Exam committee: Wee Teck Gan (representation theory, chair) Alice Chang (differential geometry) Edward Nelson Date: 05/12/00, 10am-12noon Real Analysis Convolution of functions. What are its properties? Show examples of functions that are in different L^p spaces. State Hoelder's inequality. Do you know the Baire Category theorem, and its application to open mapping theorem of Banach spaces. Can you have a function continuous at precisely the rationals (irrationals)? Show that the dual to L-infinity consists of elements not given by integrating against L^1 elements. Complex Analysis Prove the fundamental theorem of algebra. What is Schwartz's lemma? Suppose you have three meromorphic functions on the complex plane satisfying f(t)^n + g(t)^n = h(t)^n, n>=4, show that it can only be the case if they are constant multiples of each other. (I did not know this one. The point is to study the equation as a map from C to P2(C).) What are the universal covering spaces for Riemann surfaces? Algebra Prove that no group of order 132 is simple. (This one involves repeated applications of Sylow, and I did not get it without many hints.) What are the finite fields of order F(p^n), p prime? What is the Galois Group of F(p^n) over F(p) (generated by the Frobenius map.) What is the characteristic polynomial of the Frobenius map viewed as a F(p)-vector space map. Differential Geometry How to do you compute the curvature of a curve? What does the isoperimetric inequality say? Explain the Gauss-map and how the curvature is related to the differential. Why is it an intrinsic quantity? Do you know the Gauss-Bonnet Theorem? Explain it in the two-dimension case. What is the Euler Characteristic? What is the Rauch's comparison Theorem? What is the Bochner Formula? What is special about Kahler manifolds and its Ricci Form? Representation Theory What do you mean by maximal torus? What are the irreducible representations of su(2,C)? What do you know about the Bott-Weil-Borel Theorem? What are the irreducible representations of S^4? If you have two linear representations p_1, p_2 from a finite group G such that p_1(g) is conjugate to p_2(g) for every g, is it true that the two representations are isomorphic? Advice: Take some time to look at other people's generals. Most of the problems are asked over and over again. It is definitely true for general topics. Do not be overly worried about proofs and technical details. My impression is they are looking to see if you have a good understanding of what the subject is about overall.