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.