January 18, 2002 Committee: P. Ozsvath (chair), E. Stein, P. Yang Bojan Gornik's Generals Topics: Algebraic Topology, Differential Geometry Duration: 1h30 Algebra ------- Galois group of Q(sqrt(3),sqrt(2)). Hilbert Nullstellensatz. Give examples of maximal ideals in K=RxRxRx... (countably many Rs). I gave I_1=(0,*,*,*,...) or I_2=(*,0,*,*,...) and so on. Are there other maximal ideals in K ? Yes. Name one. Take the maximal ideal that includes the ideal which contains sequences with almost all (=all but finitely many) 0's. Real Analysis ------------- Talk about the fundamental thm of calculus and its proof. The Cantor function. L2 spaces, why are they complete ? Complex Analysis ---------------- The Riemann mapping thm. What are the ingridients in the proof ? How does RMT relate to uniformization ? Name some Phragmen-Lindelof thms. Topology -------- Give two manifolds (of the same dimension) X,Y that are not homeomorphic but are homotopically equivalent ? I was led to the answer: X=sphere minus three points and Y=torus minus one point (they are both homotopically equivalent to the wedge of two circles). Why are X,Y not homeomorphic ? Again, after some hinting, it turned out that their cup product structures on (relative) cohomologies aren't equal. Give an example of two compact manifolds X,Y ! I gave an answer in 4 dimensions, but was told there's an example in 3 dimensions (lens spaces). Now, why can't Z+Z+Z+Z be a fundamental group of a 3-manifold ? There were some hints, but we didn't really get to the answer. Geometry -------- What does it mean that Gauss curvature is intrinsic ? Give an example of two isometric embeddings of a surface in R3 with different 2nd fund.forms. Prove that X,Y from the topology section aren't diffeomorphic "in the spirit of diff.geom.".