Robert Harron's Generals

May 12th, 2005. Fine 608. 3:00pm to 5:45pm
Committee: Ramin Takloo-Bighash (Chair), Jordan Ellenberg, Edward Nelson
Topics: Algebraic Number Theory, Algebraic Geometry

There were a lot of questions and discussion at certain points in the
exam, so I may have forgotten some of the questions and mixed up the
order of others.

Ramin suggested we start off with the basic topics and then move on to
the special topics, and I asked me if I was okay with this. I said I'd
rather start with the special topics so we did...

1) Algebraic Geometry:

[JE] Describe the Jacobian of a curve.
     (I said I hadn't studied that. So Jordan said perhaps I should tell
him what I had studied, to which I replied Hartshorne chapters 1-4.)

[JE] Okay, but can you tell me something about the Jacobian?
     (So I said you take the Pic group and turn it into a variety. Talked
a bit about elliptic curves being isomorphic to their Jacobians.)

[JE] What's a sheaf? What's a morphism of schemes? What's the push-forward?
pull-back? What's a quasi-coherent sheaf? coherent sheaf?

[JE] Suppose F is coherent on Y when is the push-forward of F coherent on X?
     (I couldn't quite remember so I said finite type. We discussed examples
of affine schemes to see why this was wrong. I "figured out" it was true for
finite maps.)
     What about the pull-back?

[JE] What's the geometric reason a map is called finite? Give a map that
is quasi-finite but not finite.
     (We discussed several types of maps: open immersions, closed immersions,
etc. For each I was asked to give an example and we always worked with
affine schemes.)

[JE] Can you show that all plane conics are birational to a line?
     (Seems he wanted me to draw a picture.)

[JE] If I draw N points in the plane, can you tell me if there's a conic that
goes thru them?

[JE] Tell me about the analogy of the class group for function fields over F_p?
     (I incorrectly said the class number was infinite, I was thinking of
function fields in general. We worked thru why this wasn't true.)

2) Algebraic Number Theory:

[RTB] State Cebotarev Density and define the terms you use in it.
      (I had to define density and the Artin symbol. Ramin really likes this
question.)

[RTB] Prove sum 1/p diverges.

[RTB] What does Cebotarev have to do with Dirichlet's theorem on primes in
an arithmetic progression?

[JE] What is the Artin symbol in the case of a cyclotomic extension?

[RTB] State Dirichlet's Unit Theorem. Explain the spaces involved, i.e. in
what space is O^*_K a lattice?

[JE] Is the map alpha |---> (log |tau(alpha)|) (you know what I mean)
injective? What's its kernel? Show this.

[RTB] Tell me about Dirichlet's Unit Theorem and quadratic number fields.

[JE] What's the regulator of a number field?

[JE] In a quadratic number field, how would I get a small regulator?
     (I didn't really know what to do with this, so it took a while.)

[RTB] Give me a number field with class number greater than 1.
      (I, of course, wrote Q(sqrt(-5)) and said it was not a UFD. He then
asked me to compute the class number. For this field the Minkowski bound
gives it to you, so that was pretty easy.)

[RTB] Write a non-Galois extension?
      (I wrote K=Q(cube root of 2))

[RTB] Tell me how primes decompose in here?

[JE] How would you show that there are infinitely many primes in Q for
which x^3-2 is irreducible mod p?
     (We spent a while on this, I should've been thinking quicker.)

[JE] Make a comment about the number of conjugacy classes in S_n and a
known function in number theory.
     (I said it was the partition function, but didn't really know where
he wanted me to go with this, so we moved on.)

3) Real Analysis:

[EN] Let f,g in L^2, let h(x):=int(f(x-y)g(y)dy,y=-inf..inf). What can
you tell me about h?
     (Continuous, bounded, vanishes at infinity, I was asked to prove
these statements)

[EN] Let E be a set of positive measure in R, show that E-E (the set of
differences) contains a neighbourhood zero.

[EN] Suppose f_n is a sequence of continuous functions on [0,1] monotonically
decreasing to zero pointwise, what can you tell me about the convergence?
     (The convergence is uniform, this is called Dini's Theorem. This is
a really easy theorem but I, for some reason, took a while to prove it, oh well)

[EN] Suppose you're teaching a bright class of undergraduates that have
a good background in real analysis, how would you prove the fundamental
theorem of algebra for them?
     (I wasn't too clear on this one, it took me a while...)

4) Complex Analysis (maybe)/Number Theory...

[RTB] Compute the Fourier Transform of exp(-x^2).

[RTB] What does this have to do with numbers being written as a sum of
four squares?
      (I gave a blank look, and guessed that this had something to do
with Poisson Summation, which was right. I wrote out some stuff and
mentioned the theta function, Ramin reacted well to the latter and told
me to write it out. Then he mentioned that there was an m^2 in the theta
function, but that I wanted four squares, so I should raise it to the
fourth power. Then he asked what kind of function this was, I said
"Oh this is a modular function isn't it" and he asked for the weight,
which is two, and for which group is it a modular form, which is Gamma_0(2).
Then he asked if I knew the dimension formula for the space of such
modular forms, I said no. At this point, he just explained the rest
to me. Basically, you get a generating function for the number of ways
you can write a number as a sum of four squares and you can find an
explicit basis of the space of modular forms of weight two for Gamma_0(2)
et voila... Afterwards, Ramin told me he was just messing with me on
this last question, just to see what I would do, apparently Shalika
asked him this question on his generals at Johns Hopkins... I guess
he's trying to pass on the tradition. It is quite an elegant solution.)

5) Algebra

Well there was no algebra part of my exam, they decided that I had done
enough in the Number Theory and Algebraic Geometry sections.

Throughout the exam everyone was very kind and helpful. They were willing
to wait on me while I thought and would provide hints when necessary.
When I was wrong about things, they didn't mind, they would lead the
discussion until I noticed my mistake.