Wei Ho May 16, 2005 2pm-4:40pm Manjul Bhargava (chair), Ramin Takloo-Bighash, Edward Nelson Algebraic Number Theory, Representation Theory ----------------------------------------------------------------------- ADVICE/REFLECTIONS: (I'm putting this first, since those of you with different special topics and examiners probably won't read this transcript.) Take the right attitude towards the exam and the requisite studying. You will probably pass, so try to actually learn some math and enjoy studying for it. Don't bother memorizing proofs! (They won't have the patience to sit through a 5-minute proof of some basic fact anyways.) You should know the main steps in most big theorems and have a sense of what results imply what other results, but that's about it. They really are testing your understanding of the material, so they'll ask lots of examples and some random questions that are impossible to practice ahead of time. You won't be able to answer most of your exam off the top of your head. Stay calm during the exam. This was my biggest problem. I got very frustrated for not knowing something or messing something up, and then couldn't think at all. Of course, knowing that you're not supposed to panic doesn't mean you won't, so if you think you're liable to do this, take mock generals with older students to get used to not having the answer. I think that they aren't really interested in a question once it's clear you can solve it, so you will spend the majority of time trying to work out problems that aren't easy for you. Also, if your committee knows that you are pretty comfortable with a particular topic, they may not even ask about it. Picking a committee chair that might want to work with you helps. While it is intimidating to think that your future advisor might not think so well of you after your generals, it's more important that someone on the committee is 'on your side' to some extent; he/she can temporarily steer the conversation to things you know if things are going poorly. And make sure you talk to everyone on your committee before your exam. Just try to relax and get a lot of sleep before your exam :) Good luck! ----------------------------------------------------------------------- I'm sure I've forgotten many things that happened (or rather, blocked them out of my memory), but here's a version of my general exam. Most of the time, they were happy to give hints when I needed them (though even happier when I didn't need them!). I've included some of them, but there were many many more. Throughout, they were all very nice and they tried to create a relaxed atmosphere. I was too nervous in the beginning to realize this, but the second part of the exam, in particular, was more like a discussion than an exam, and I actually didn't mind being there. They asked me to pick the topic I wanted, so I thought I'd just get analysis over with. (bad idea in my case!) REAL ANALYSIS: (all Nelson) N: Talk about absolutely continuous functions and functions of bounded variation. I defined the former and mentioned the fundamental theorem of calculus. He asked for a Lebesgue version of the fundamental theorem that didn't look like the usual calculus one, and I didn't really know what he wanted. He asked me about absolutely continuous measures, and how they related to the functions of the same name... I still wasn't sure where he was going, so I threw out a lot of vaguely related things (Lebesgue decomposition, Radon-Nikodym) but I definitely got the sense that this still wasn't what he was looking for. Eventually we moved on, but by this point, I was completely flustered, and that lasted for most of the real and complex analysis sections... N: If you have a sequence of continuous functions on [0,1] with integrals converging to 0, do they have to converge to a continuous function? (no) Give an example. Does the limit function have to be continuous somewhere? (yes, use the Baire category theorem. I started explaining this in detail, and he cut me off once it seemed clear I knew the answer.) Does a pointwise-convergent sequence of L^1 functions on [0,1] have to have a limit function that's Lebesgue integrable? (no) Give an example. Somewhere in here I said something about nice things happening with monotone or dominated convergence. He didn't seem to care though... There was also a discussion at some point about Riemann integrability. He asked me what made a function Riemann integrable. I first said you needed only countable discontinuities, but he said I was missing something (the obvious part!). He suggested we move on to complex. I wasn't sure if the committee was satisfied or not (I wasn't!), so I tried to indicate that we could do more real, if anyone wanted... COMPLEX ANALYSIS: N: If a power series has radius of convergence 1, can it be continuous on the entire unit circle? Can it be analytic? I didn't process the question at first, and asked him to clarify, but I think it just confused me more. At some point, we left a very awkward discussion, but returned to it at the end, when I was able to think a little better and answered it. As a warning, there was a bit of confusion about 'Taylor's theorem' -- I was referring to the theorem in Ahlfors, but he thought I meant the real-variable version... N: Suppose you have a function analytic on the top half of the unit disk, continuous on the real interval [-1,1]. Can you define an analytic extension to the whole disk? How do you prove it? T: Compute the Fourier transform of e^{-\pi x^2}. (I tried to just say, well, it's basically the Gaussian, but he wanted me to compute it from the definition directly. I didn't remember the computation, but fortunately got it with help.) This also led to a discussion of the Fourier transform, and why one would want to normalize. I said 'inversion formula' immediately, but I think I should have also said something about the isometry for L^2. N: Consider all germs of meromorphic functions at 0. What kind of structure does this have? (C(t), so a field) Is it algebraically closed? Why not? B: Talk about conformal mappings. This led to Schwarz's lemma. (At some point Ramin told us about some generalization of Schwarz's lemma, but I have to admit that I didn't really process any new information since I was trying to prepare myself for the next question.) At this point, they wanted to move on, with the possibility of returning to real and/or complex. (I was hoping they would return, so I could redeem myself.) About one (excruciating) hour had passed at this point. (The rest of the exam flew by!) ALGEBRA: B: Classify groups of order... umm... how about 14. (I started to say something about applying the Sylow theorems, but was immediately told to do it an easier way.) Why do you know that there's a group of order 7? (I said Lagrange's theorem, to some laughter... They told me the correct name, which I have now forgotten!) Are all index-2 subgroups normal? T: What's Cayley's theorem? (again, I didn't know which theorem the name corresponded to, but fortunately they told me it had something to do with permutation representations...) Give an example where you don't need to use S_n for an order-n group. Give an example where you do. (I said Klein four group, since that's the first thing that popped into my head, though he was looking for the more obvious answer of Z/pZ) B: What is an integral domain? Prove that all finite integral domains are fields. T: Let's do some representation theory (finite groups). Tell me about the representations of D_4. (I essentially wrote down the character table, which I had worked out pretty recently, and was periodically asked to slow down and explain what I was doing.) What is the representation that corresponds to the second line of your character table (a nontrivial 1-dim rep)? Oh wait, that's too obvious. What is the 2-dimensional representation? How can it be interpreted geometrically? T: What's an induced representation? Write down the definition. I don't like your definition; do you know another? (no...) Well, here's a better one (in terms of functions). Now why are these equivalent? T: Does induction preserve irreducibility of a representation? (no, e.g. inducing a restricted rep gives a direct sum of the original rep.) What can you say about the kernels of the irreducible representations? (normal subgroups) How about kernels of direct sums of irreducibles? What kind of functor is induction (e.g. left-exact or right-exact)? B: We should do some Galois theory... What's a splitting field? Give an example of something that's not. What's its Galois closure? Draw its lattice of subfields. (Since I chose the easiest example of Q(cuberoot(2)), this came back during number theory of course...) ALGEBRAIC NUMBER THEORY: B: Do you know the ring of integers of Q(cuberoot(2)) offhand? Given that it's Z(cuberoot(2)), what's the discriminant? Which primes ramify? Wait, what happens at infinity? (Oops, forgot about it) What happens to the prime 5? What proportion of the other primes decompose like that? What proportion of primes split completely? What's left? T: (in the middle of the discussion above) What theorem are you using? State Chebotarev. (I asked if they wanted me to define things like Artin symbol, but they declined.) T: What classical theorem... (I didn't even listen to the rest of the question to know that he wanted Dirichlet's theorem on arithmetic progressions. I offered to prove it via class field theory but they weren't that interested in the details.) B: What's the key step? Why doesn't L(1,\chi) vanish? (I gave the same answer Manjul had given in his generals (!), that the product over all \chi of L(1,\chi) is essentially the Dedekind zeta function. But Ramin raised some objection about how that might only work for even characters, so I also gave the class field theory reason.) B: What's a Hilbert class field? What's the Hilbert class field of Q? Why? T: Here's a crazy question. What's Spec of the adeles? Wait, do you know what the adeles are? (yes. But he didn't ask me to define them.) Ok, then take 30 seconds to think about it and give me your best guess. They were happy with number theory, even though I was surprised that I wasn't asked to compute a class number, prove the units' theorem, or state the main theorems of class field theory. REPRESENTATION THEORY (Lie algebras): B: Talk about the representations of sl_2, say over C first. (I immediately said that the representations all looked like wedge^k V, for V the standard representation. Oops! He asked a few very careful questions, until I realized that I had meant Sym...) What's a root? What's a weight? B: What are the weights of Sym^k V? (I said something stupid again, that the weights were k to -k with everything in between. Oops again!) T: Can you decompose Sym^2 (Sym^2 V)? B: What is the trivial representation in there? T: Describe the Weyl unitary trick. How do you know there's a compact group associated with your complex Lie group? (construct the split form and take the invariants of the conjugate linear to obtain the compact form) Take the standard representation of su_2. What happens when you tensor with C to get sl_2? How about for other representations? Wait, this question is too easy... T: What does SU2 look like algebraically? What does it look like geometrically? (S^3) Why? (I didn't know why, so he made me write it down and actually do the computation.) The determinant condition doesn't quite define a sphere, so what's wrong? (I chose a bad embedding) B: What is a root system? Where does the 2(b,a)/(a,a) come from, in terms of Lie algebras? (I commented that I always got the beta and alpha mixed up when writing down the reflection formula, and they said that they did too. I felt better.) B: What are all the root systems of rank 2? (I started drawing Dynkin diagrams, so...) What is a Dynkin diagram? I drew the rank-2 Dynkin diagrams and root systems, and named the corresponding Lie algebras. They were pleased that I knew that sp_4 was isomorphic to so_5, which was apparently crucial in Ramin's Ph.D. thesis. And they were amused that I kept forgetting to fill in the root system, since I was drawing the simple roots off the top of my head. But in doing this without working any details out, I mixed up the lengths for sp_4, so I asked if it was ok if I worked out the roots on the side. They said that I probably should've done that first anyways! Luckily I managed to fix it... T: How do you get the Lie algebra from the Lie group? For an infinite dimensional representation, what could go wrong for a representation of the group when going to the algebra? Like if it's not smooth? (I had a lot of trouble with this, especially since it turned into a real analysis question when we tried a concrete example.) We agreed to go back to complex since they had also noticed how nervous I was in the beginning, and we tied up some questions that I hadn't finished earlier (and this time around, they seemed so obvious!). I still felt unsatisfied with my analysis performance, but they asked me to leave. A few minutes later, the door opened and they congratulated me!