Laszlo Erdos
Generals questions
April 29, 1991 1:30 - 3:40
Committee: E. Lieb, J Mather, E. Nelson.
Special topics: 1, Functional analysis
2, Statistical mechanics, ergodic theory.
ALGEBRA:
Structure theorem of finitely generated Abelian groups. Modules over
p.i.d. Why is a torsion free module free module (not using the
structure theorem)? Proof of this last statement.
How many automorphisms does the complex field have? How to extend
a simple automorphism (e.g. sqrt(2) -> -sqrt(2) ) of an algebraic
field into C. How to extend a general subfield isomorphism?
What feature of C allows it to do? Plus plenty of similar subquestions
(from Nelson) since I was not quite sure about the subject. I
mentioned the transfinit induction, but I forgot that before it
I should extend to the algebraic closure. How to characterize
the extensions. How can it happen that a subfield of C is isomorphic
to C.
COMPLEX:
Vitali convergence theorem. Montel theorem. Uniqueness theorem (all with
proof)
What do you know about several complex variables (I told
polydiscs, Cauchy formula, domain of holomorphy, extension
from a polydisc with a hole into the whole polydisc, and that
seemed enough. I mentioned proofs where I had some valuable ideas,
but they did not require).
REAL ANALYSIS:
Take f in L1(R). Let g(y) = integral (from -infty to infty) of
exp(-iyx^2)f(x)dx (so not the Fourier transform). What can
you say about it? I had to prove the existence, continuity,
and that g tends to 0 as y tends to infinity.
Is it enough to prove the theorem for a dense subset of L1?
Take f in Lp, g in Lq (not conjugate exponents). What
can you say about f*g (convolution) (Young inequality).
What if they are conjugate exponents? I had to prove
that f*g is continuous and tends to zero at the infinity.
Suppose E has a positive Lebesgue measure. Prove that
E-E (as a set) contains interval.
FUNCTIONAL ANALYSIS:
Take a sequence of functions in H1 (Define H1, topology, norm).
Suppose they weakly converge (definition). Does it follow that
the limit of the derivative is the derivative of the limit (proof).
Long discussion about the notion of weak convergence. Banach-
Alaoglu theorem. What topics do I know about functional analysis
(I listed something, they didn't ask anything)?
STAT. MECH. ERGODIC THEORY:
What is a Bernoulli shift. Measure, sigma algebra. Entropy.
How to define the entropy in general. Kolmogorov-Sinai's theorem
on generating partitions. Kolmogorov-Ornstein theorem about
the isomorphism of Bernoulli shifts. (No proof.)
What is the free energy of a lattice gas. Define the thermodynamic
limit. When does it exists. Sketch the proof. What is the
dimension of the Boltzman constant.
I always tried to say something, even if it was vaguely connected
to the question. Seemingly it worked. One must pay very much attention
to the professors' helping questions, remarks. If you can catch
the idea before they finished the sentence, it seems as you
found out the half of it. Mention everything what you know,
even if it is not perfectly relevant.
One more remark: the real and functional analysis questions
were mixed, both were asked basically by E. Lieb and therefore
there was not a definite borderline between them.