Martin Niepel January 25, 2001, 10:00 am - 12:20 pm, Fine 801 Committee: Wee Teck Gan, Stalker, Szabo (Chair) Special topics: Representation theory, Algebraic topology Complex analysis St: State Schwarz reflection principle. Can you prove it ? I got confused because I tried to prove it's analytic version and that hadn't worked. In harmonic case (after many helpful questions) I finally got something what looked like a proof. Sz: State Riemann mapping theorem. Brief sketch of the proof ? Sz: Can you state Picard theorem ? Example of an entire function which doesn't attain one value ? Do you know any generalization ? I stated Big Picard. (I am not sure if Szabo wanted this, but this was the last complex analysis question and we moved to algebraic topology) Algebraic topology Sz: Give an example of a space with non-commutative fundamental group. I started to compute the fundamental group of the torus and realized soon afterwards that it is commutative, so my next answer was the double torus. Sz: Can you prove that that group is non-commutative ? I couldn't do it on the spot, so I gave the third example - '8' - two pinched circles. Sz: What theorem can you use ? (Van Kampen) It holds only for open subsets ... Sz: What about the fundamental group of the space consisting of a circle pinched to a sphere ? What is pi_2 of that space ? What is the universal cover ? Sz: Give an example of a space with fundamental group Z/3Z. I started thinking of an action of Z/3Z on a simply connected manifold, but did not get anywhere. (after exam I realized, that there are such actions on S^3 providing Lens spaces.) Sz: What if you glue the boundary of the disk n-times around the circle ? (fundamental group is Z/nZ) How to prove it ? (again Van Kampen, but I could not do it then) What is the homology ? Sz: Homology of RP^2xRP^2 ? Cell decomposition ? Homology over Z_2 ? Write down at least one boundary map (over Z). Cohomology ring (over Z_2)? Representation theory G: What kind of representation theory have you studied ? (mostly finite groups and Lie algebras) G: Character table of S_4, say something about all things you are using. (Permutation representation, inner product on characters, decomposition of the regular representation, orthogonality of rows and columns.) G: Representations of SO(2) ? G: What is the maximal weight of an irreducible representation ? I mentioned ideas and terms involved - Cartan subalgebra, roots, weight spaces, Weyl group, etc. G: Representations of sl2 ? Real Analysis St: State Holder inequality. St: This will be a non standard question - having a convex function, its second derivative is positive. How could you generalize this for functions not having second derivative ? E.g. for f(x) = |x| (absolute value) ? I did not know what Stalker wanted, and he asked me if I heard about 'positive Borel measure'. I said 'no'. (hell, what kind of measure ???) But I tried to define weak derivatives, wrote down the formula for the integration by parts, said something about boundary conditions (compact support of test functions). And that was what he wanted. Algebra They did not want to ask anything in algebra, but finally ... G: What is Hilbert's theorem No. 90 ? (I said: "I know that there is such a theorem ..." Szabo asked Gan: "What is it ?") G: What is the Galois group of x^4 - 3 ? ******************** comments: It is important to know the examples well. It's quite embarrassing when you do not remember the conjugacy classes of the symmetric group and have to work them out on the blackboard, disturbed by questions like "do you know what the conjugacy class is ?". Know how to apply theorems to those simple examples. If you get a problem which you have not seen before it is more probable, that you choose the correct way to solve it, they usually do not ask difficult things. My committee told me after the exam that they expected more smooth passage from 'simple examples' to the 'general theory', mainly in my special topics and that was the only complaint. Afterall, they were very nice, patient and helpful. And the passed me. ;-)