General Examination of Matthew Kerr

January 11, 1999
Topics:  Differential geometry, Algebraic topology
Committee:  Browder (chair), Fefferman, Hsiang


Topology

Browder -- tell me about homotopy groups of S^3.

[I proved Hurewicz to get pi_3(S^3), then said I wanted to discuss the 
Hopf invariant as an iso. from pi_3(S^2) since there is a beautiful proof 
that pi_4(S^3) has to be Z2 or 0 using Freudenthal suspension theorem and a 
couple of these Hopf results.  Though I was eventually allowed to 
continue, this got me into trouble because having showed H:pi_3(S^2) --> 
Z was well-defined, onto (Hopf fibration), a homomorphism (S^3 a 
coH-space; linking #'s), I realized I couldn't show the kernel was 
zero.  Then Hsiang asked what other situations had maps with H(f)=1; I said
pi_15(S^8) and pi_7(S^4); he said, can you use the same proof as above 
and show these are Z?  I said, I don't seem to know all of the proof, so 
I can't tell whether, um, THAT part would extend.  Indeed THAT part does 
not extend, I was told.  So I concluded my original argument and asked 
them if they would like to see a proof by Postnikov fibrations that 
pi_4(S^3) was Z2, and they declined.]

Hsiang -- what is Lefschetz fixed point theorem? [wrote it down]
       -- can you prove it? [no, I said I hadn't studied this area recently]
       -- what is Hopf index theorem? [stated it]
       -- can youy prove IT? [I again replied no.  This was apparently 
not a good enough answer so they dropped some hints and a started putting 
together some facts about Thom class, euler class, diagonal embeddings; 
this was probably the most stressful part of the exam, though the 
examiners were always very kind.  speaking of the euler class,...]


Differential geometry

Hsiang -- state and prove Gauss-Bonnet.  

[he meant the version for a piecewise smooth curve bounding a region in 
a 2-manifold; I extended this to the G-B formula for the euler 
characteristic and then talked about the Chern-G-B theorem, i.e. the 
analogue for higher dimensions; while I could not write down the 
Pfaffian of the curvature matrix, I said how one would 
calculate it and that was fine.  then I decided to grab some tea, so it 
was 10 mins or so before we started analysis.]


Complex Analysis

Fefferman -- state the Riemann Mapping Theorem and give the "executive 
summary" version of the proof [the important thing is to know why the 
limit is onto has the same characteristic as the functions that approach it, 
i.e. is 1-1]
	  -- write down conformal maps from the upper half-plane to the 
unit disk and the upper right quadrant to the unit disk.
	  -- how would you compute the integral from -inf to inf of  
cos(x)dx/(1+x^6)?


Real Analysis

	  -- write down the Laplace Transform F(t)
	  -- suppose the derivatives of f are bounded; what can you say 
about F(t)?  [this led to a discussion of dominated convergence and 
differentiation under the integral sign; actually I don't remember 
finishing this problem after I found the dominating function...]


Algebra

Hsiang -- What is Sylow? [I stated it a gave a summary proof, saying what 
group acted on what set in each case and wroite down the general analogue 
of the class equation.  Then hsiang commented that this was similar to 
cartan's existence theorem for maximal tori.]
       -- How do you use Galois theory to prove the impossibility of (a) 
trisecting the angle, (b) squaring the circle, (c) doubling the cube 
(with ruler and compass)?  [I just talked my way through this.]
Fefferman -- what about solving equations?
Hsiang -- What is an example of an inseparable extension?  Prove it [I 
started to do so, and this led into a discussion out of which the 
following questions emerged]
       -- why is GF(p^n) unique? [I said, look at it as the splitting 
field of x^(p^n[-1]) - 1.]
       -- and you know that because ...? [ah, F* is cyclic.  That's what 
they had really wanted to hear the first time.]
       -- prove that. [I proved it and got myself so confused I didn't 
at first realize I had proved it]
       -- (for fun) is finite division ring necessarily commutative? [Yes.]
       -- why? [here I thought for half a minute and gave up, Hsiang said 
that was fine.  This is where the exam stopped; it was only a little over 
two hours.]