Tim Hsu generals questions committee: browder (chair), mather, prato (the questions are not verbatim but are more or less correct) analysis: state the fundamental theorem of calculus. can you give exact conditions when it holds and counterexamples relating to it? when does a fourier series converge to its original function? what can you say about the fourier series of a smooth function f? (the coefficients go to 0 as n -> infinity.) does this condition imply that f is smooth? consider the series 1 - 1/2 + 1/3 - 1/4 + .... what does this converge to? can you show why this converges? can you rearrange the series to converge to something else? what's a general statement about the convergence of alternating series? state cauchy's theorem for a triangle. can you give a converse to this theorem? state stokes' theorem for a region in the plane. use stokes' theorem to prove cauchy's theorem. show that a holomorphic function always has a taylor series around every point. can you prove the integral formula? can you prove the riemann mapping theorem? algebra: state the structure theorem for abelian groups. state the main theorem of galois theory. talk about canonical forms for linear transformations. differential geometry: if a manifold M has a smooth map f to the real line, when is the inverse image of a point c a submanifold of M? state the frobenius integrability theorem. prove the integrability condition is necessary if integral submanifolds exist. what is the fundamental lemma of riemannian geometry? explain the conditions. algebraic topology: state the poincare duality theorem. consider a map of some degree (1 is easier) between 2 manifolds of the same dimension. what effect does this have on the fundamental group? can you show that rp^2 can't be embedded in R^3? talk about interesting mappings from S^3 to S^2. (hopf bundle) talk about the homotopy effects of this bundle. can you say anything about any other higher homotopy groups of spheres? can you say anything about the effects of suspension on higher homotopy groups? can you say something about the hopf invariant? can you show that cp^2 can't be *immersed* in R^6? (a theorem of massey) ----- i missed a considerable amount of the questions and still passed. (i answered more than a few "can you?" questions with "no", which is generally a fine answer.) also, all three of the examiners were very nice, very friendly, and always willing to help me along when i got stuck (as i did a lot). tim