Jun Su’s Generals (Jan 21, 2015, 1:00PM-3:10PM)
Special Topics: Algebraic Number Theory, Algebraic Geometry
Committee: R. Taylor (Chair), A. Ionescu, Z. Patakfalvi
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There might be some questions that I’ve forgotten. Moreover, the exam was actually filled with hints and I‘ve only included a few of them here.
Questions were asked by:
[T] - Taylor
[I]- Ionescu
[P] - Patakfalvi
At the beginning the professors asked what I would like to do first. I said analysis (but now I think starting with a subject which you are familiar with and often talk about with other people would make you feel less nervous, especially for non-native English speakers…) Professor Ionescu said it seemed that’s his job.
Real Analysis
[I] - State Egorov’ theorem. Can you think of an example for which we can use this theorem?
(I thought for a while and came up with nothing. So Professor Ionescu said we can come back to this later.)
[I] - State dominated convergence theorem.
[I] - Talk about L^1(\mathbb{R}^n).
(I mentioned completeness and was asked to prove that. Then I mentioned separability and was asked to prove that smooth functions with compact support are dense in L^1. I used convolution but didn’t figure out all the details but it seemed the professor didn’t mind.)
[I] Talk about the Fourier transform.
(I defined the Fourier transform on L^1(\mathbb{R}) and described few properties of the image, and then state the Plancherel’s theorem to extend the definition to L^2. Then I was asked to prove the inversion formula and one more time I was not required to figure out all the details but just reminded that knowing the Fourier transform of the Gaussian is important in the proof.)
Then Professor Ionescu said (for me) algebra is more important. So Professors Taylor asked me which would I prefer, linear algebra or finite group theory? I chose the later.
Algebra
[T] Suppose that you are given a finite simple group of order 168. How many elements of order 7 are there?
[T] How many conjugacy classes do them form?
(I didn’t manage to see the answer immediately, so I followed Professor Taylor’s hint to look at the normalizer of a 3-sylow subgroup, and showed in this way the the normalizer of a 7-sylow cannot be commutative.)
Then Professor Taylor asked Professor Patakfalvi if he had any algebra questions.
[P] Talk about (finite purely) inseparable extensions.
(At that time I was a too nervous to realize that I could mention that these extensions can be obtained by adding p^n-th roots of elements. So Professor Patakfalvi asked a few more specific questions such as how the minimal polynomial of an element in the extension looks like and whether these extensions are always simple.)
Then Professor Taylor allowed me to choose between the two special topics. I chose to take algebraic number theory first.
Algebraic number theory
[T] Talk about the field \mathbb{Q}(\sqrt{10}): calculate the class group, the group of units…
(I calculated the Minikowski’s bound and looked at the factorization of 2 and 3, but I was too nervous to write down the factorization of 3 explicitly. Then I followed Professor Taylor’s hint and realized that the fact that 6 is a norm implies that the product of a factor of 2 and a factor of 3 is always principal, so the class group is isomorphic to \mathbb{Z}/2. And then, still following hints, I looked at positive units greater than 1 and saw that 3+\sqrt{10} is a fundamental unit.)
[T] Consider the maximal abelian extension unramified at all finite places.
(I was not sure how big the difference between this field and the Hilbert class field is and made a incorrect guess at first. Professor Taylor asked me to write down the main theorem of global class field theory-and then he came to the blackboard and led me to go further! I realized that the Galois group of the maximal abelian extension unramified at all finite places over the Hilbert class field is isomorphic to the quotient of the elementary 2-group generated by real places modulo the image of the group of units, and in this case it’s trivial. So for \mathbb{Q}(\sqrt{10}), the Hilbert class field equals the maximal abelian extension unramified at all finite place, which is actually \mathbb{Q}(\sqrt{5}, \sqrt{10}).)
[T] Now consider the maximal abelian extension unramified at all finite places other than 3.
(I made another incorrect guess here by writing a such extension which is still abelian over \mathbb{Q}… Usually I draw commutative diagrams for these kinds of problems but I didn’t do so at that time so I was whether I should calculate some kernel or some cockerel… Perhaps seeing that, Professor Taylor came to the blackboard again to help me. After the exam I drew a commutative diagram and saw clearly that the Galois group of this field over the the maximal abelian extension unramified at all finite places is isomorphic to \mathbb{Z}_3\times\mathbb{Z}_3 modulo the image of totally positive units, which should be \mathbb{Z}/2\times\mathbb{Z}_3.)
Then Professor Taylor gave me a standard question to end this section:
[T] What’s the Chebotarev's density theorem? What the density of those primes modulo which 2 has a cubic root?
Algebraic Geometry
[P] Consider vector bundles over the projective line over an algebraically closed field, what can you say?
(I said they are direct sums of line bundles. So Professor Patakfalvi continued to ask:)
[P] Is every short exact sequence of vector bundles split?
(I answered “No” immediately, and tried to get an counterexample from my memory. But what I wrote down was \mathcal{O}(2)\rightarrow\mathcal{O}(1)\oplus\mathcal{O}(1), and it looked strange and I soon realized such a map must be zero… But professor Patakfalvi encouraged me "That’s already very close.” and asked me when there exists a nonzero map from \mathcal{O}(i) to \mathcal{O}(j). The I wrote down \mathcal{O}(1)\oplus\mathcal{O}(1)\rightarrow\mathcal{O}(2).)
[P] Suppose you have such a surjectivemorphism. What can you say?
(I claimed that the kernel is a line bundle, but only came up with PID arguments which only work in 1-dimensional case. But professor Patakfalvi smiled and said “That’s a perfect proof for our case.”)
[P] Then how do you know if there is such a surjectivemorphism? How do you tell if such a morphism is surjective?
(I constructed a surjectivemorphism explicitly, but didn’t manage to answer the second question in a concise way. I think I could have said (in this case) “the images of global sections generate the sheaf”.)
[P] Now that you know the kernel is a line bundle, what it is?
(I came up with calculating the Euler characteristic and showed it’s \mathcal{O}.)
Finally they asked Professor Ionescu “Don’t you want to ask some complex analysis questions?” So...
Complex Analysis
[I] Suppose you have a holomorphic function on the complex plane, bounded by |z|^2. What can you say?
[I] How do you prove the Picard’s little theorem?
(I gave a proof based on the universal cover of \mathbb{C}-{0,1}. It seemed Professor Ionescu wanted to lead me to find another proof, but I didn’t get it…)
[I] One last question: do you know Hadamard’sthree circle theorem? Is the condition that the function is bounded on the strip region necessary?
(After a few failed attempts, I arrived at the counterexample f(z)=exp(exp(iz) on {z: -\pi/2