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?