Simon Marshall's Generals
May 2007
Committee: Sarnak (chair), Bhargava, Fefferman
Special topics: Lie groups and symmetric spaces, number theory
Most questions were asked by Sarnak, but I can't remember exactly who asked what.
The answers the committee were looking for involved main ideas but few details, and
I was able to stop and ask for help when I couldn't remember something.
Algebra
Describe a group representation and characters. Why is the character table a square?
What are the representations of S_4?
What is a variety? When is k[X_1,?,X_n] a PID or UFD? What is the nullstellensatz?
What does it mean for a variety to be irreducible, and why is the notion important?
I spent a little while groping around for places where irreducibility is significant.
Sarnak eventually steered me towards looking at the number of points on the curve when
you reduce mod p, got me to write down the number for the example of a reducible
variety I had given (xy = 0), and talk about the number on the irreducible example
(x2 = y3+5). After mentioning the Hasse bound and Jacobi sums, we finished by
observing that the number of irreducible components roughly determines how many
points mod p there are.
When can you solve e^B = A for a given complex matrix A? Why can't you always do this
for real A and B, and give a counterexample.
Describe rational normal form for a matrix in terms of modules over the PID k[X].
Analysis
What does analytic mean? Suppose you have an analytic function in a semicircle whose
absolute value is bounded by 1 on the curved edge and 2 on the straight edge. What
bound can you give for the function inside? I said it was bounded by the harmonic
function in the semicircle with the same boundary conditions. Can you write down this
harmonic function? I said I could conformally map the semicircle onto something
simpler and use the fact being a harmonic function is conformally invariant, but Sarnak
said I could do something much simpler from high school geometry. After floundering for
a bit, he said to look at the angle a point inside the semicircle subtends at the
diameter, and I showed this gives the function we wanted.
What are the rank, order and genus of an entire function? What is the order of the
zeta function? Why does it have to have infinitely many zeros? How do you bound it
in the critical strip? Give an elementary bound for the sum (-1)^n/n^s using summation
by parts on the line Re(s) = 1/2.
Lie Theory
Prove the Peter-Weyl theorem. Talk about compact Lie groups, the tori, restriction of
representations, weights, and the Weyl integral formula. What are the characters of all
irreducible representations of SO(3)? Describe them in terms of spherical harmonics or
polynomials. Is SO(3) simply connected? What is its universal cover?
What is the Iwasawa decomposition? What piece of undergraduate linear algebra is it related
to? What is the volume form on GL(2,R)?
What is a Riemannian manifold? A Riemannian globally symmetric space? Show that the
isometries of such a space M are transitive, so set-theoretically we have M = G/K for G and
K Lie groups. Do we see the differential structure of M this way? Why do the isometries of
M form a Lie group?
Suppose g is a semisimple Lie algebra. What is a Cartan involution of g? How does this relate
to the global picture of M as G/K? What is the rank, in terms of the manifold M and algebra g?
What is the rank of hyperbolic space? What two groups is H_3 the quotient of? I wrote
SO(3,1)/SO(3), and gave the Lie algebra of SO(3,1) with a description of the Cartan involution
and it's two eigenspaces.
Number Theory
Manjul began by asking how x4+1 reduces mod p, and if it can ever split completely or be
irreducible. I was a bit vague, so Peter asked the question for a general polynomial f.
I said you could show there were infinitely many primes which split in the extension generated
by a root of f, either by looking at the zeta function of the field and showing it had no pole,
or directly by taking primes dividing f(n) for n an integer. He then asked if there were
primes which were inert in the splitting field of (x2-2)(x2-3) and I said there were no
unramified ones, because the Frobenius would generate the Galois group but that was impossible
as Z/2+Z/2 wasn?t cyclic. I also showed it directly by saying that if 2 and 3 were both
nonresidues mod p, then 6 was. Manjul asked the question about primes being inert in the
splitting field of x4+1 again, and I used the same argument by noting that the multiplicative
group of primitive 8th roots of unity isn?t cyclic. Peter finished by asking me about Chebotarev.