General Exam Carolina Araujo 1/26/01 2:00pm Topics: Algebraic Geometry and Model Theory Committee: Janos Kollar (Chair), Simon Kochen, Edward Nelson --------------------------------------------------------------------- Real Analysis (Nelson) - Let h(x) = \int_R f(x-y)g(y) dy. What can you say about it if both f and g are in L^2? (I proved it existed for all x, was bounded, continuous, and vanished at infinity) - Define different notions of convergence for functions on [0,1]. Which modes of convergence imply others? When does convergence in L^p norm imply convergence in L^q norm? - Suppose you have a sequence of continuous functions on [0,1], pointwise convergent to f. Does it have to converge uniformly? (No) What if the functions decrease to zero? (Yes) Prove it. - Consider the problem: (i) y' = f(y) (ii) y(x_0) = y_0. What are sufficient conditions for the existence of a solution? (First I showed that f Lipschitz implies existence and uniqueness of a solution, then I gave a vague idea of how to prove that continuity of f is enough to guarantee existence) Complex Analysis - (N) What can you say about a holomorphic function whose real part is bounded? - (N) What is your favorite proof of Liouville's Theorem? - (N) Suppose you have a holomorphic function on { 0 < x < 1}, continuous and bounded in absolute value by 1 on the boundary, and bounded in the interior of the strip. What can you say? (It's bounded by 1 on the strip) Prove it. - (Kol) What is special about simply connected regions in the plane? - (Kol) Prove the Riemann Mapping Theorem. Algebra - (Kol) Compute the Galois group of p(x) = x^7 - 3. - (Koch) Classify finite fields, their subfields, and field extensions. Which are the automorphisms of a finite field? Model Theory (Kochen - he had helped me make the program for this topic) - What exactly did you study? (The basic stuff, plus Ultra-products, Morley Theorem, Kochen's paper on Model Theory of Local Fields - as an application, plus some general logic, like Godel's Incompleteness Theorems and Tarski's book on undecidability. He didn't ask me almost anything about these things...) - Define Ultra-Products. (I defined them, and stated the Ultra-Product Theorem) - Is there any property of Ultra-Products that goes beyond the given models? (omega-saturation) Do you know how to prove it? (I said yes, and proceeded to prove it, but he stopped me and said he just wanted to know if I knew it) - Define model completeness. Use the model completeness of Algebraically Closed Fields to prove Hilbert's Nullstellensatz. - Define ordered fields, real fields and real closed fields. Use the model completeness of Real Closed Fields to solve Hilbert's 17th Problem. Algebraic Geometry (Kollar - I had talked to him the whole semester before the exam, and he had given me lots of questions to work on. I believe that's why he asked me so little here) - Compute the genus of the curve x^3 = y^6+1 (I did it by blowing up the curve until I got a nonsingular one, and computed its arithmetic genus. I had to prove the formulas I used: the genus-degree formula and how the arithmetic genus changes when you blow up the curve. He also asked me what happened in characteristic 3, and I took a long time to realize the curve was nonreduced. That was really embarassing...) The exam took about 1h 50min.