Josh Greene vs. The Generals Committee: Benny Sudakov (chair), Zoltan Szabo, Natasa Pavlovic Special Topics: Algebraic Combinatorics and Algebraic Topology April 22, 2005, 1:30-3:40 PM ------------------ ALGEBRAIC TOPOLOGY (Zoltan) What books did you study? (I didn't want to be responsible for too much, so I just said Hatcher and Bott + Tu.) Suppose that you have a space that fibers over a sphere with spherical fiber. What can you say about its homology? (My first thought was to use spectral sequences, but I was surprised that would arise in the first question on my exam. So I mumbled on about the LES in homotopy of a fibration and how you could use Hurewicz to recover the first non-zero homology group from that. Zoltan sat non-plussed, so I did end up using the Serre spectral sequence, which might have been overkill, but it turned out to be useful in thinking about the follow-up questions. I also finally realized that the Hopf fibration would be useful to mention here and did so, hence...) Now let's specialize to the case of base S^2 and fiber S^1. Write down the Hopf fibration. (OK.) Can you give an example (of a space as in the first question) with non-trivial homology in dimension 1? With homology Z/p in dimension 1? (I figured out that a suitable lens space will do.) How do you classify S^1 bundles over a given space? (I declared that I all I knew was line bundles were classified by homotopy classes of maps into RP^infty. He mentioned that CP^infty was the relevant space here, but what he was looking for was for me to say "the second cohomology group." Which makes sense -- that's the same as homotopy classes of maps into CP^infty = K(Z,2). And now I understand why lens spaces give all the examples in response to the previous question.) What is CP^infty? Discuss its homotopy and co/homology. (I mentioned it was a K(Z,2), and that Hurewicz gave the first non-vanishing co/homology groups. I thought I was going to have to use spectral sequences again to deduce its cohomology since I was in the K(G,n) frame of mind, but suddenly the right neurons fired and I remembered the story is much simpler in this case.) How about RP^infty? (Similar. I computed the boundary map in homology explicitly, talking about degrees of maps. Therefore...) What is the degree of a mapping? (First from a sphere to itself, then to arbitrary manifolds. I had to state all the typical assumptions -- smooth, oriented, closed...) What is Poincare duality? (OK.) What if the space is not closed or orientable? (You can manage.) What is the Euler characteristic of a 3-manifold? (The alternating sum of ranks was my first response. Then Zoltan asked it again, I realized what he wanted me to say, and I enthusiastically and inexplicably announced "Three!" before sheepisly correcting myself.) What is the classification of surfaces? (I started fumbling here for no real reason...) How does the Euler number enter in? (Yeah, yeah...) How does it behave under the connect-sums? (I completely blanked. Definitely the easiest question so far, so it was embarassing to require all the nudging that I did.) How do you know that RP^2 # RP^2 is not the same as RP^2 # some orientable surface? (Well, I'll use the formula I finally managed to scratch out just before...) What is the relationship between #^3 RP^2 and RP^2 # T^2? (He asked this after I had already written that they were homeomorphic. I said you use some polygon cutting-and-pasting, and that was enough.) ALGEBRAIC COMBINATORICS (Benny) Can you state the oddtown theorem? (I decided to start botching things in this section right off the bat. So it took me a minute to formulate it correctly.) What if all sets have even size and even intersections? How many can sets can there be? (I gave the obvious construction of pairing people up and taking all subsets of pairs. I didn't recall how to show that gives you the maximum number, but eventually remembered the subspace generated by the incidence vectors mod 2 is contained in its orthogonal complement, and that gave it.) What if all the intersection sizes are the same? (Fisher's inequality. Couldn't quite kill the proof at the end, but it was just some trick I couldn't recall.) Discuss more general results along these lines; i.e. (non-)uniform, (non-)modular Ray-Chadhuri - Frankl - Wilson theorems. Prove the modular non-uniform version. (OK.) What is the importance of the hypothesis that set sizes are precluded from belonging to the list of intersection sizes? (The eventown construction.) Can you describe some applications? (Explicit constructions in Ramsey theory, the Borsuk conjecture, both of which I screwed up immensely. At one of the many points I got stuck I pondered aloud, "What would Noga do?" to no one's amusement.) What is Hilbert's Nullstellensatz? (Stated it.) What is the version used in combinatorics? (Noga Alon's combinatorial nullstellensatz.) What is the corollary you use? (If a certain coefficient is non-vanishing then the polynomial doesn't vanish identically on a certain product set.) Give an application of your choosing. (I started to talk about list colorings of graphs, but Benny told me that was the wrong choice, so I talked about Cauchy-Davenport instead.) COMPLEX AND REAL ANALYSIS (Natasa) What does it mean for a function to be analytic at a point? (I said that meant it had a power series at the point, and she asked how I would prove that. Prove my definition? I knew what she wanted, and said it was equivalent to being C-differentiable.) What are the Cauchy-Riemann equations? Why do they imply that the real part of a holomorphic function is harmonic? Cauchy's integral formula and its generalization to higher derivatives. Deduce Liouville's theorem as a corollary. Apply this to prove the Fundamental Theorem of Algebra. What is Cauchy's residue formula? How can you use it to compute integrals of real functions on the real line? What is the L^p norm? What about p = infty? (I answered this and all previous questions, but nothing noteworthy arose.) Why do you need to consider equivalence classes of functions in order to define L^p as a normed linear space? (I had forgotten. She told me to write down the definition of a normed linear space, and just as I was about to write down the first axiom for the norm I figured it out.) How do you prove that it is complete? (This I severely screwed up. I could see the page from Rudin, but couldn't write it down. She prodded me: show that completeness is equivalent to every Cauchy sequence possessing a convergent subsequence. This got the ball rolling a bit more, and we basically did the proof.) What are the basic facts about Fourier transform? (There was the standard discussion about how to normalize properly.) What is the transform of the convolution of two functions? (I stated what.) How do you prove it? (I said it was straightforward, which almost satisfied her, but finally I had to mention you need Fubini-Tonelli.) Suppose that you had a calculus student who was stubborn and wasn't satisfied with the Fourier transform on L^1. How could you extend it to L^2? (I didn't ask, but I wanted to know what this person was doing in a calculus class. I mentioned Plancherel, starting with L^1 \cap L^2 and going from there. She mentioned that there is also an approach using Schwarz spaces; I asked her for a reference on that, and she referred to the book (by Duoandikoetxea) she had suggested I look at prior to the exam. Whoops.) ALGEBRA (Benny) What is Sylow's theorem? (I wrote it all out.) Show that there is no simple group of order 132. (I was surprised that counting up elements of different orders actually takes you all the way through here.) What can you say about finite fields? (Just the standard stuff. I asked how you know there are irreducible polynomials of every degree, Benny said you just count them, and then I figured out a different response.) What is the structure theorem for abelian groups? How many of order 2000? (OK.) -------------------- REFLECTION AND ADVICE (me) At first, I didn't want to write anything on the board, much to the chagrin of my committee. I was also quite timid in getting into details about anything, failed to mention important things that I thought were just obvious, and also very sloppy in talking through proofs in logical order (especially in the combinatorics section; my performance there was a major disappointment, which Benny more or less confirmed afterwards). When it got to Natasa's turn she had to keep insisting I write on the board for every question. So I would aim to be a better communicator on my next generals exam. Very little in the way of proofs were requested throughout. Basically anything that could fit, in large handwriting, on the back of an index card was sufficient. Understand the moral arguments to proofs. Much more important to the committee was that I could apply what I knew in concrete situations. If you are nervous about the exam, talking to your upperclassmen is a very consoling experience. These written accounts are limited in how much they can convey all the bumbling mistakes you make and prodding you receive throughout your exam. Once you get going you start to calm down, the time flies, and you just accept the fact that you're going to make mistakes when you think on your feet. A mock exam would be a good idea in case you are very nervous. Take your exam as soon as you feel there is nothing more you could do to prepare -- prolonging the wait will just strain your nerves. These accounts usually don't speak to the extent to which the material you are examined on is culled to you. Choose and talk to your examiners prior to the exam so that there won't be any surprises. Zoltan told me early on that knowing the basics from Hatcher would be sufficient, and I confessed knowing really nothing about classifying spaces and characteristic classes. I prepared for combinatorics by using a syllabus for a course Benny had taught in the past, although "algebraic combinatorics" could mean lots of other things. Also, a former student of his -- Peter Keevash -- wrote this wonderfully condensed set of notes on the Babai-Frankl book and Alon's Combinatorial Nullstellensatz that I studied from, and Benny even kept referring to his copy throughout the exam! So if you elect to do this as one of your topics, ask Benny or Peter or me for a copy. There are some bizarre questions on the standard questions list (as of the time I took the exam). Just be realistic and don't stress out over them. You're not going to have to know about the Laplace transform if your topics are combinatorics and topology, and your committee is there to provide lots of hints. I think it's pretty easy to identify what is core material from the available resources, and knowing that is sufficient for passing. There's no need to get intimidated by more advanced material that you read in some of these generals accounts, although it is nice to know something more than the bare minimum. Bringing a water bottle is a great idea. Good luck!