My Generals Exam -- Zhiwei Yun May 5,2005, 1:00pm-2:15pm Committee: MacPherson(Chair), Gunning, Pandharipande Special Topics: Algebraic Geometry, Algebraic Topology Complex Analysis: (Gunning) If you have a function on the unit disk which is holomorphic outside the origin, what can you say about it? ---I said 3 types of singularities. What's the behavior near an essential singularity? ---Big Picard. Give an example of an essential singularity and prove it. ---exp(1/z). Suppose a function is holomorphic on an annulus, what does its Laurent series look like? ---It can be written as a sum of two functions. One is holomorphic near infinity and the other holomorphic near 0. Algebra: (P)Over the finite field F_p, are there irreducible polynomials of each degree? ---Yes and I computed the number of them. (M)If you have a Z/5 action on a complex vector space, what does this action look like? ---The vector space splits into 1-dim weight spaces. (M)What about an S_3 action? ---I described representations of a general dihedral group. (P)Classify groups of order 8. ---abelian,dihedral and quaternion. Real Analysis: (Gunning) We have various notions of convergence for a sequence of functions, say pointwise, L^1, uniform, etc. What are the relations between them? ---I state the relations and give examples to show those are all the implication relations. L^1 convergence implies... ---a.e. convergence of a subsequence. And I gave the proof. Algebraic Geometry: (Pandharipande) Consider a cubic surface in P^3, what are its Hodge numbers? ---The only interesting number is h^(1,1)=7. What's its intersection form? ---diag(1,-1,-1,-1,-1,-1,-1). Give an example of a genus 4 curve. ---I write down a hyperelliptic one. Are there any non-hyperelliptic ones? ---Yes. Compute the dimension of moduli spaces. Give a concrete example. ---I thought about complete intersection of two surfaces in P^3, but somehow convinced myself that it wouldn't work. Actually it does, given by the complete intersection of a cubic and a quadric. Algebraic Topology: (mostly MacPherson) What's the relation between \pi_1 and H_1? ---Abelianization. Are there any space with trivial H_1 but nontrivial \pi_1? ---Pick any space with \pi_1 a non-abelian simple group. How do you construct a space with \pi_1 isomorphic to a given group? ---Pick one loop for each generator and attach one 2 cell for each relation. If the Chern class of a complex line bundle is trival, is the bundle trivial? ---Yes. And I gave a proof using Cech cohomology. What's the first Stiefel-Whiteney class of the tangent bundle to the Klein bottle. ---I computed the mod 2 cohomology of Klein bottle and said this w_1 is non-zero. But one cann't distinguish nonzero mod 2 classes. (P)What's the cohomology ring of the Grassmannian? ---I wrote down in terms of Chern classes of the canonical bundle. (P)Give a projective embedding of Grass(2 planes in C^4). ---Using Plucker coordinates. (P)What does it look like? ---It's a quadric hypersurface in P^5.