Zhiren Wang's generals.
Committee: Lindenstrauss (chair), Bhargava, Szeftel
Topics: ergodic theory & representation theory
May 1, 2007
REAL
* What's the dual of L^p? How do you prove it? What happens to L^\infty?
* Why are continuous functions of compact support dense in L^p? C_0^\infty functions?
How do you approximate the characteristic function of [0,1] by C_0^\infty?
* Is there a probability measure \mu on [0,1] such that
log\mu([x-\epsilon,x+\epsilon])
lim_\epsilon ------------------------------- > 1.5 almost everywhere?
log\epsilon
(No.)
* State Vitali covering lemma.
COMPLEX
* State and prove Moreira. What condition do you need besides this equality?
* What can it be used to prove? (reflection lemma, passage to limits)
* Gamma function, define it in both product and integral forms and use both to show
\Gamma(n)=(n-1)! .
* What is a conformal map? What condition for a function to be a conformal map? Define a
conformal equivalence between {Re z>0, Im z>0} and the unit disc.
* What can you say about (1/N)\sum_{n=1}^N exp(2\pi i\alpha n^2) when \alpha is irrational?
(this was rather an ergodic question, but they posed it here)
ALGEBRA
* What's the orientation preserving symmetric group of the cube? Why?
(After a long sequence of hints, I finally guessed it's S_4, the permutation group of
the four diagonals.)
* What's S_4? Character table.
* How large is the symmetric group of the regular dodecahedron?
* How can you find a normal subgroup from the list of conjugacy classes?
* Without looking at the group, can you tell if the union of several given conjugacy
classes is a normal subgroup from the character table?(Yes, any normal subgroup is the
intersection of the kernels of several irreducible representations.)
* The Galois group of Q[2^{1/3}]. Can it be extended to a Galois extension? Write down all
the intermediate fields of this Galois extension.
* Do you know how to construct an irreducible polynomial over Q whose Galois group is S_5?
REPRESENTATION THEORY
* Representations of sl(2,C).
* What is a symmetric quadratic form on C^2? (It is Sym^2(V^*) as a representation of
SL(2,C))
* What's Sym^2(Sym^2(V^*))? How does it decompose? How can you find the trivial
representation in it?
(I had no idea. They told me two ways to do this: 1. Sym^2(V^*) is in fact the adjoint
representation sl(2,C) and there is a fixed subspace in Sym^2(Sym^2(V^*)): the one
spanned by the Killing form. 2. It's just the determinant of the symmetric quadratic form.)
* Why can you decompose all these into irreducible representations?
* Say something about Cartan subalgebra and root system.
* List all the 2 dimensional root systems.
ERGODIC THEORY
* Why is a Bernoulli shift ergodic? Mixing? Its entropy?
* What can you say about its factors? (also ergodic and mixing...)
* What systems cannot be a such factors? (I said those who are not ergodic or not mixing
to some laughter.)
* Give a system which is ergodic but not mixing. Prove it.
* Suppose in a system (X,T,\mu), for some function f, \sum_1^N f(T^n(x))-->+\infty
almost everywhere, what can you say? (I failed to figure out this; the correct answer
is that the order of the growth must be linear everywhere.)
* Do you know what joining is? (No, but I know what disjointness is if you want...)
* Ok, what's disjointness?
* Why are a Bernoulli shift and a zero-entropy system disjoint?