Francis Fung's generals: Oct 18, 1993, 2:00 PM
Committee: Browder (chair), Conway, McNeal (was to be Fefferman but he
was suddenly called out of town because his mother-in-law was ill)
I am already forgetting my questions, but I will try to remember...
The examiners were very pleasant, and when I came into Browder's
office, we sat and chatted for a minute or two. Then we got started.
The questions may be somewhat out of order, but I think I have almost
all of them.
Representation theory was first:
Conway asked me to tell him about representation theory, so I went on
a spiel about representations and characters; mentioned orth, char
tables, etc.
Why is the number of irred. reps equal to the number of conjugacy
classes? (I said I could prove it but that the "real" reason was that
both were the dimension of the center of the gp. alg)
What can you say about a character induced to G and then restricted?
(I didn't get it but he said the answer was easy; he said, of course,
that I should be talking about Frobenius reciprocity, so I did)
What about the Galois conjugates of rows and columns? (I mumbled about
conjugates of representations and cyclotomic fields a bit, and the
power maps)
What about G/N? (quotienting the char table; I had a little trouble
since I was slightly fuzzy, so I worked out the example of S3)
What about the abelianization? (linear characters tell you the index
of the commutator)
Then complex: (McNeal)
Let's go to infinite groups: What is the group of conformal auts of a
disc? How do you prove it? (Schwarz' lemma) What is a winding number?
What sort of number is it? How do you prove it's an integer? (I
rambled on about taking continuous logarithms; he said that was a good
heuristc, and then wanted to know how I would PROVE it; so I started
constructing the cont. log, and got a little lost; in spite of my
befuddled answer, he was kind and accepted it)
What is Brouwer's fixed point theorem? (D^2 case) What if, instead of
that, I give you that |f| < 1 in the unit disc? (I floundered a bit,
saying something about Banach's Fixed pt theorem, and then the hint of
Rouche's theorem was given, which, after I overcame "blackboard
blindness" made it obvious; f(z) - z and (-z) have the same number of
zeros in D)
Do you know anything about degrees of maps or integral representations
in several cplx variables? (I did not, I mentioned Hartogs' thm as
the only thing I had even heard of)
How do you know that the inverse of a 1-1 holo fn is holo? Can you
write down a Cauchy formula for the inverse? (I couldn't quite answer
this question, but I went on a spiel about the open mapping theorem
and m-to-1 ness; eventually it wandered by the inverse function
theorem, which I somehow remembered to state, and the d/dz-bar
operator, and how to show that the inverse function was holomorphic
using real-differentiability and d/dz-bar (f(f^-1(z)) was 0). When I
began contemplating the unpleasant thought of working it out, he told
me that it was something that I might like to think about after the
exam, but that I was not obliged to work on it then. In the middle of
that, I was asked why I know that the zeros of f' (or a holo. fn in
general) have no limit point in the interior, and I said that power
series in general have this property.
Then alg top. (Browder)
Returning to fixed points, what about a map f:R^2 -> R^2? (no, say a
translation) what if f has finite order? (I had no clue; turns out
such maps have fixed points for n = 2, not for high n like 5 and up,
but the order of f must be composite)
What is Poincare' duality? (I stated it in its easiest form: remember
that "cup" and "cap" are very easy to confuse when you are pronouncing
them rapidly!)
What is the Lefschetz Fixed Point Theorem? I stated it for
cpt. oriented manifolds; remember that it's the TRACE of H_i(f)!!! I
volunteered something about Euler chars, but he didn't really want to
hear much about it.
Do you know anything about homotopy groups of spheres (I said I knew
they were very hard (then he wanted to know if I knew something more
about small ones), and then said something about pi_n(S^n) and
pi_3(S^2)) Do you know an easy way to prove Pi_3(S^2) is Z? (I said
no, just the long exact seq. in homotopy)
Classify homotopy classes of maps from the torus to the sphere. (My
first thought was smash product, so he hinted to look at cell
structures: then I was told to guess, so I guessed Z and it was right)
What about S^2 to the torus T^2? I was quite unsure, so I looked again
at the cell structure and was told to guess; I guessed 0, which was
again right; later that night I realized that the prod "What is the
universal cover of T^2?" was the big hint, but I was a little too
nervous to pick up on it)
Do you know anything about vector bundles? I hedged a bit, so... What
about fiber bundles? (I defined them) Differentiable fiber bundles?
(I said diff. trans. fns) And vector bundles? (GL-n transition fns)
Then real:
What is Lebesgue's Monotone convergence theorem? Can you sketch a
proof? (I did)
How do you know that you can approximate from underneath by simple
functions? (I floundered a bit on this question, sort of going "well,
you just construct the approximation"; I guess the point was to talk
about measurability)
This wandered into something about Lebesgue's motivation for his
integral, including: Which do you think is more natural, Lebesgue or
Riemann? (I rambled about how Leb. was a generalization of Riemann,
admitting a wider class of admissible preimages of open sets)
And a little more about the concept of measurability.
Then a question: if int_1^infty f(x) x^n dx = 0 for all n>=2, what can
you say about f? (I floundered a bit, was given a hint to use a
compact set so substitute 1/x) Remember your Jacobian when changing
variables!!!! Anyway, you get to a point where you get integral
against y^n is 0 for all n>=0, I got stuck, was told to use linearity,
and then I realized that Stone-Weierstrass came into play, so it made
f a.e. 0 since it integrated to 0 against all polynomials)
Then another question on measures, Given f in L_1 of X, and eps >0,
can you find delta such that mu(A) < delta => int_A f < eps? (I
rambled about abs. cont., and then they proceeded to specialize the
cases, asking me to prove it if f was bounded, and finally culminating
in "say f is less than 5...!"; then I mumbled something about
approximating by a C_c(X) function and they seemed satisfied)
Then yet ANOTHER question on measures: if E_n are a seq. of meas. sets
in [0,1], and lim (n-> infty) m(E_n)=1, what can you say about the
measure of their intersection? (I floundered totally, though my first
instinct, inclusion-exclusion, was essentially correct; I never did
answer it though; eventually it was refined to "can you find a
subsequence whose limit was close to 1? There was some mumbling about
eps/2^n, and then, after noticing that we had moved quite far afield,
he suggested that we put this question aside)
Conway asked me about Borel sets, what they were, whether all such
sets were G_del's and F_sig's (I said I wasn't sure), and I mumbled
about inner and outer regularity and all Leb. sets being Borel + meas
0.
And then they wondered if there was any area left,
and there was algebra, so I launched into a spiel about Galois Theory;
said something about Fund. Thm, splitting fields, etc.
McNeal asked if I could exhibit an irred. polynomial with Galois group A3,
and so I gave the nec and suff condition that its discriminant be a
rational square (and Conway said "Well, technically that only means its
Gal. gp is *IN* A3" but of course I then specified irreducibility)
And then Conway asked about a poly over Q and its splitting field.
Then he began to ask what condition on the group was given by
irreducibility of the polynomial, and I immediately blurted out "transitive
subgroup of S_n where n is the degree of the poly", at which he was
suddenly at a loss for another question, and at that point McNeal put
forth the motion that we quit (at 3:07.55)
The committee I had was extremely nice to me, and when I got stuck,
they generously offered hints and suggestions.
At the end of each subject, that person would say something like
"well, I think that's enough of that; should we go on to something
else?" and then another topic would be found.
The examiners told me that I was answering the questions faster than
they could think them up! I guess I was quite nervous....they also said
that I was very good at communicating to them my nervousness.
The exam lasted about an hour and 5 minutes. It didn't hurt that Conway
had a train to catch at 3:26 (or 3:56 but the earlier one was better)...
Well, hope you have enjoyed reading this rendition of my generals!