Tim Hsu
generals questions
committee: browder (chair), mather, prato (the questions are not
verbatim but are more or less correct)
analysis: state the fundamental theorem of calculus. can you
give exact conditions when it holds and counterexamples relating to
it?
when does a fourier series converge to its original function? what
can you say about the fourier series of a smooth function f? (the
coefficients go to 0 as n -> infinity.) does this condition imply
that f is smooth?
consider the series 1 - 1/2 + 1/3 - 1/4 + .... what does this
converge to? can you show why this converges? can you rearrange the
series to converge to something else? what's a general statement
about the convergence of alternating series?
state cauchy's theorem for a triangle. can you give a converse to
this theorem?
state stokes' theorem for a region in the plane. use stokes' theorem
to prove cauchy's theorem.
show that a holomorphic function always has a taylor series around
every point. can you prove the integral formula?
can you prove the riemann mapping theorem?
algebra: state the structure theorem for abelian groups.
state the main theorem of galois theory.
talk about canonical forms for linear transformations.
differential geometry: if a manifold M has a smooth map f to the real
line, when is the inverse image of a point c a submanifold of M?
state the frobenius integrability theorem. prove the integrability
condition is necessary if integral submanifolds exist.
what is the fundamental lemma of riemannian geometry? explain the
conditions.
algebraic topology: state the poincare duality theorem. consider a
map of some degree (1 is easier) between 2 manifolds of the same
dimension. what effect does this have on the fundamental group?
can you show that rp^2 can't be embedded in R^3?
talk about interesting mappings from S^3 to S^2. (hopf bundle) talk
about the homotopy effects of this bundle.
can you say anything about any other higher homotopy groups of
spheres? can you say anything about the effects of suspension on
higher homotopy groups? can you say something about the hopf
invariant?
can you show that cp^2 can't be *immersed* in R^6? (a theorem of
massey)
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i missed a considerable amount of the questions and still passed. (i
answered more than a few "can you?" questions with "no", which is
generally a fine answer.) also, all three of the examiners were very
nice, very friendly, and always willing to help me along when i got
stuck (as i did a lot).
tim