Blair Dowling's General exam: Jan 6, 2005, 2:00 PM, Dr. Seymour's office Special Topics: Algebraic Number Theory, Selected Topics in Discrete Math (namely Matroid theory & Combinatorial Optimization) Committee: Paul Seymour (chair), Ramin Takloo-Bighash, Robert Gunning Notes: within each topic, things may be out of order, and I feel certain questions were omitted in most categories due to my failing memory. Matroid Theory (Seymour) -Defn of matroid (and some examples) -Example of non representable matroid (I gave the Fano) -Discussion about largest binary matroid of rank k -Matroid union theorem (without proof) -Matroid intersection theorem (prove easy direction) -Konig's theorem (with proof from Matroid Intersection Thm) -I know there were more questions here, I just have no idea what. Combinatorial Optimization (Seymour) -Lucchesi-Younger (with proof outline) -Give all the associated definitions (nu, tau, etc) -What if we took the hypergraph of directed cycles? Is nu = tau? (I think he gave K(3,3) as a counterexample here - uncertain). What if G was planar? Draw a small digraph & take its dual on the board. If we take the dual twice, what happens? -State Edmonds polytope matching theorem -What is the convex hull of perfect matchings? Matchings? -how does edge-colorability relate to matching theorems? (no clue) -If G is 4-regular, not 4-edge colorable, and we put weight 1/4 on every edge, is that in the convex hull of matchings (or PMs)? What if we take Petersen with 1/3 on every edge? "If you can't draw the Petersen graph, you fail" What are the perfect matchings in the Petersen graph? -Define a perfect graph - give some examples...more examples -What is the strong perfect graph conjecture? -Dilworth's Theorem (+ proof idea) -How would you prove that G is perfect iff its complement is? Alg. Number Theory/Algebra (Takloo-Bighash) -Integers of K= Q(sqrt(-6)) -Discrimant of K -Class number of K - state Minkowski bound (didn't actually finish my ideal factorizations...he pointed out that (2) and (3) ramify as they divide the discriminant) -Cebotarev density theorem (statement) -Define all the words you used in stating that (frobenius, density, conjugacy class). This went on for a while...talking about decomposition groups, inertia groups, how they related to this, etc. -Give (in my case vague) definitions of ray class group, etc -Apply Cebotarev to a cyclotomic field & get a famous theorem (Dirichlet's theorem on primes in arithmetic progressions) -Dirichlet's Unit Theorem (state & outline proof) -Apply the unit theorm to a real quadratic field. What is the cyclic group? Complex Analysis (Gunning) -Take a function which is analytic in the punctured unit disc. What can we say about its behavior at the origin? If it's bounded? If there's an essential singularity? (I mentioned Casorati-Weierstrauss) -Prove Casorati-Weierstrauss (on board, in detail) -What if f is analytic in the punctured disc & it never takes on any value in the neighborhood of some point? -How do analytic & conformal mappings relate? -(vague memory of being asked to state a famous result of Riemann somewhere in here - he wanted the Riemann Mapping Theorem, which I gave...then he asked why the whole plane wasn't equivalent - I said someone told me stuff about universal covers - he said it was easier & explained some until I said Liouville's theorem) -How would you prove Liouville's Theorem? - if we have a map of the upper half plane to, say, a triangle... (Blair: "I don't know Schwartz-Christoffel"...Ramin: "I don't either, and I taught it"...we moved on). -Define Laurent series -Say you had a function in the annulus between 1/2 and 1. If a function has an infinite Laurent tail, must it have an essential singularity (no - 1/(.25+z)). Can you write down the Laurent expansion of that (I messed up, then sorta corrected mistake and he said it was fine...used expansion of 1/(1-z)). Real Analysis (Gunning) -Take the integral of e^(ixy)f(x)dy = g(y). If f is in L1, prove g is continuous. You can use any famous complex analysis theorems you want. I stated the one that the fourier transform is continuous & goes to zero for L1 functions. Gunning said any theorem except that one. They made me do this in detail. - Now, is g differentiable? (guess:no?) What if f has compact support? (uh, yes?) Prove it...again, in detail. - State Lebesgue's Dominated Convergence Theorem. Apply it here (on both the above things). He called this "Lebesgue's Integral Theorem", which confused me for a bit) - L2 is a Hilbert space - what do we know about Hilbert spaces in general (he wanted me to say they're all equivalent to L2(mu) for some mu, which I did -I gave the example of Bergman space). - Lp is complete - how would you prove this? - What can you say about convergence (L1 -> measure -> subseq a.e) - Give an example that converges in L1 but not almost everywhere. - Why do you get a subsequence converging a.e.? The exam took about 1.5 hours, and was generally quite relaxed. They were quite willing to help me when I fumbled things (or just completely forgot in some cases).