Peter Varju's generals (May 15, 2008)
Topics: Harmonic analysis, Ergodic theory
Committee: E. Lindenstrauss, S. Klainerman, R. Gunning
Duration: 2 hours
Real analysis:
L: Define the Lebesgue measure on the [0,1] interval.
Me: I started to give the definition through the extension
theorems, but was asked to give another definition.
It turned out that he meant the inf of the total length of
intervals needed to cover the set.
L: Now prove that [0,1] is not the union of finitely many
sets of measure 0.
Me: I gave a proof.
L: State and prove Lebesgue's monotone convergence.
Me: First I wrote the inequality in Fatou's lemma in
the opposite way, and I was corrected.
Then I had to give several counterexamples to show
the necessity of the conditions in Fatou's lemma
and Lebesgue monotone convergence theorem.
L: Baire categories?
Me: I stated the theorem.
K: Can you give an example of an application.
Me: Banach-Steinhaus.
Algebra:
G: Abelian groups of order 9?
Me: C_3xC_3 and C_9.
G: Can you prove that they are not isomorphic?
Me: C_9 has an element of order 9.
G: Groups of order 9?
Me: I proved that they are commutative.
G: 27?
Me: I could only say that if it isn't commutative,
then its center is of order 3.
G: 12?
Me: I gave the information one can obtain from
the Sylow theorems.
G: What is the Jordan canonical form.
Me: First I stated it over the complex numbers,
than over general fields. I was told that the
second one is called the rational canonical form.
K: What can you say, if the matrix is orthogonal or
symmetric?
Me: Then diagonalizable.
Complex analysis:
G: Classify singularities.
Me: I had some problems proving that if the function
is bounded, then the singularity is removable. First
I tried to do it with Morera, so I needed continuity.
Then i got a hint and finnished the proof (with the
Cauchy integral formula).
G: Riemann mapping theorem? What happens at the boundary
in the case of a polygon?
Me: I stated the theorem, and proved the extendibility
to the boundary by the Schwartz reflection principle.
G: What is your favourite non-elementary function?
Me: I gave the definition of the gamma function
via the integral formula, and proved the recursive formula.
Ergodic theory:
I don't remember much of the details, but first
I had to talk about the pointwise ergodic theorem,
then about entropy.
Harmonic analysis.
I got questions about singular integrals and the multiplier
theorems.