Andrei Jorza's Generals
May 05, 2006 1pm - 3:30pm
Committee: Wiles (chair), Kollar, Stein.
Topics: Algebraic Number Theory, Algebraic Geometry
Wiles: What would you like to start with?
Me: Analysis.
Stein:
(Complex Analysis)
- Talk about the zeros of a holomorphic function?
- Now what sequences of zeros are permissible?
- Can you write the function on the blackboard?
- Are those coefficients right?
(Probably not, had to expand log.)
- Do you know any theorem that talks about the behavior of the function at
infinity based on the distribution of its zeroes?
(I had no clue.)
- Do you know about the Gamma function?
(Wrote the product formula, correct up to some factors.)
- Is that correct?
(No, added e^{\gamma z}.)
- That's still not correct.
(I got confused and wrote the integral definition.)
- So what is the behavior at 0.
(I messed up, got embarassed. More embarassement later.)
- Let's do the Riemann mapping theorem. How does the proof go?
(This was horrible. I knew the way the proof goes, but completely forgot
why the sequence of functions converges to a conformal map. This lasted
about 30 painful minutes.)
- I think I'm satisfied with complex.
Wiles:
(Algebra)
- Can you show that all groups of order $p^n$ are solvable?
(I started with some nonsense.)
- Perhaps you should look at the center of the group.
(That's enough of a hint.)
- Do you know how to do this for groups of order p^r q^s.
(For small values of the exponents, yes. They smile and say that it's
harder in general.)
- Take a finite field extension $\F_{p^n}/\F_p$. What is Frobenius? What
is its characteristic polynomial?
(I wrote a multiple of the polynomial and after a while Wiles said
enough.)
Kollar:
(Algebra)
- What is the main theorem of Galois theory?
- Take a quadratic extension of a field of characteristic 0.
(I write x^2-a.)
- There are more quadratic polynomials than that.
(I write $x^2-ax+b = (x-a/2)^2...$.)
- Ok, ok. Is it Galois?
- Take a degree 2 extension on top of that. Does it have to be Galois
over the base field?
(Plenty of embarassement and confusion for about 5 minutes of failed
attempts at showing it has to be. They stop me and give me a
counterexample.)
- What statement in group theory can you think of that reflects this?
(Revelation. I say a normal subgroup of a normal subgroup need not
be normal.)
Wiles: Should we do some real analysis?
Me: Sure. (Bad idea.)
Stein:
(Real Analysis)
- What is the fundamental theorem of calculus?
(I write some permutation of a large number of terms from real
analysis.)
- Are you sure that's what you want for conditions.
(No.)
- So start the other way around.
(I start with the function to integrate.)
- What conditions on the function need to be imposed so that this
works.
(I write some L^1 condition, then get confused completely about what
the conditions should be. After some horrible 10 minutes of me having
no clue what I was doing, I admit I had no idea how to prove it).
- Ok, ok.
- So what does L^1 mean?
- What does the integral mean?
(I write the definition of the Lebesgue integral.)
- What's a simple function?
- What are measurable sets?
(I write the definition of the completion of Borel sets.)
- What are Borel sets?
(I write what they are.)
- That makes no sense because all subsets are measurable according to
you.
(I got really confused, thought I got the definition wrong. After 5
minutes it turns out that my phrasing: "the open sets are Borel, then
add complements and countable unions" was bad, I should've said "the
smallest \sigma-algebra with these properties.)
- So let's get back to our problem.
(I offered Radon-Nikodym, but was refused.)
- If you have a continuous function almost everywhere differentiable with
the derivative almost everywhere 0, is the function constant?
(Got confused, forgot Cantor.)
- So do you know the Cantor set?
(Yes.)
- How might you use this to give a counterexample.
(I construct the Cantor function, which is a counterexample. I fail to
prove it's continuous, though.)
- Alright.
Wiles: Shall we do some number theory?
Me: Yes, perhaps that would boost my morale. (Wiles smiled.)
Wiles:
(Number Theory)
- Let's start with some local fields. For what primes $p$ does $\Q_p$
contain a cube root of 3?
(I use Newton polygons to settle the question for $p=2$, then embark on
a nonsensical process of expanding (1+x)^(1/3) in power series.)
- Perhaps you should reduce modulo $p$.
(I realize my idiocy, then explain Hensel's Lemma and reduce the question
to the splitting of $p$ in the extension $\Q(\sqrt[3]{2})/\Q$.)
Me: Is there a better answer?
Wiles: You can't really have a better answer than this.
- What is the splitting field of this field?
- How about the Galois group?
- What is a Frobenius element?
(I define decomposition group, inertia, Frobenius, etc.)
- Can there be primes that are inert in the splitting field?
(After a hint I show that for unramified primes the decomposition group
is cyclic of order 6 and generated by Frobenius. Mention it's a subgroup
of the Galois group, $S_3$. Then I stared at the blackboard for about 5
minutes, not seeing the contradition; Wiles finally said the obvious.
Puzzled look on my face.)
- Talk about abelian extension.
(I write down the global Artin map.)
- Ok, but what about ray class fields?
(I explain what the conductor of an extension is and how the Artin map
gives a ray class field whose ramification is bounded by some ideal.)
- Take a representation of this Galois group $S_3$ on $\GL_2(\C)$. Can you
define the Artin $L$-function?
(I define it.)
- Do you know what the Artin conjecture says?
(I say what it says.)
- Can you prove it in this case?
(My reactions were: what?; hold on; what?; hm; Then I mentioned that I
can show meromorphicity.)
- How?
(I write Brauer's theorem and explain how to use it. I explain how Tate's
thesis gives holomorphicity for the characters in Brauer's theorem. This
gives meromorphic continuation.)
- So how do you do this for our representation?
(I say I have to show the coefficients are nonnegative and proceed to draw
Young diagrams, the only way I know about representations of $S_3$.)
- Ok, that's good. Shall we go on to some algebraic geometry?
Kollar (lying flat on his blue mat):
(Algebraic Geometry)
- Talk about lines contained in hypersurfaces of degree 3 in the projective
space.
(I write out the incidence variety projecting on the Grassmanian and the
universal hypersurface.)
- So why is this a variety.
(After a while I manage to write down some equations that satisfy him. I
compute the dimension and then the dimension of the Fano variety in a
general hypersurface.)
- What theorem are you using?
(I struggle to quote the theorem correctly.)
- So why can you use the theorem?
(I realize that I need to show that the image is dense, i.e., that a general
hypersurface contains at least a line. I end up trying to show that the
general hypersurface composed with an element of PGL_n contains the line
[x,y,0,\ldots,0])
- You can't use that because PGL_n need not be projective? Is it affine?
- Ok, ok.
Some comments:
The exam lasted a lot longer than this account suggests and was full of silent
moments when I had no idea what was going on. The professors were very patient
and understanding and gave lots of hints that made the exam fluid (when I knew
what I was talking about). The side effect of this was that I got embarrassed
at their need to be merciful, which made me lose focus even more.
A few suggestions: learn the Riemann mapping theorem and the fundamental
theorem of calculus. Also, when you quote theorems make sure you quote them
precisely, or else you run the risk of having to prove them. Make sure you
look at past exams, often professors reuse (unknowingly; professor Wiles said
he could not remember past exams) problems.