Nike Vatsal's General.
Topics: Alg. Number Theory, Al. Geometry.
Comittee: Washnitzer, Wiles, Rudnick.
Questions:
Real Analysis -- most of these questions were asked by Rudnick.
Define Fourier Transform.
Does it map L^1 to L^1?
What can you say about the image?
Show that the Fourier transform of an L^1 function vanishes at infinity.
What is the transform of the char. fn. of an interval?
What can you say about the Fourier transform of a compactly supported function?
CAn you find functions other than the Gaussian that are equal to their Fourier
transform?
What about real valued functions with positive Fourier transform?
State the Poisson summation formula. Prove it.
Define the Riemann zeta function. SDoes it have analytic continuation to
all of C? Why not? What is the behaviour at s=1? Show that it has a meromorphic
continuation to the complex plane. (This can be done with the summation
formula.)
Algebra:
Talk about groups of order 55. Display a non-abelian one.
Give an example of a ring that is not a UFD. I gave Z[sqrt(-5)].
Factorize 6 in two different ways in this ring.
Is there any way to "explain" the two facorizations?
Factor the ideal generated by 6 into prime ideals.
Talk about the Galois theory of X^3 - 2 = 0 (over Q).
Describe the spilitting field. What are the subfields of the splitting field?
The above questions were asked by Wiles.
What is a skew-symmetric real matrix? What can you say about it's eigenvalues?
Show that if A is skew symmetric then U = (I+A)(I-A)^(-1) is orthogonal. Can
you recover A from U?
These were asked by Washnitzer.
Complex Analysis:
Give a conformal map of the right half plane onto the unit disc.
What can you say about an entire function that maps into the plane minus the
negative real axis?
What about the plnae minus a segment?
CAn you generalize the above cases? (i.e. Prove the small Picard Theorem.)
Prove Liouville's theorem.
What is the gamma function? Talk about its meromorphic continuation. I started
to give a proof using contour integrals, but they wanted to do it using the
functional equation, and started asking me about functional equations.
Consider a function analytic in the unit disc satisfying
f(2z) = f(z)/(1+f(z)^2). and f(0)=0
If such a function exists can it be continued to a meromorphic
function on C? Does such a function exist?
Prove the meromorphiccontinuation of the gamma function.
If an analytic function is represented by a power series with radius of
convergence one, can it ever be continued to a neighborhood of the disc in
which the series converges?
Number theory:
Waht does the Artin reciprocity law say for the p-th cyclotomic field?
Can you use this to prove quadratic reciprocity?
What is the Tchebotarev density theorem?
Consider a monic irreducibe equation of primce degree with rational integer
coeffs. Show that there are infinitely many primes q such that p has no roots
mod q.
What is the maximal abelian extension of Q_p? What about the maximal unramified
extension? Describe the maximal tamely ramified extension. What can you say
about the structure of the unit group of number field? P-adic field?
Sketch a proof of Dirichlet's theorem on primes in a progression.
What is the residue of the zeta function of a number field? How can this be
used to get a formula for the class number? For which fields can this be done?
Algebraic Geometry:
What is the Jacobian of a curve? (I defined it over C). What is the Jacobian of
P^1? Write down a curve of genus one. I gave y^2=x^3-1 but was not required to
prove that it actually had genus one. What is it's jacobian? Describe the group
law on this curve. Write down a holomorphic differential on it. (Proofs were
not required). Find the points of order two on the curve.
I assumed throughout that he ground field was C. The words "scheme," "sheaf,"
and "cohomology" did not enter the conversation.
The exam lasted just over two hours.