Committee members: Professor Paul C. Yang (Chair), Professor Peter Constantin, Professor Claus Sorensen (1:00pm-4:15pm, May 14th, 2012) Elliptic PDEs of Second Order 1. State and prove weak maximum principle, Hopf's lemma and strong maximum principle. 2. Talk about some properties of the spectrum of elliptic operators (positivity, number of eigenvalues, multiplicity of eigenspaces) 3. De Giorgi-Moser's estimates for divergence form. Give the main steps to get Harnack's inequality. Explain why it is so important? 4. Talk about John-Nirenberg's inequality and try to find a function which belongs to BMO but does not belong to L^{\infty} 5. If u is harmonic in B. Assume the boundary of B is U\cup V (disjoint), u=0 on U and du/dn=0 on V. Show that u=0 in B. 6. If u is harmonic in B\{0}, and u=o(|x|^{2-n}), show that u can extend to the whole B. Real analysis 1. Fundamental theorem of calculus 2. Is L^1[a,b] compact? Find a precompact subspace of L^1 3. Assume we have a function f in H^1. If grad f belongs to L^2, is f bounded? Try to give an example Complex analysis 1. State and prove Rouche's theorem 2. State and prove Jensen's formula 3. Talk about the growth of numbers of zeroes for an entire function of given order, and prove it 4. State and prove Laurent's theorem (expansion and power series), also prove Cauchy's integral formula Algebra 1. Talk about cyclotomic field 2. Given any finite cyclic group G, let Q be the field of rational numbers. Is there a Galois extension K/Q such that Gal(K/Q)=G? Riemannian geometry 1. Gauss-Bonnet formla for surfaces (local version and global version) 2. Gauss-Bonnet-Chern's theorem for closed manifolds (higher dimension), basic ideas of moving frames (connection and curvature forms, Maurer-Cartan) 3. Give an applications of Gauss-Bonnet's formula 4. State Cohen-Vossen's theorem for complete surfaces. (It's an inequality version of Gauss-Bonnet. Give an example such that the inequality is strict) Talk about the main ideas of Cheeger-Gromov's theorem for chopping Riemannian manifolds. 5. Given a sphere, if we remove 3 points, can we construct a complete negatively curved metric? 6. State and prove Gromov-Bishop's volume comparison theorem (by Laplacian comparison) 7. State and prove Shiu-Yuen Cheng's maximum diameter theorem Comments: Don't worry! The professors were very nice. They had been smiling during the 3 hours, and they tried to let me calm down when I was stuck. If you have no background for some questions, just let them know and they would switch to another question. You don't have to know about everything. It will be okay if you can figure them out with some hints. Good luck!!!