Ko Honda's generals Committee: Shimura (head), Katz, Nelson Topics: Algebraic Geometry and Algebraic Number Theory Talk about factorization of a prime in a polynomial ring. What is irreducibility? For what rings R is it true that R[x_1,...,x_n] is a unique factorization domain? (I said R UFD). Talk about PIDs. What is wrong with unique factorization if we don't have a domain? Now, PIDs are Noetherian, but are there UFDs which are not? Define function of bounded variation. How can you associate a measure to a monotonically increasing function which is bounded? Take f entire. What are the conditions for the existence of a square root function? Look at the case where the region is bounded and simply connected. What are the conditions there? How would you extend a locally constructed square root to all of the region? Talk about analytic continuation on simply connected regions. Next, you can do the same problem by using Weierstrass factorization.... Talk about Fourier transform. What is the image of L^1? Talk about the L^2 theory of Fourier transforms. (Since I used the Schwartz space to show that the image of L^1 is C_0,) what is the dual of the Schwartz space? Do you know anything about Fourier analysis on L^p (L^p <-> L^q duality, interpolation theorem)? In connection with the taking the square root function, can you take the square root of f(z)=z(z-1)(z-2). (I talked about the Riemann surface associated to f(z), forgetting about the Riemann surface not sitting in projective space.) What genus does the Riemann surface have? What is the genus? (I gave the dim H^1(C) definition, which was part of my downfall.) For the original curve in C^2 contained in CP^2, what is H^1(C)? It's the same as H^1(C). Now take C'=C-{p_1,...,p_n}? What is H^1(C')? (Here I said, "I don't know too much about these things," but Katz responded, "Well, you should know it anyway.") Talk about residues of meromorphic differentials. State the residue theorem. How can you use residues to compute the homology of H^1(C') via integrating mero differentials? =========================================================================== (Since I could only give vague generalities,) Talk about Euler characteristic. What is it on a surface? Look at the simplicial decomposition and tell me the Euler char. in terms of the k-simplices. Now use this to compute H^1(C'). Need H^2(C')=0. How does one know this? Talk about Poincare Duality. What are the conditions you need for Poincare Duality to hold? What is the Fermat curve? Compute its genus. Give a basis of its holomorphic differentials. Talk about the Hilbert polynomial. (I gave a definition in terms of H^i(F(m)) with F coherent.) What does this have to do with the classical definition of the Hilbert polynomial? What is the arithmetic genus? What are the degree, the leading coefficient, and the last coefficient of the Hilbert polynomial? What is an algebraic group? Give one (O(n)). Give another. (I gave U(n), which was not a good choice.) Why is what you wrote down wrong? Give some more algebraic groups. GL(n), SL(n)... Which of them are rational varieties? (GL(n), SL(n) are.) Prove it. You defined O(n) as xx^t=1. Give a more intrinsic definition of this, via nondegenerate symmetric bilinear forms. Give intrinsic definitions of Sp, U, etc. in this way. Which ones are connected? Talk about Dirichlet's theorem on arithmetic progressions. Define I(C)/P(C). (This is where I got majorly bogged down.) Talk about the case where we have Q(z_m)/Q. So C=mv_infty in this case, but what happens when C=m only? (....by this time my brain was complete mush.) Exam took 3 1/4 hours.