Sr] Suppose you have a holomorphic function on upper 
       half of the unit disk, and you know that it is
       at most 2 on the semi-circle, and at most 1 
       on 1 on the real axis part of the boundary.
       Bound the function in the interior.
       (I said Phragmen-Lindelof, but could not see
        the right multiplier. After lots of help,
        I got it. I won't spoil it here)
Sarnak to Kohn: "You should ask him some stuff on dbar".
       (I tried to define dbar operator, but was
        stopped)

Real analysis:
- [K] What theorems do you know about selecting a convergent
      subsequence?
      (Being hesitant to name one, I eventually chose 
       Arzeli-Ascoli)
- Prove it.
  (I was stopped near the end of the proof)
- What about for complex numbers?
  (Montel's theorem)
- Define Fourier transform
  (I defined Fourier transform on R^1 and said
   that I can define it more generally.  Got approval from 
   Sarnak for putting 2pi in the right place)
- For which function does it make sense?
  (As written for L^1, but can be extended)
- Define L^1.
- What is a Fourier transform of something in L^1.
  (Continuous and decays to zero).
- Why?
  (Started proving Riemann-Lebesgue, but was stopped
   within a couple of seconds).
- Can both function and its Fourier transform be compactly
  supported?
  (Said no, but blanked on the proof, and got a hint)
- [Sr] A function is smooth and is rapidly decaying,
       what can you say about its Fourier transform?
       (Do not know).
- [Sr] What does "rapidly decaying" mean?
       (Defined)
- [Sr] What is Fourier transform of derivate? 
       (Answered, with omission of the pesky 2pi 
        factor, got corrected).
- [Sr] Suppose a function in L^1 has one derivative
       in L^1, what can you say?
       (Decays like 1/x)
- [Sr] So, if a function has all derivatives in L^1?
       (That is what he meant when he said "smooth"!)
       (Answered).
- [Sr] Central limit theorem.
       (It was completely unexpected. I was heavily helped 
        with it)
- [Sr] Which functions are Fourier transforms of themselves?

Algebra:
- [Sd] State Sylow theorems.
- [Sr] What is a simple group?
       (Defined, and said that I could define only A_n and Z_p)
- [Sr] What about groups of matrices?
       (I said that PSL(2,F_5) is simple because it is A_5)
- [Sd] Show there are no simple groups of order 132.
- [Sr] Let O be the ring of integers of Q(sqrt(d)) d>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.