Alexandros’ Generals, 05/05/2015 (1:30 PM -- 4:30 PM)
The examiners were Assaf Naor (chair), Paul Yang and Zsolt Patakfalvi. My special topics were Functional Analysis and Differential Geometry.
Algebra:
[ZP]: State the fundamental theorem of Galois theory. What is a Galois extension. A short discussion about separable and normal extensions followed.
[ZP]: Write an extension that is not Galois. What is the Galois group of the polynomial x^3-2?
[ZP]: Prove that a finite p-group has non-trivial center.
[ZP]: Classify groups of order 8. (This never actually finished.)
[ZP]: What is a Noetherian ring? Give an example of a ring that is not Noetherian.
[AN]: Prove that the free groups of rank 2 and 3 are not isomorphic.
Differential Geometry:
[PY]: What does it mean that the Gauss curvature is intrinsic? How would you prove that? Something elementary: in the proof you mentioned that (usual) partial derivatives of smooth functions commute. How do you prove that?
[PY]: What are the compatibility equations for the first and the second fundamental form? How would you prove them? Do you know any related theorem? (Bonnet’s theorem)
[PY]: Do you know what are the conformal automorphisms of R^3? Define inversion. Do you know that every local conformal automorphism in R^3 is the restriction of a global conformal automorphism? (not really)
[PY]: State the Gauss-Bonnet theorem (local and global versions). What happens in the non-compact case? (Cohn-Vossen’s theorem) Give an example in which the inequality is strict. A discussion followed on what I believed that the difference in the inequality is.
[PY]: State the Schwarz Lemma. What are you using to prove that? (Maximum principle) Can you guess any other manifolds where similar statements might hold? (Manifolds with bounds for the Ricci curvature, such that you can compare Laplacians with the spaces of constant negative curvature - i.e. the disk with the (rescaled) Poincare metric.)
Functional Analysis:
[AN]: Suppose I give you a closed convex set in a Hilbert space and a point outside the set. What can you say? (There exists point of minimum distance.) Prove it. How is this described geometrically?
[AN]: State and prove the Radon-Nikodym theorem.
[AN]: State and prove the closed graph theorem. (Somehow I managed to do terribly in this one.)
[AN]: State and prove the Grothendieck inequality. Give an application. (I gave the restatement for bilinear forms in C(K)xC(L) spaces.) This is the same thing - give one actual application. (I said that operators from L_1 to L_2 are absolutely summing.) Something related to factorization theory? (Operators from C(K) to L_1 factor through Hilbert space.) Can you give bounds on the norms of the factors?
[AN]: State and prove John’s theorem on convex bodies.
Real Analysis:
[PY]: Do you know any Sobolev inequalities? Suppose I give you a function on the disc that vanishes on the boundary and whose gradient is in L_2. What can you say? (not much) What if we suppose it is radial? (After some hints we got to the fact that it must be in every L_p, p<\infty.) Must it be in L_\infty?
[AN]: State and prove Baire’s category theorem. Give an application. What can you say about pointwise limits of continuous functions? (After a while he told me that the set of points of discontinuity of the limit is of first category and I sketched the proof.)
[AN]: Prove the existence of non-measurable sets. Where do you use the fact that you work with the Lebesgue measure?
Complex Analysis:
[PY]: Can you write down a formula for a conformal map from a triangle to the disk? What is the idea?
[PY]: Suppose I give you an entire function whose order is bounded by a polynomial. What can you say? Prove it.
[AN]: You mentioned Cauchy’s integral formula. Can you prove it? Prove Cauchy’s theorem for triangles.
Comments: All three examiners were very friendly and provided me with hints when needed. Several times I wasn’t asked to give full proofs, but just sketch the main idea or give an explanation (especially on the Geometry questions that involved computations).