Peter Milley
Special topics:  Algebraic Topology, Differential Geometry
Examiners:  David Gabai (chair), Zoltan Szabo, Alice Chang


Real Analysis:

Prof. Chang started with the questions on this topic.

Is there a function in L^1([0,1]) which is not in L^p([0,1])?  (1/x^a
where -1 < a < -1/p) Vice-versa?  (No, not on a finite measure space.)
In a finite measure space, how do you show that the continuous
functions are dense in the L^p functions?  (First, show continuous
functions are dense in the simple functions, then that the simple
functions are dense in the L^p function.)  Then we discussed
convolutions and how the convolution of two integrable functions is
continuous, and how to construct g_n such that the convolution of f
and g_n will tend to f as n->infinity.

Next:  State Fatou's lemma and give an example where strict inequality
holds.  (Characteristic function of [n,n+1].)  An example defined on
the unit interval.  (Char. function of [1/n, 1/n+1] times a suitable
constant.)

Then Prof. Gabai jumped in and asked me to define the Cantor set and
give its measure.  Then:  if the Cantor set is homeomorphic to another
subset of R, does that set also have measure 0?  (No.)

There was, to my surprise, nothing about Fourier transforms or
coefficients.  All in all the real analysis section was much easier
than I was expecting.  Not so for:

Algebraic topology:

First, Dave asked for a statement of the Van-Kampen theorem.  (I
forgot the requirement that the intersection be connected.)  After I'd
finally written that down on the board, Prof. Szabo asked for the
cohomology of CP(n), including the ring structure.  Also, how do you
show that S^2 cross S^4 isn't homotopy equivalent/homeomorphic to
CP(3)?  (The ring structures aren't the same.)

Then we talked about higher homotopy groups for a while.  I stated
that pi_r(S^n)=0 if r<n (by the simplicial approximation theorem), and
when I stated that I wasn't sure how to show that pi_n(S^n)=Z
Prof. Szabo led me through the argument involving degrees and
cancelling out the pre-images of regular values and so forth.  Then
there was the usual question about pi_3(S^2).

The Prof. Gabai asked what I knew about the homotopy groups of an
irreducible three-manifold.  This was something of a stumper, as I did
not know the statement of either the topological Sphere theorem or the
Hurewicz theorem relating pi_n to H_n.  Eventually it came out that if
M is a compact irreducible three-manifold with infinite pi_1, then all
of the higher homotopy groups vanish (pi_2 by the sphere theorem and
the rest by Hurewicz and the fact that H_n of the universal cover
vanishes for n>=3).  Not my best moment.

Next Prof. Szabo asked for a statement of Poincare Duality and the
conditions under which it holds.  What happens if you have a
non-orientable manifold?  (Still works, but you have to use Z/2 as

your coefficient ring.)  Similarly for the Lefschetz fixed-point
theorem.  Asked for a proof, I outlined the algebraic proof; after
some pointed questioning it was clear they were expecting a
geometrical argument.  Then Dave asked if the Lefschetz theorem still
held in a non-triangulable topological space.  I had no idea and said
so, but I guessed it would still hold for compact spaces; to my
suprise it apparently doesn't.

Algebraic topology was definitely the hardest part of this exam.

Algebra:

Prof. Szabo started on this one.  First:  how many abelian groups of
order 200 are there?  What's the classification of finitely-generated
abelian groups, and an outline of the proof.

Next:  define a simple group and give an example (A_5).  How can you
show that A_5 is simple?  (Some traitorous part of my brain was so
tempted to answer "GL(3,Z/2)" instead of "A_5" for the first question;
thankfully I didn't listen!)

Dave then asked what common geometrical object has A_5 as its symmetry
group (the icosahedron, or dodecahedron if you prefer, although I
don't know how to show this off the top of my head).

Dave then asked how to calculate a matrix in a different basis, and
whether two real matrices which are conjugate in M_n(C) are conjugate
in M_n(R) as well.  (Oddly enough, I'm not sure...)

Prof. Szabo then asked me if there were any non-abelian groups with 15
elements, and we were done this section.

Complex Analysis:

Prof. Chang started out, asking to name the types of singularities
that a meromorphic funcition can have, with examples.  This led to me
giving the statements of the Little and Big Picard theorems.  Then
Prof. Chang asked me how I would calculate the residue of a function
at a pole of order 2.

After this I was asked for a statement and proof of Liouville's
theorem.  Then: does this imply that a bounded _harmonic_ function on
the entire plane is constant, and why?  Thankfully, I was not asked to
prove that a harmonic function must be the real part of an analytic
function.

Next, Dave asked for a statement of the Riemann mapping theorem, and
Prof. Szabo asked me what I could say about extension to the boundary.
Prof. Szabo provided an example of a region conformally equivalent to
the unit disk, but where an uncountable # of points on the boundary of
the unit disk get mapped to the same point on the boundary of the
region.

Then Prof. Chang asked me about some Weierstrauss theorem that I
hadn't heard of.  When I blanked on that, she settled for rough
discussion of how I would construct an analytic function with a given
set of zeroes, and a statement of the condition under which an
infinite product will converge.  (product of a_n converges absolutely
if the sum of (1-a_n) converges absolutely.)

Differential Geometry:

At this point, the examiners seemed stumped for questions.
Prof. Chang finally started with a basic one: state Gauss-Bonnet and
describe roughly how you go about proving it.

Dave asked how to find the area of geodesic triangles in hyperbolic
space and spherical space (angular defect and angular excess,
respectively).

Next: what could I say about the homotopy groups of a manifold of
negative sectional curvature (by Hadamard's theorem the universal
cover is R^n, so the higher homotopy groups vanish and pi_1 must be
infinite).

Dave then asked what I could say about the sectional curvature of a
surface minimally embedded in a three-manifold.

Then they asked me how many different kinds of curvature I could
define (the curvature tensor, sectional curvature, Ricci curvature,
scalar curvature...)  There followed some discussions amongst the
examiners over different definitions of Ricci and scalar curvature
that appear in the literature.

And that was that...