Generals Transcript- Evan Huang April 29, 2025 Committee: Peter Ozsvath (chair), Zoltan Szabo, Susanna Haziot Special topics: algebraic topology and differential topology Start: 10:30 am End: 1:10 pm Topics and questions are in order of when they were asked. Real Analysis: [H] What is a measurable set? What is a Lebesgue-measurable set? What is a Lebesgue-integrable function? [H] State Fatou's lemma. [O] Given an example where strict inequality holds. [H] If a sequence (f_n) of functions converges pointwise, when can we say that the integral of the limit of f_n equals the limit of the integral of the f_n (in other words, when can we switch the integral and the limit)? [H] Sketch the proof of Holder's inequality. [H] What is L^p? Is L^p complete? Sketch the proof. What is a Banach space? [H] What is a test function? What is a distribution? How would you define the distribution associated to the function 1/x, and why is it well-defined? Complex Analysis: [H] Suppose f is a holomorphic function on a punctured disk. What can you say about f? [H] If f has an essential singularity at z, what happens as f gets close to z? [H] State the residue theorem. Sketch the proof. State Cauchy's integral formula. [H] Compute the Fourier transform of 1/cosh(pi*x) using the residue theorem. [H] Give an example of a conformal map from the upper half plane H to the unit disk D. [S] Why does SL_2(Z) act on H? Is the action free? What if we divide out SL_2(Z) by I and -I; are there still fixed points? What is the quotient H/SL_2(Z)? [S] Let Lambda = Z + Zi and Lambda' = Z + Z(2i) be the lattices inside C spanned by 1 and i, and by 1 and 2i. Then C/Lambda and C/Lambda' are tori with holomorphic structures. Are they biholomorphically equivalent? What is the (conformal) automorphism group of C? [H] State the Riemann mapping theorem. Why can the region not be all of C? [O] Do you know the Riemann mapping theorem for annuli? Prove it. What is the Schwarz reflection principle? What is Schwarz's lemma? [S] If I have four distinct points p_1, p_2, p_3, p_4 on the Riemann sphere S^2, and I have another four distinct points q_1, q_2, q_3, q_4 on S^2, is there always an automorphism of S^2 taking each p_j to q_j? What is a number associated with four points on S^2 that tells me when this is possible? Algebra: [O] Let F_3 be the free group on 3 letters, and let H be a subgroup of index 2. What is H? What if H is index d? Is there a subgroup of F_3 isomorphic to F_4? [S] What is a simple group? Do you know examples? Prove A_5 is simple. [O] Peter hands me a regular dodecahedron and asks me to compute its symmetry group. [O] Can you construct a Galois field extension with Galois group S_5? What about A_5? Algebraic Topology [O] Is there anything you want to tell us about? I talked about the Borsuk-Ulam theorem. [O,S] Consider the equation y^2=x(x-1)(x-2)(x-3)(x-4) in C^2. This defines some topological space X. What is its Euler characteristic? Its fundamental group? Is its one-point compactification a manifold? How would you show X is a manifold? [O] What is a K(G,n)? Why is CP^infinity a K(Z,2)? [O] What can you say about homotopy groups of spheres? Why are pi_n(S^n) and pi_{4n-1}(S^{2n}) infinite? Define the degree of an element in pi_n(S^n). Define the Hopf invariant of an element in pi_{2n-1}(S^n). What is a generator of pi_n(S^n)? Give examples of maps S^7->S^4 and S^15->S^8 with Hopf invariant 1. Construct a map S^11->S^6 with Hopf invariant 2. Why is the Hopf invariant 0 on pi_{2n-1}(S^n) when n is odd? [O] Sketch the proof of Serre finiteness (of homotopy groups of spheres). Differential Topology [S] What books did I read? I said Morse Theory, Lectures on the H-Cobordism Theorem, Topology from the Differentiable Viewpoint (all by Milnor), and Differential Topology by Guillemin and Pollack. [S] State the h-cobordism theorem. Sketch the proof. [O] Sketch a cartoon of a cancelling pair of critical points. [S] What is a Morse function? Why do they exist? [S] State the Poincare conjecture. How is the h-cobordism theorem used to prove it? What about in dimension 5? [S] State Sard's theorem. [S] What is a self-indexing Morse function? Construct one for T^2. Write down a self-indexing Morse function for CP^n.