May 15, 2002: Fine 602. (1:30 to 3:30)
Committee: Wiles (Chair), Gunning, Pandharipande.
Special Topics: Algebraic Number Theory, Algebraic Geometry.
Wiles asked me what I wanted to start with: I said complex-
1) Complex Analysis:
[G] Classify singularities. How do functions behave near poles and
near essential singularities? (I mentioned Big Picard) Do you know
how to prove it? (no) Do you know how to prove that the range is at
least dense? (yes. He didn't ask me to prove it though.) Give an
example of a function that has essential singularity at 0. (exp(1/z))
[G] How do functions on annuli look like? (Laurent series) Give an
example of a function that has infinite laurent tail but has no
essential singularity… (hmm) What about 1/(1-z) on the annulus with
inner radius 1 and outer radius 2? ok)..
[G] Suppose we have a function whose first n-derivatives
vanish at the origin. How does it look like (geometrically)?
Gunning said he would like to finish the real analysis part too,
so that he can take a nap.
[P] What are the topics he has to be interviewed on? (after all
it was also his first generals!)
2) Real Analysis:
[G] What are the different kinds of metric one can define on [0,1].
(L^p's) How are they related? Give an example
of a sequence of functions that converge in L^p but not in the
sup-metric. Why are people interested in L^p's?(they are Banach
Spaces) Can you prove it... first define Banach space. (I gave a
sketch of the proof.) What is special about L^2? (Hilbert space)
Why aren't the others Hilbert spaces? (I mentioned that the others
are not self dual!) Give an example of a sequence of functions
that converge in all the L^p's but not pointwise.
[G] How will you generalize fundamental theorem of calculus for the
Lebesgue setting? What is absolute continuity? Define functions of
bounded variation. What is the relation between functions of
bounded variation and monotonic functions?(!? oh!)
Gunning looked happy: Wiles asked whether Pandharipande is
willing to ask any analysis question. He said "no". but:
3) Algebra:
[P] How many proofs of fundamental theorem of algebra
do you know? (3-galois theory, liouville, winding number.) ok,
lets do a new proof using lie groups! Gunning laughed, Wiles
looked amazed. I stood petrified - and readily confessed my
ignorance about lie groups. Wiles said "we can't expect him to
know lie groups". Anyway, I didn't have to do anything. Rahul
described how one gets a proof using some structure theory of lie
groups!
[W] What are the eigenvalues of Frobenius? (this time I
got totally confused. Wiles remarked that this is the simplest
linear algebra question with which you can confuse anybody!)
[W] What are the groups of order p^2? What about order pq? (it
depends.. ) What if q=1(mod p)? (then it may not be abelian)
[P] give an example- (S_3).
[W] how will you generally describe the structure - (semidirect
product: Wiles made me write it down completely on the board.
[W] Isn't it the other way? (I don't think so!))
[W] I guess we can ask algebra questions while continuing with
special topics. What do you want to start with? I said: number
theory.
4) Algebraic Number Theory:
[W] How do primes split in Q(sqrt(-23))? (quadratic reciprocity).
[W] What is the class number of Q(sqrt(-23))?(I mentioned about
Minkowski bound: and blabbered "in this case it is less-than-
equal-to 2") what? It's surely greater than 3. (oh! Yeh, yeh… it
is less than four. Then we looked at the primes over 2 and 3
separately.) How will you show that the primes on 3 are not
principal? ... use norm.
[W] What is the ray class group of the conductor 7? Define ray
class group. How is it related to abelian extensions? (ray class
groups are galois groups of ray class fields which are abelian
extensions unramified outside c) How is ray class group related to
ideal class group?(it is exactly the ideal class group when we take
the trivial cycle) what more? (ideal class groups are quotients of
ray class groups.) Hence? What can you say about the kernel? (I got
confused between principal ideals and primes.) what??
(embarrassment.)
[W] Is there any other way to compute class number? (class number
formula) write it down. What is the class number formula like in
our case of quadratic extensions?
[W] Do you know any bound for the class number,.. discriminant?
(Minkowski gives a bound) what else? (Wiles wanted to hear about
Brauer-Siegel formula.)
[W] Do you know about the class number problem? How many imaginary
quadratic fields are there with class number one?(finitely many)
What about real quadratic? (isn't it open!?)
[W] What is Artin map? Write it down. What is the kernel?
[W] Do you know about non-abelian L-functions. Wiles pulled
himself back and with a smile said: "I guess I am very happy with
number theory!" (how!?)
5) Algebraic Geometry.
[P] Compute cohomology of the tangent bundle of P^n.
[P] Suppose f is a morphism from P^1 to P^1 of degree 2. What is
the direct image of O(1)?(its coherent) and what? (it took me a
long time to figure out that it's locally free of rank 2) Can you
write it down explicitly. (with lots of help from Pandharipande I
finally figured out that its just O+O.)
[P] Are there any non-hyperelliptic curve of genus 2. (no) What
about other genus? (there are) Give an example for genus 3. (plane
quartic)Why is it nonhyperelliptic?(canonical bundle is very
ample) In the hyperelliptic case how many morphisms to P^1 of
degree 2 are there. (essentially one) Why? (I said I can not prove
it- he didn't mind. But he pointed out that it's true only for
genus greater than 1.)
[P] If we blow up the point (x,y) in the plane, what kind of
variety do we get? (nonsingular..) What happens if we blow up
(x, y^2) instead? (singular) Describe the closed subscheme (x, y^2).
[P] One last question. What is Riemann-Roch theorem? Suppose we
pick up a point P "at random" from a curve of genus g>1, what can
you say about l(P)? What happens if we pick d (d