Brian Conrad's Generals.
(total time: 2 1/4 hours)
Committee: Aizenman, Faltings, Wiles
Special topics were algebraic geometry and algebraic number theory
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Real Analysis:
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If we integrate from 0 to 10, for what p is
x^p integrable? What about x^p cos(1/x)?
In sense of Riemann vs. sense of Lebesgue.
(I embarrasingly bobbled this one a bit, but they didn't seem to mind).
If f_n is a seqeunce of integrable functions, when is int(f_n)
convergent? Can you give an example where this fails?
Let E be a measureable set in R with positive measure. Do there
exist points x,y in E whose average (x+y)/2 is in E?
(I tried to slip in that Rudin question about E + E containing
an interval, but they weren't fooled). Perhaps use metric
density. Discuss the existence of such points.
(When I said the magic words Fund Thm of Calc, they suddenly backed off)
These were all asked by Aizenman.
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Algebra:
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Discuss groups of order 55.
Give an example of a ring where unique factorization fails. Explain.
(I gave Z[sqrt(-5)], the canonical example with the canonical explanation)
Discuss Galois group for X^5 - 2 over Q.
These were asked by Wiles.
Do you know what a module is? What is a projective module?
What are these good for? (defining derived functors)
Can you explain what Ext is. How does one interpret elements of
Ext^1? Give a non-trivial example (I gave 0 --> Z/2 ---> Z/4 --> Z/2 --> 0).
Can you compute Ext^i(Z/2,Z/2) over the base ring Z/4 for all i >= 0.
(Faltings prodded me a bit to write down an actual proj resolution; it's
periodic and so from that you get the answer).
Can you have Ext without having such resolutions? (I babbled about usefullness
of notion of universal delta functors, which seemed to satisfy Faltings).
What is finite projective dimension? What can you do with it? (prove
Serre's theorem on regular local rings, though I didn't prove this of course)
What rings besides fields have finite injective dimension? (Faltings
told me regular rings worked; that was news to me at the time)
These were asked (obviously) by Faltings.
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Complex analysis:
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Suppose that you have a sequence of analytic functions on a region and
they converge pointwise on some simple closed curve inside the region. What
can you say about the seqeunce of derivatives on the interior of the
curve? (I said one should assume some L^1 boundedness on the curve
to apply dominated convergence theorem to Cauchy formula).
Can you map an open square to an open disc analytically? What about
continuous extension to the boundary? (I claimed there was no clear
obstruction, Aizenman claimed conformalness was relevant; as
there's no analyticity condition AT the boundary, I disputed
his point of view, but then they changed questions)
Faltings followed up with some suggestion to use Schwarz reflection for this
problem, but I didn't follow his point, and he shortly gave up the chase,
much to my relief.
Can you prove the Picard theorem? ("Which one?" "Either one." Whew.)
Do you know about elliptic functions? Tell about integrating
1/sqrt((1-x^2)(1+x^2)) (concepts, not calculations).
Tell us about the Dirchlet Principle? Does it relate to
a variational problem? (I babbled about log singularities and taking
limits, and then mentioned the magic phrase "harmonic functions" and they
were satisfied)
These were asked by Faltings and Aizenman.
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Algebraic Geometry:
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What is a scheme? Give examples.
What is an elliptic curve? Compute the genus of y^2 = x^5 - x (use
Hurwitz). Can a genus 2 curve embed the plane? Prove what you need
(i.e., say how to compute arithmetic genus, and I added in why
it equaled geometric genus).
Give a lower bound for the degree of a map from a genus g curve to
P^1. Can there be an upper bound? (no, look at hyperelliptics).
How would you write down all the holomorphic differentials
on the Fermat curve? (this was Faltings' attempt to be cute at
Wiles' expense)
Give an example of a non-hyperelliptic curve. Explain. (degree 4
plane curve)
What is the Jacobian of a curve? How do you determine its dimension?
What is the theorem of the cube? What is it used for? (I blanked on the
latter)
Compute the cohomology of projective space. (conceptually--Faltings
wanted to hear me say "Cech cohomology")
Does every curve embed P^3? Why are all curves even projective?
What about embedding a curve into P^2?
What is a regular scheme? Is regularity the same as smoothness over
any base field? (I said it was true over perfect base fields
but not otherwise)
Do you know what Neron minimal models are? Determinant of
cohomology? (I said "no")
These were asked by Faltings.
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Number theory:
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Tell about class field theory, Artin L-functions. Discuss CFT in
terms of abelian L-functions. What should a non-abelian analogue be?
(I mentioned L-functions coming from L-series of automorphic forms)
Discuss the unit theorem. Can you make it effective? (I didn't know how)
What about embedding the units into a p-adic completion? (I didn't
know)
What about the group of rational points on an elliptic curve?
(I proved the Mordell-Weil Theorem to keep them happy).
Can you make the computation of this effective? (I mentioned a little
about Tate-Shafarevich group)
Tell about l-adic Tate module representation coming from an elliptic curve.
What can you say about the characteristic polynomial of Frobenius
as l varies?
What is a Frobenius element?
They asked a couple more about CM-curves (explicit examples
and fields of definition).
Why is exp(pi*sqrt(163)) nearly an integer? (q-expansion of J-function)
How do you generate abelian extensions of imaginary quadratic fields?
These were asked by Wiles and Faltings.
Can you give a heuristic proof of Fermat's Last Theorem?
(this last one was asked by Aizenman, the physicist!! Wiles and Faltings
seemed only slightly amused).
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They were all very polite, and quite understanding when I bobbled
things, and very willing to suggest hints.