Kiran Kedlaya's generals (October 1996)
Topics: algebraic number theory, algebraic geometry
Committee: de Jong (head), Katz, Nelson

Unless otherwise specified, proofs were not requested.

ALGEBRAIC GEOMETRY

dJ: Define an affine scheme. Define a variety. ("A reduced scheme of finite
type over a field...") What additional conditions would some people
impose? (I suggested irreducibility, algebraic closure of the field--he
wanted geometric (absolute) irreducibility.) Can you give an
irreducible variety that is not absolutely irreducible? (I gave
Q[x,y]/(x^2+y^2).)

dJ: Define the genus of a (projective) curve.  (We briefly discussed what I
should define as a "curve", and finally I settled on "a dimension 1
subvariety of a projective space."  I first discussed geometric genus in
terms of differentials.)

K: What is a differential?  (I constructed the sheaf of differentials.)
What can you say about this sheaf?  (It's locally free of rank 1, i.e.
invertible.)

dJ: Give another definition of genus. (I defined arithmetic genus using the
Hilbert polynomial.) Define the Hilbert polynomial. (I first defined it
classically, then using cohomology. Along the way, I blurted out that the
Hilbert polynomial was intrinsically defined--when Katz said
"What??", I realized that my definition as Euler characteristic of O(n)
depends on the choice of O(1), and hence an embedding in projective space.)
Why are these equivalent? (H^0 is the Hilbert function and H^1 is
eventually trivial.)

K: Can you state Serre duality?  (I tried and failed.  Embarrassing, since
I had anticipated such a question.)  Can you state it for curves?  (I was a
bit befuddled at this point.  With some help, I cleared my thinking and
said I needed a pairing between 0-forms and 1-forms, and suggested
residues.)  What can you say about the residues of a 1-form on a curve?
(They sum to 0.)

K: How would you compute H^1 of a curve? (Use Cech cohomology, putting the
curve in projective space and using the canonical affine subsets.) That
would be way too complicated in practice. What's the fewest open sets you
can use? (Since a projective curve minus one point is affine, two open
sets suffice.) Describe Serre duality in terms of this cover. (To pair a
0-form and a 1-form, take the residue of their product at one of the two
points not lying in the intersection of the affines.)

K: Define an algebraic group. Give some examples. (GL(n), SL(n), O(n), Sp(n)).
Why is one of these not a variety in the sense we discussed earlier? (O(n)
is not connected, so I changed it to SO(n).) Do you know what a rational
variety is? Are any of the examples you wrote down rational? (GL(n) is,
because it is A_k^{n^2} minus a hypersurface. SL(n) is because it admits a
map to A_k^{n^2-1} given by writing out all but one of the matrix entries.)
Are the others rational? ("I don't think so." "Wrong.") Do you know what
the Cayley transform is? (I hadn't heard the name, but realized he meant the
mapping sending a skew-symmetric matrix S to (I+S)(I-S)^{-1}.)

I commented, "I should have guessed 'Yes' before because I can't think of
how to prove that something is NOT rational."  Consequently...

K: How can you show that a curve is not rational?  (Its genus is nonzero.)
In fact, if you have a rational map between two curves, what can you say?
(By Riemann-Hurwitz, the genus of the first is no less than the genus of
the second.)  Now what about higher-dimensional varieties?  (The arithmetic
genus is a birational invariant.  Katz protested that this was too deep,
but de Jong noted that it's proved in Hartshorne.)  Can a rational variety
map to a curve of positive genus?  (That induces a map from P^n to the
curve, and the restriction to each P^1 inside P^n would be constant.)

At this point, Katz asked: "Are there any other topics in algebraic geometry
that you know about which we haven't covered?" I suggested elliptic
curves (I did go to Harvard, after all), and consequently...

K: Define an elliptic curve.  What is Weierstrass form?  What is special
about it?  (The origin is a flex.)  What is a flex?  How many flexes are
there?  (9) Can a line touch an elliptic curve to order higher than 3?
(No, by Bezout.)  How would I tell whether a given point has a given finite
order. (I suggested division polynomials--not what he wanted.)
Prove that the curve (with the geometric group law) is
isomorphic to Pic^0.  (They prodded me to use Riemann-Roch. Afterward, I
could answer the previous question--check whether n(P) - n(O) is the
divisor of a function.)

NUMBER THEORY

K: Give me a quick sketch of the proof of Dirichlet's theorem about primes
in arithmetic progressions. (I defined Dirichlet density, stated the
theorem, mentioned the Fourier analysis on (Z/mZ)^* and noted that you
needed to show that L(1, chi) is nonzero.)

REAL ANALYSIS:

N: Speaking of Fourier...  talk about the Fourier transform.  (I defined,
said it mapped L_1 to C_0 and L_2 to itself.)  What can you say about an
L_2 function and its transform?  (They have the same L_2 norm if you
normalize correctly--which I hadn't done.)  What other classes behave well?
(The Schwartz space.)  What is a function of rapid decrease?  (I said f
decays faster than any polynomial--eventually he got me to add that its
derivatives do likewise.)

K: Can you rewrite the rapid decrease condition in a more "symmetric"
fashion?  (I had mentioned that Fourier transforms interchange
multiplication by x with differentiation; he wanted me to note that x^m
(d^n f/dx^n) is in L_2 for all m, n.)  Is this condition sufficient?  (I
had no idea; I think he wanted me to say "Sobolev".)

N: What does it mean for a function to satisfy a Lipschitz condition? What
does this imply about the distributional derivative? (I wrote down |x| as
an example of a Lipschitz, nondifferentiable function, and he finally
coaxed me
to say the derivative was bounded, even though I didn't know what it means
for a
distribution to be bounded. Luckily, he didn't ask.) If f: R -> R^n
is Lipschitz, what can you say about the equation dx/dt = f(x)? (The
existence and uniqueness theorem for solutions of ordinary differential
equations holds.) Do you know the idea of the proof? (Rewrite as an integral
equation and find a contraction mapping.) Does the proof work if R^n
is replaced by a Banach space? (Yes.)

COMPLEX ANALYSIS:

N: Suppose you have a holomorphic function in a strip, continuous and
bounded in absolute value by 1 on the boundary, and bounded everywhere.
What can you say?  ("Oh dear, I don't really remember Phragmen-Lindelof..."
I first suggested the function might have to be constant, but was told to
map the strip to a disk to see why that need not be.  Then I suggested
correctly that the values inside should be bounded by 1, truncated the
strip above and below, wrote out the Cauchy integral formula, but was
unable to bound the integral on the long sides.  I did note that the
contribution of the short sides went to 0.)

K: What might one use Phragmen-Lindelof for? (I stated Hadamard's
three-circles theorem and gave a very poor description of why it was
related.)

K: Can you draw a picture to illustrate what "genus" means? (I drew some
tori with different numbers of holes.) What is the Riemann surface
associated to an elliptic curve? (The analytic submanifold of CP^2 cut out
by the defining equation.) How would Riemann have answered this question?
(The usual two-sheeted cover.) What's the fundamental group of a torus?
(Eventually, this led to a homomorphism from the fundamental group to C
given by integrating a differential, giving an isomorphism with C mod a
lattice.) How would Weierstrass describe this isomorphism? (The pe
function.)

dJ: Can you describe an open region of C^2 such that every holomorphic
function defined there extends to a larger region?  (I wasn't expecting
several complex variables!  I remembered a construction from Gunning,
though not the justification; he didn't pursue it.)

NUMBER THEORY (again):

K: State Cebotarev's density theorem. What other theorem is it related to?
(Applied to a cyclotomic field over Q, it implies Dirichlet's theorem.)

ALGEBRA:

dJ: Start constructing the character table for S_5. (I listed the conjugacy
classes, wrote the trivial, alternating and standard representations, then
suggested using the wedge product to generate more.)

dJ: Compute Ext^1(Z, Z/2) and Ext^1(Z/2, Z) over Z. (The first is zero because
Hom(Z, .) is exact--it's the identity functor!--while the second is Z/2,
computed by taking the long exact homology sequence of the exact sequence
0 -> Z -> Z -> Z/2 -> 0.)

K: What's the nontrivial extension corresponding to the nontrivial element
of Ext^1(Z/2, Z)? (The same exact sequence: 0 -> Z -> Z -> Z/2 -> 0!)

Total time: 2.5 hours

Tips: Pay attention to who's on your committee when preparing--I had two
algebraic geometers and it shows. Also listen closely for hints--my
committee was generous with them (sometimes inadvertently), and my lack of
indication of hints does not mean I answered a question unassisted. Above
all, stay calm. If you can say anything relevant about a question, do so,
but don't be afraid to admit you don't know something.