Runpu Zong
Committee members: Professor János Kollár (Chair), Professor Weinan E,
Professor Michael Aizenman
(1:00pm-3:15 pm, May 11th, 2011)
Some of the questions are vague due to my bad memory, sorry for
this:-)
Real analysis
By Professor Michael Aizenman:
1. Some questions about absolute continuous functions,
differentiation, e.g. commutativity of limit and differentiation.
2. Some questions about convex function (I failed and I cannot
remember)
3. State and prove Lebesgue Density Theorem(the density function of a
measurable set is a.s equal to its characteristic function, proof see:
page 21 of the online note of professor McMullen:
http://www.math.harvard.edu/~ctm/home/text/class/harvard/212a/03/html/index.html)
Complex analysis
1. State and prove schwartz lemma
2. State the geometric interpretation of schwartz lemma (comparison of
the curvature of the hyperbolic metric of the unit ball in C and its
pull back after the conformal map)
3. state and prove Poisson's formula(seperate out the real part and
imaginary parts of cauchy formula)
Mathematical Physics
by Professor Weinan E(essentially some basic PDE and the poincare
inequality of chapter 5 of evans' book)
1, write down the heat kernal, derive the t->0 and t-> infinity
behavior of a heat kernal representation of a solution, with given
innitial condition.
2. Use Variational Method to get the equation \delta u + f =0, with
boudary condition, f|_{boundary}=g
3. Guess a form of the lagrangian which will give the same equation
with another more boudary condition, \gradient f|_{boundary}=v, use
variational principle to derive it(guess the form by Green's formula,
if you don't remember Green's formula, the professor will help you to
derive it out by yourself)
4. Guess a sufficient condition for the sovability of this system of
equation (a integral formula that make g and v compatible), prove it
(by a cauchy-schwarts inequality argument, without the integral
formula condition, one will get an infinity in one side bounded by a
finity number another side, contradiction)
by Professor Michael Aizenman: 5. (by Professor Michael Aizenman)
State your favorite theorem in quantum field theory (I said the
non-renomalization theorem of super-symmetric theory which I just
learned in the course of physics department I attended)
Algebra
By professor János Kollár
1. Write down two successive Galois Extension which is not Galois
after composition (add square of 2 and forth-square of 2 to Q is
enough)
2. State and prove Weistrass Preparation theorem(I failed this
problem)
3. State and prove that polynomial ring over a UFD is again UFD (Gauss
Lemma)
Algebraic Geometry
By professor János Kollár
1. When is the function ring of an affine curve UFD (if and only if it
is a rational curve, using Jacobian of Curves and some well-known
exact sequence relate class groups of open variety to its completion,
if class group is zero, then jacobian should also be zero, hence the
curve is genus 0)
2. Deduce that all cubic surface are blow up of six points of
\mathbb{P}^2 ( I begin with the existence of 27 lines, but failed in
the middle of argument, finally I can only prove that generic cubic
surface are of such kind by dimension count, for reference to this
problem, see: Professor J .Kollár, A. Corti, and K. Smith's book
"rational and nearly rational varieties" which contain a specific
session on cubic surface, or the last chapter of GHT 52, Hartshorne's
"Algebraic Geometry" )
Comments: The professors are super nice in the whole exam, just ask
them when you are not clear with definitions, and just state your
ideas if you are not technically assure of your argument, thery will
always appreciate it as long as you show your potential to solve the
problem, good luck!!!