Dan Grossman's generals
Gunning (chair), Hewitt, Mather
Compact Riemann surfaces, Differential geometry
COMPACT RIEMANN SURFACES (All asked by Gunning)
State and explain (don't prove) the Riemann-Roch theorem.
(I gave the version using l(D), no line bundles.)
What does it mean for g=0? for g=1?
for effective divisors in these cases?
What does it mean for a divisor D=r.p (p a point, r>0)?
what is the generic (wrt p) situation for r=1?
(this shows that the set of holomorphic differentials
do NOT have a common zero; all points are generic)
what is the generic situation for r=2?
State and explain (don't prove) Abel's Theorem.
State and explain (don't prove) uniformization.
How do you know that for genus g>1, the covering is the disc?
ALGEBRA (All asked by Hewitt)
Which finite groups are Galois groups for some field extension?
(All.)
Now let's work our way towards a characterization of n such that
a regular n-gon is constructible.
(This goes on for some time; one derives the necessary condition
(n is a product of a power of two and some Mersenne (sp?) primes)
and then proves that it is sufficient using some group theory. I
am prompted helpfully all the way.)
State the structure theorem for f.g. modules over a PID,
describe application to a linear operator on a f.d. vector space,
and then further application if the field is algebraically closed
(ie, Jordan canonical form).
How do you know that an operator satisfies its characteristic
polynomial? What is the name of this theorem? (Cayley-Hamilton)
How would you prove the Fundamental Theorem of Algebra?
(Liouville's Theorem, giving a nice transition into...)
COMPLEX ANALYSIS (All asked by Gunning)
State (don't prove) the Riemann mapping theorem.
What happens at the boundary?
(Well, that depends.)
Okay, say in the case of a polygon mapping to the half-plane.
(Schwarz-reflect across the sides to extend.)
State the Schwarz reflection principle.
What does it look like at the corners?
(...messy?...)
Okay, say it's an equilateral triangle.
(Boy do I feel stupid. z->z^3.)
Suppose you want an entire function with prescribed zeros.
What condition is there on the zeros?
How do you define it in this case?
(Canonical products, of course).
What is the condition for [finite genus]?
(I worked this out by example, with the Gamma and P-functions,
and didn't have to prove the general statement).
Which meromorphic functions (on some open plane set) have
meromorphic antiderivatives?
REAL ANALYSIS (Mather and Gunning)
(Mather)
What is the Lebesgue measure of the middle-thirds Cantor set?
Give an example of a strictly increasing function with derivative
zero almost everywhere? (uh-oh, I have no idea) okay, not strictly?
(Cantor-Lebesgue function.)
State the fundamental theorem of calculus.
Give an example of a function differentiable at every point of [0,1]
with derivative NOT in L^1.
(Gunning)
What is a Hilbert space? Give examples.
How would you prove that L^2(measure) is a Hilbert space?
(I don't even get to prove it; I just say I would prove it the
same way I would prove it for L^p for all p>=1; Gunning says,
"Hey, those aren't Hilbert spaces!" and I say, "Blah blah blah.")
Is L^{1/2} a Hilbert space? (I return Gunning's complaint.)
Is it complete.
(Well, my proof won't work for p<1, because it uses Holder's ineq.,
but I don't really know...)
DIFFERENTIAL GEOMETRY (Mather)
Define Gauss curvature.
(For a surface in E^3, please?)
Yes.
(I annoy him by saying it is the sectional curvature.)
Define sectional curvature.
Define Riemannian curvature.
Define Riemannian connection. Don't say it, write it out.
NOW what is Gauss curvature, in terms of curves in the surface?
Do you know what an osculating circle is? (Yeah, right.)
Gauss-Bonet Theorem for surfaces in E^3 (with proof, but he stops
me before the end because he is lost).
Describe a compact surface in E^3 having negative Gauss
curvature. (I have no idea.) That's okay, never mind.
(At this point Gunning's efforts to stop Mather succeed,
they send me out, I run to the water fountain (bring a TALL glass
of water with you), and return to be told I've passed. They were
very nice, and didn't seem to care about my knowing any particular
thing, as long as they could eventually discover something I did
know.)