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).