Manjul's generals Committee: Conway, Wiles, and Fefferman Special Topics: Algebraic Number Theory & Representation Theory ------- C: "So who is the chair of this committee?" F (scratching head): "I think it's one of you two." W: "Is it me?" C: "I don't know... Perhaps it's me." W: "How will this be determined?" F: "Does it matter??" C: "Let me find out from the office downstairs." (Just as he was about to leave, I informed Prof. Conway he was chair. A similar discussion then ensued as to what my special topics were!! It was quite entertaining. But at last, we were all ready to begin.) C: "So who wants to ask the first question?" (Everyone kept looking at Conway. (He was chair after all.) ) C: "Hey, why is everyone looking at me?!?" (Another entertaining dispute followed. Finally, Prof. Fefferman agreed to start.) COMPLEX ANALYSIS: F: What is the Riemann mapping theorem? Why isn't the complex plane conformally equivalent to the unit disk? Why are all automorphisms of the unit disk linear fractional transformations? What can you say about conformal mapping of multiply connected regions? When are two annuli conformally equivalent? Prove it. How would you integrate $\int_{-\infty}^\infty \frac{dx}{(x^4+1)^2}$? (concepts, not calculations) C: Classify the types of singularities. Describe each. How does a function behave near an essential singularity? (it goes crazy) In what sense? (big picard) Talk about the singularity log has at 0. (This led into a discussion of Riemann surfaces... Things like: "Define Riemann surface." "Define branch point." "Define sheet." "What happens when you wind around a branch point?") F (to C + W): Well, I think I'm quite satisfied with complex analysis. Do you agree? W: Oh yes, definitely. C: Yes, indeed. Shall we move onto real? (Agreement. Uh oh...) REAL ANALYSIS: F: Given a sequence of functions converging pointwise, when does the limit of their integrals converge to the integral of their limit? C: Given a sequence of functions converging pointwise, when does the limit of their derivatives converge to the derivative of their limit? What is Fatou's Lemma? (There were a couple more questions, but I've forgotten what they were, I'm afraid. Then the motion was made to move on to algebra. There was universal agreement, even from me. Prof. Wiles was nominated to ask the first question, as he had been rather silent until now. Wiles declined.) ALGEBRA: C: OK, I'll get us started. Describe Galois theory! W (with a smile): Is Q(cube root of 2) normal? What is its splitting field? What is its Galois group? Draw the lattice of subfields. C: And draw the corresponding lattice of subgroups too, upside down. W: Classify finite fields. Why do they exist? Why unique? Why are finite field extensions separable? What is the structure of the multiplicative group of a finite field? Prove it. C: How many irreducible polynomials of degree six are there over F_2? (9) C: Tell me a condition on the Galois field which is implied by irreducibility of the polynomial. ("Huh ?" He repeated the question. I wondered whether he might be asking about the Galois group, so I mentioned it must act transitively on the roots.) Oh yes, yes, that is what I meant to ask. (Embarrassment.) I .. er.. Sorry.. about ..that... I h..aven't b..een getting mu..ch sleep.. lately... (Yay! I wasn't the only one...) F (in a very comforting voice): Oh, don't worry, John! You are doing just fine. There's no need to be nervous... (laughter) C: Thanks... Ok, state the Sylow theorems. Classify groups of order 35. Why can you conclude that the group is the direct product of its two Sylow subgroups? (they commute) Do any two normal subgroups commute? (no; they do if they intersect trivially) C: Do you know what the quaternion group is? How many elements are there of each order? Suppose I have a field extension of the rationals with Galois group the quaternion group. How many quadratic extensions does it contain? (3) Can any of them be imaginary? (I started to think... he interrupted.) Oh, I don't expect you to answer that right away. We'll work through it together. (He gave me a hint, and suddenly I understood what he was getting at, and completed the proof that none of the quadratic subfields can be imaginary. He seemed pleased.) (I thought about it another second, then said, wow, neat problem!) C: Hey! You are in no position to be complementing me! (laughter; slowly moving into some representation theory) Speaking of the quaternion group, construct its character table. (He offered help with the last row, but I politely declined, not understanding what he was talking about. Luckily, I was able to finish without any trouble.) In the two dimensional character you just wrote down, what matrix does the group element "-1" correspond to? F: Is the character matrix of a finite group always square? Why? C: What can you determine about a group just by looking at its character table? (normal subgroups, abelian quotients, ...) If a character table of a group has all real entries, what can you conclude? (all elements are conjugate to their inverses) Ok, suppose we take the character table of a finite group, and adjoin all its elements to the rationals? What can you say about the resulting field extension? (it's abelian...) Say I now apply an element of the Galois group to the character matrix. What happens? (it permutes the rows) Does it permute the columns? (He gave me a hint, which made me realize that the answer, remarkably, is yes.) Now let's talk about the case of complex conjugation. What does it do? (switches rows in pairs, and switches columns in pairs) Is the number of row switches the same as the number of column switches? (where does he get these questions!?!) (he gave me another hint, and i realized that, again, the answer is yes! this was really neat! but i refrained from complementing him this time...) (We were all getting really interested; what could be said for general Galois automorphisms? Conway explained that although there is a lot to say about this, the answer in general is that, unfortunately, the results don't extend very well.) C (to W): Anyway, we should be getting on to some algebraic number theory. Why don't you start? (Wiles accepted this time.) (but more rep. theory later!) ALGEBRAIC NUMBER THEORY: W (with a smile): What is the Artin map? Define decomposition group. Please write it on the board. (All my answers so far--except for the subfield lattice and the character table-- had been completely verbal. So I think he wanted to find out once and for all whether I knew how to write!) Define inertia group. (I stopped writing again.) What does the Artin map say in the case of a cyclotomic extension? (I said what it does; I mentioned Chebatorev density implying Dirichlet's theorem in this context.) C: Prove that the cyclotomic polynomial is irreducible. C: Discuss the splitting of primes in the ring Q(sqrt(-3)). (I happened to remember when -3 is a square off-hand, so I just stated the answer.) C + W: Hey, no fair! We want you to derive the answer, using quadratic reciprocity! (oh... Oops.) C: You mentioned the Chebatorev Density Theorem earlier, which by the way I think was irrelevant to Andrew's question. What does it say? (hey!! was it really that irrelevant?? I asked Wiles if he thought it was, but he laughed and chose not to enter the dispute.) W: How would you prove Dirichlet's theorem on primes in an A.P.? (I didn't know where to start, so I started from the beginning, defining L-functions, and reducing the problem to the nonvanishing of L(1,\xi). I had to use the blackboard for this one.) Why doesn't L(1,\xi) vanish? (because if you take the product of L(1,\xi) over all nontrivial \xi, you essentially get the zeta function of the cyclotomic field... He seemed to like that answer, so decided to up the level a bit.) W: Do you know about non-abelian analogues of these L-functions? (...) How do you show they are holomorphic? (use Brauer's theorem...) (At this point he decided to up the level a little more. he was toying with me!) W: [some crazy question about Artin representations that I can't even remember] (I have no idea! Wiles laughed. He proceeded to tell us some very interesting things about them...) REPRESENTATION THEORY: W: So what is Brauer's theorem? (hadn't actually stated it before) (the lattice of virtual representations is generated by characters induced from elementary subgroups) C: What's an elementary subgroup? W: Say, by the way: what did you mean by representation theory? Finite groups? Lie groups? Lie algebras? (I informed him I had actually meant Lie algebras!!) (Conway therefore proceeded to ask another question having absolutely nothing to do with Lie algebras.) C: What is Maschke's Theorem? Can you prove it? (the group algebra of a finite group over a field of characteristic p is semisimple iff p doesn't divide the order of the group. gave a quick proof. Conway informed me afterwards that I had actually given him a generalization of Maschke--the original theorem was for characteristic 0.) Well, does anyone want to ask any other questions? (F + W indicated they were satisfied.) (what happened to Lie algebras?? of course, i didn't complain. :) ) (We adjourned.) (The examiners were really pleasant, and so wonderfully polite, both to me and to each other! The exam lasted only about 1 hr 20 minutes, but it was really intense (once it began)! ) (Oh, do be careful to state all the hypotheses correctly when you are stating a result--they were very picky about this with me, and would make me repeat things if I ever tried to sweep anything under the rug!) (Good luck!)