Ian's generals - examiners Hsiang, Christodoulou, Conway; special
topics Alg. Top and Diff. Geom.
Algebra.
At first, Conway couldn't think of a good question, so murmured `What
is Algebra, anyway?' I was allowed to take this as a question, iff I
didn't take it seriously - so I replied `Category theory is Abstract
nonsense, so Algebra is Abstract sense'. Whethe r this was remotely
relevant to the rest of the exam is another question, but...
Started properly with Jordan Canonical form (which I couldn't
remember, so we went to other canonical forms and established what it
must be.) After the statement, I had to state the minimal poly. and
dimension of eigenspaces from the Jordan form. I was then asked what
other thm is this most related to (Class of Ab. Grps.)
Then they wondered what other branches of Alg. there were, so I
mentioned non-ab. groups, and proved Sylow for them. This was
voluntary.
Then they asked about Galois theory, so I stated the main results. And
they asked me to prove the impossibility of trisecting the
angle. (*)(Here, I `proved' that the extension field must be of dim a
power of 2, and stated the rest).
Then they wanted me to classify finite fields; I thought (incorrectly)
that I needed cyclicity of the *-groups for this, so they asked me to
prove that too.
These were mostly asked by Conway, although Christodoulou asked the
starred question.
Complex
Conway asked me to classify poles, so I did. At `essential
singularity' he asked me what big result was true here, (Big
Picard). I also mentioned the little one, but that I couldn't prove
it. He told me that I could, and gave me a hint; so I did.
Then Hsiang asked me for statement of Riemann mapping, and what
happens with annuli. I didn't offer proofs, and he didn't ask for
them.
Real
Christodoulou asked about the Arzela-Ascoli, but I looked very
blank. He kept trying to lead me on to it, but without success. He
gave up after a while.
Then he tried functions of bounded variation, so I gave a def. Then he
wanted to know about their points of cty; I had no clue, but
constructed counter e.g's for a while. Then Conway mentioned
monotonicity; so I `proved' that they are the sum of 2 monoton e fcns
(by constuctin one el. of the sum); and stated the result wanted (cts
a.e.).
Christodoulou then wanted to know about nec & suff conditions for the
Fund. theorem. I mentioned C2, (we already had a counter e.g. on the
board from the last section). Conway then led me to a cts counter
e.g.; but I knew not much else.
Then Christodoulou (think!) wanted to know about Fourier trans., so I
stated what it maps to what, and said something about it being useful
for DE's, and proved what it maps f' to. (they didn't mind me not
knowing the statement here).
Diff Geom.
Christodoulou started with Ricci curvature, which went very badly for
some time. Then Hsiang moved on to vector bundles, which I defined
(although I forgot he wanted a vector bundle, not just any old bundle
for some time). He then asked me to define the t angent bundle, and at
this point I basically froze for the rest of the exam. From here on I
refused to admit I knew anything (even 2+2); they tried to think of
things I knew for some time, but without success.
Alg Top (see above, nothing remotely non-basic was asked, except when
they asked me what I could do).
Comments: Well, I passed, though I don't know how (given the last
section). Apparently, my performance on the basic subjects was better
than average, and they liked me being able to think on my feet (when I
didn't know the answer). Not much else to say, t hat everyone else
hasn't said.
Yours,
Ian