Chenyang Xu's generals
My committee: Kollar[K] (chair), Pandharipande[P], Tian[T]
Advanced topics: Algebraic Geometry, Commutative Algebra.
[T]:talk about the definition of L^p space, prove it's complete. on R^n, is L^2
contained in L^1, in which spaces does this hold?
[K]:state the Galois fundamental theorem, is abelian Galois extension
transitive. Give me a counterexample. (I gave a number field example.) How about
a function field example? (I couldn't figure out a good example at the time, but
Kollar hinted to compute the ramification point of a morphism....)
[T]:Prove the Riemann mapping theorem, when will two annuli be conformally
isomorphic, why?
[P]:talk about the representations of S^4, how many irreducible repsentations will
there be? Talk about one of the 2 3-dimensional representations.
[T]:talk about the bounded harmonic function on R^n, how about the bounded
harmonic function outside a ball (which I didn't really know the exact answer
except the fundamental solution, but they didn't mind)
[K]:talk about the polynomial ring over a field (I said regular, Nagata, at which
thay all laughed, since they wanted me to say it's Noetherian). Prove the
Hilbert basis theorem. Give an example of a regular ring which is not UFD (I gave the
Q(sqrt(-5))), give me one over an algebraically closed field. (I gave the example of
P^n minus some hypersurface) why? Can you compute the coordinate ring of this
variety (what a shame, I spent much of the time figuring out what it is)
[P]:are all injective endomorphisms of a finitely generated module over a Northerian
ring surjective? how about if it's surjective, is it also injective? How about the
coherent sheaf on a projective variety?
[K]:which is your favorite method to resolve the singularity of a
surface? (albanese). Can you describe it? (which I don't remember well)
[P]:if there is a degree 2 morphism, what's the sheaf of f_*(O)?
Comment: I believe I'm the first one who took commutative algebra as an advanced
topic, so before the test I didn't know how to prepare for this, but they only
required some basic knowledge. And all of them were very kind throughout the
exam. But since I was pretty nervous at the time, when I faced a new problem for
which I hadn't prepared, it was hard for me to work it out in front of them. Good
luck to everyone.