Margaret Doig's Generals

subjects: algebraic topology, differential geometry
committee: Szabo (chair/topology), Tian (geometry), Grushevsky
length: 2 hours

I was allowed to choice the order. Asked for a special topic first, and
Szabo and Tian each politely insisted that the other should start. I
finally interrupted and voted for...



Algebraic Topology (questions mostly by Szabo and based on a theoretically
complete understanding of Hatcher):

-Define a CW complex. Need it have a finite number of cells?

-Give a cell decomp of CP^n.
 (answered correctly but proved it for RP^n, which no one noticed!)
-What's the cohomology? Cup product structure? (mentioned intersection
 numbers and said H^*(CP^n) = Z[a]/(a^{n+1}); was asked to elaborate)

-Derive pi_n(S^n).
 (I messed up horribly and spent twenty minutes trying, with Zoltan's
 help, to rederive everything I should have known about degree)

-Do you know any other homotopy groups of spheres?
-Calculate pi_k(S^n) for k<n, pi_2(S1 v S2) (covering map; asked to
 explain the relation between pi_k of a space and of its cover)
-What about pi_3(S2)? (Hopf fibration)
-Then several other similar examples which I don't remember.



Differential Geometry (Tian; based on do Carmo's 'Riemannian Geometry')

-Say something about spaces of nonpositive curvature. (covered by H^n)
-NONpositive, not negative. (oh, or R^n)
-prove it? (much bumbling, way too much help - and, in the process, we
 covered the exponential map, the fact that it's a local diffeomorphism,
 conjugate points, the formula for a Jacobi field in terms of
 exponential maps, fact that a Jacobi field has positive first derivative
 on a nonpositively curved manifold; also mentioned geodesic completeness,
 Hopf-Rinow, Hadamard (the proof of which I also blanked out on - Tian
 said it didn't matter), what happens to the Jacobi in a manifold of
 constant curvature...)

-Show that the fundamental group of a compact Lie group is finitely generated
 (even worse. couldn't remember anything about Lie groups, but we painfully
 derived the existence of a bi-invariant riemannian metric, the equation
 <u,[v,w]>=-<[v,u],w>, Ricci curvature, etc; eventually found formula for
 the connection in terms of Lie brackets. also talked briefly about
 Bonnet-Myers and about finding a geodesic representative for each
 homotopy class and showing that pi_1 really is finitely generated)



Algebra (Grushevsky; I studied Dummitt and Foote):

-Define a representation. (said I didn't know any representation theory)
 That's fine. We'll do groups instead.

-What's a Sylow p-group? The Sylow theorems? (didn't have to prove them)

-What's a simple group? Is A_4 simple? (yes)
-Really? (well, yes, I'm pretty sure)
-Ok, what are the conjugacy classes of S_4?
 (told him which things commute, but he laughed and fixed it)
-Are they all normal subgroups? (yes)
-Are you sure they're all even subgroups? (yes)
-*smile, followed by long silence* (oops, no)
-Do you still think A_4 is simple?



Complex Analysis (I used Narasimhan and Conway):
-[Szabo] What's the Riemann mapping theorem? How do you prove it?
 (gave a sketchy proof)
-Doesn't that show that an annulus is conformal to the unit disk?
 (looked embarassed and mumbled about which step was wrong; didn't work
 out a full correction, though)

-[Szabo?] What is an entire function? What do you know about them?
 (Liouville)
-Anything else?
 (I thought they took almost all values at infinity, similar type of thing
 to what happens in Picard)
-What's Picard? Must entire functions take all values at infinity? (no; e^z)

-[Grushevsky]-Can I find a conformal map from the plane minus a point to the
 plane minus any other point? (yes!)
-What about two points? (fractional linear transformation)
-three? (didn't know; apparently, it's not always possible - the cross
 ratio is invariant under conformal maps)



Real Analysis (used Folland; Grushevsky asked):
-Is it possible for a series of functions to converge in some way but not
 almost everywhere? (yes, in about a dozen different ways! L1, in
 measure, almost absolutely...)
-Give an example for functions over the interval that converge in L1 but
 not a.e. (take characteristic function of, say, width one; move it around
 on the interval so you get a sequence that converges pointwise nowhere.)



Comments: There were a lot more intermediate questions, especially small ones
I could answer verbally. In fact, they didn't want board work or details for
most questions unless my verbal description was really sketchy. And they really,
really gave me a lot of help. I spent part of topology (the discussion of degree!)
and just about all of geometry muttering and writing random things on the board
without giving evidence of any higher mental processes at all. The only thing that
saved me (besides a merciful committee) was being able to spit out theorem
statements and equations and give definitions without having to think about them.