Brent Doran
2:30 - 4:30 pm, January 20, 1999

Topics: Algebraic Topology, Differential Geometry
Examiners: Robert MacPherson (Chair), Alice Chang, Emmanuel Kowalski

When asked about order of topics, I decided we might as well
proceed in alphabetical order.  MacPherson said this was a
rational way to proceed. I commented that rationality is good,
because then we don't have to worry about messy torsion phenomena.
Naturally enough, MacPherson proceeded to ask me some questions
involving torsion later in the exam :-) A number of the questions
went by very quickly, especially when I could answer immediately,
so I have unfortunately forgotten some of these (but they were
standard questions that appeared on other peoples' generals).

ALGEBRA: (Kowalski)
- Define a representation of a group (on finite dimensional vector space
over C, say; also, throughout let "group" mean finite group)
- Define irreducible representation, explain Schur, construct invariant
inner product, Maschke's theorem, ...
    - (MacPherson: what can go wrong if over R?  Give examples)
- What can you say about characters?  Definition, orthogonality
relations, using them to determine if a given irreducible representation
is a subspace of another given representation...
He wasn't interested in character tables at the moment, because...
- Let (rho, V) be a faithful finite dimensional representation of G.
Show that, given any irreducible representation of G, the nth tensor
power of GL(V) will contain it for some large enough n.
- Consider the space of functions from the natural numbers to C endowed
with the usual law of addition and the following analog of the
convolution product: f*g(n) = sum_{d | n} (f(d)g(n/d))
Show this forms a ring.  What does this ring remind you of and what can
you say about it?
After some attempts to explain and extend the question, we moved on to:

ANALYSIS: (Chang)
- Discuss the relation between L^p spaces on [0,1] and 1/x^a (that is,
when is this function in L^p?).  What about 1/(x^a*Log^b(x))?
- Show that L^p is (L^q)^* if 1/p + 1/q = 1, 1<p<infinity
    - Explain what goes wrong and give examples for p=1
- Lebesgue decomposition of measures and Radon-Nikodym derivatives
- Discuss Fatou's Lemma, give a counter-example
    - Relation with Monotone and Dominated convergence theorems
    - give a sequence of continuous functions on [0,1] such that
taking limits and integration cannot be interchanged
- Discuss uniform convergence and some useful criteria related to it
- Let {f_n} be a convergent sequence of functions on a compact interval
whose derivatives are bounded.  Is the convergence uniform?  Prove it or
give counter-example.
- Prove that continuous functions are dense in L^1, on a compact space
- Discuss the Fourier transform and its key properties.
    - L^1 to continuous vanishing at infinity, interchanges
differentiation and multiplication by x, L^1 /cap L^2 mapped into L^2
with unique extension to isometry by Plancherel,
    importance of Schwarz space
- What about how it affects convolutions?
- Other facts about convolutions: increasing smoothness properties
- What can you say about convolution kernels and their properties (e.g.,
Poisson kernels)
- Brief discussion about "Dirac-delta functions"
- What is the Hilbert transform, and what can you say about it.

COMPLEX: (Kowalski)
- State the Riemann mapping theorem (not interested in the proof)
- Write down a conformal map taking the upper-half plane to the unit disk
- Write down a conformal map taking a horizontal strip to the unit disk
(A couple questions I now forget, then...)
- Prove Borel-Caratheodory Inequality (at least, I think that is the name
he gave)
    Hint: use Riemann Mapping theorem
    (The gist was to show the functions in question had image
contained in some half plane, then explicitly use conformal maps to the
unit disk, and finally, apply Schwarz Lemma)
- Let f, g be analytic functions, say Re(f - g) = 0, show Im(f-g) = const
- Explicitly show a particular class of functions, described as power
series and with certain growth constraints, is log convex (sorry, I can't
remember the conditions; the proof was relatively messy and actually
required working with the series, multiplying, combining terms,
reindexing, all done on the blackboard)

GEOMETRY: (Chang and MacPherson)
- (C) Define Gauss curvature, say for a surface in R^3.
- (C) Why is it an intrinsic quantity? (Theorem egregium, and I begin to
sketch a proof via the structure equations [probably the cleanest way to
show it], but a more intuitive answer was requested; I talked about role
of positive curvature and negative curvature, interpretations in terms of
geodesic deviation.  As to why the quantity itself might be intrinsic,
and not just the sign, I resorted to giving plausibility arguments with
simple examples)

- (M) State Gauss-Bonnet; what does it say about something simple, like
triangles
- (M) Discuss Frobenius integrability (since I already have the structure
equations on the board), interpret the vanishing of the curvature form in
terms of integrability
- (M) Give an example of when integrability fails.  (I explain that the
one-dimensional case is integrable just by virtue of the existence and
uniqueness of solutions to ODE's, so the simplest counter-example is in
k=2, n=3.  A counter-example drawn there is really the "only" sort of
counter-example: just flow one way, then another, then back the first,
back the second, and end up, say, above the original point)
- (M) Define principal bundles, and connections on a principal bundle

I was actually hoping for some Riemannian geometry questions, but they
decided to move on to...

TOPOLOGY: (MacPherson)
- Give an example of a compact surface with non-abelian fundamental group
(genus >= 2)
- How would you compute the fundamental group of the Klein bottle?
(Polygonal complex picture).  What does Hurwitz tell you? (abelianization
of pi_1 is H_1, so here H_1 is Z + Z/2Z)
What is H_*(Klein bottle)?
- Give an example of a topological space with Z/3Z first homology group
(disk with triangular boundary where boundary edges are identified in
order and direction induced by the orientation)
- Compute H_* of this last space.
- What is the homology with Z/3Z rather than Z coefficients?
- Classify complex line bundles over S^2.  (I identify with CP^1, talk
about first chern class, classifying bundles by degree)
- What about higher rank complex bundles?  (Chern classes no longer are
sufficient to classify)
- How would you classify bundles in general, say over some nice CW
complex? (Homotopy classes of maps into the classifying space)
At this point I comment that one can do a more direct argument in the
case of S^2, taking a cover by northern and southern hemispheres, and
consider homotopy classes of maps of the intersection (homotopically a
copy of S^1) into the structure group.
- What is the classifying space for complex rank 1 bundles?  (CP^infinity)
- What is the classifying space for complex rank 2 bundles? (Infinite
grassmanian of 2-planes)


ADVICE:
    The summary above is much smoother than the exam was, or at least
much smoother than the exam *felt*.  I had many long awkward pauses when
trying to answer questions, and sometimes began a problem in entirely the
wrong way, but they were very patient and helpful.  They were most happy
when I could answer, after a struggle, a question I had clearly never
thought about before.
    The advice others have given on these "generals help" pages is
quite good, but you shouldn't expect that the questions you get are just
repeats of questions from old generals exams.  You have to be on your
toes!  On the other hand, the committee drops many quite explicit hints,
and sometimes builds to a question (although at the board it can be hard
to see the links).  Don't be scared to admit you're totally stuck on a
problem, at least once you've made some serious attempts.  Also, although
it's helpful to say what you know about a problem, be aware the committee
picks up on what you say and may steer the questioning in that direction,
so be sure to have a good understanding of any auxillary concepts you
introduce (e.g., structure equations in my Differential Geometry section
helped lead into integrability discussion).
    Finally, I have two important (although perhaps "obvious")
suggestions for preparation:
(a) WORK EXAMPLES.  It's always good to have an example for every
occasion.  At the very least you can use it to test conjectures at the
board, and having them at your fingertips, ready to use, is occasionally
seen as a sign of maturity. Besides, it makes preparation for the
generals MUCH more fun, fulfilling, and (in my opinion) of lasting
long-term benefit.  And best of all, you can impress folks back home with
them!  You'll be the life of every party!
(b) PRACTICE ORAL EXAMS WITH OTHER STUDENTS.  I wish I did this more.
I've lectured classes before, and so had board experience, but lectures
are prepared.  This is different.  The little bit I did with others in
the days before the exam proved very useful.  Learn to use the board
effectively (my computations during the exam were sprawled all over the
board, and so difficult to correct when I made an error).  Most
important, learn to recover from mistakes in front of a friendly
audience, and to clarify points they find confusing without getting
tongue-tied.