Allen Knutson's generals, 10/93, length about 2:20
Examiners: Hale Trotter, Dennis McLaughlin, Lisa Jeffrey
Special topics: Lie algebras & algebraic topology
Complex analysis (McL)
Draw a Riemann surface in the polygonal representation.
Let f be a meromorphic function on it. How can I define a log f?
(Have to cut it until it's simply connected.)
Wait, what about the zeros and poles of f?
(Oops! Make cuts to those too.)
Now let g be another meromorphic function on here.
What's \int log f dlog g?
(I computed it using the residue theorem, with some prodding, having
forgotten the properties of dlog g, but was given time to rederive them.)
Okay, now compute it the other way, directly.
(Complete failure. Eventually we gave up.)
Algebra (T)
If M is free abelian, how can I put quotients of M in some standard form?
(Talked about row and column reduction of an integer matrix, occasionally
adding auxiliary rows.)
Okay, I believe that you could fill in the details. What was crucial about Z?
How does the procedure simplify if it's an ED, not just a PID? (Since my
proof had just used PID-ness.)
Do you know a PID that's not an ED?
("Yes." He almost let it go at that, too.)
How would you start finding the groups of order 56?
Is there in fact a way for Z_7 to act on a group of order 8 nontrivially?
What does square discriminant of a polynomial indicate about the Galois group?
What's a Galois group that's not S_n or A_n?
Real analysis (J)
Routine application of Baire category - but know the equivalent statements!
Show Fourier series of L^1 functions are continuous.
If \sum m(A_i) is finite, show B = {those points in infinitely many A_i}
is measure zero.
If a sequence of functions converges uniformly to 0 on a finite measure space,
do its integrals also do so? On an infinite measure space?
Let phi: B -> {complexes} be a Banach algebra hom. (She defined these, just in
case.) What can you say about its norm? (I needed a hint.) What if some
element of norm < 1 goes to the number 1?
Do you know a function continuous on only the irrationals?
Do you know a sequence of functions on [0,1] whose integrals converge to
zero yet converges nowhere?
Alg Top (McL)
Talk to me about S^2 wedge S^1.
(I computed pi_1, pi_2, and all the (co)homology groups. Started on pi_3.)
Well, let's do pi_3 of an easier space, do you know pi_3(S^2)?
Okay, let's do CP^n. What's a cell structure on it?
What do the attaching maps look like? (This was more fun.)
What's the (co)homology? What's the ring structure?
(I used big machinery I couldn't prove to compute this, ill-receivedly.)
Okay, you're using Poincare' heavily, can you prove that?
(Not pleasantly, just via Mayer-Vietoris. Yes, I know there's a cellular way
but I don't remember it. However, I am comfortable with the De Rham end.)
What's the De Rham theorem? State the isomorphism.
Why is that a chain map? ("Stoke's theorem.")
For what more general kind of spaces could you set such a thing up?
(I needed lots of prodding here, and was eventually led to "paracompact".)
Is S^1 homotopic to a loop space of something else?
Lie algae (J)
How can you compute the Killing form on the Cartan in terms of the root system?
What are the eigenvalues of [1 0,0 -1] in sl_2 in an irrep?
If alpha is a root, why is -alpha?
How do I find an sl_2 corresponding to alpha?
Why are +/- alpha the only multiples?
What do root strings look like? Why no holes? Why irreducible?
What about angles between roots?
If (beta,alpha)>0, why is beta-alpha a root?
Let alpha be a simple root in some positive system. What happens when you
reflect some other positive root through alpha?
(It's still positive.)
What's the reflection through alpha applied to the sum of all positive roots?
Reflections...
My exam appears to have been a bit atypical in that only two or three times
was I even given the chance to say "No, I can't, I don't know" - all by
Dennis McLaughlin. In many ways it was a culture test: "Do you believe this
naive thing?" "No, because here's the standard counterexample."
It was hard to feel at the time that I was doing well, because I'm used to
written tests, where you can trap a problem, play with it for a while, and
then finally kill it. As soon as it was clear that I was going to kill a
problem, my examiners would take it away! So I had to be quick, and only
really got that chance during the algebra section.