Committee: Gang Tian (chair), Peter Ozsvath, Robert Gunning
Special Topics: Complex differential geometry, Symplectic topology.
Duration: 2 hr 10 min.
Date: May 8, 2014
Complex Analysis
[G]: -What can you say about holomorphic functions on the disk minus the origin?
-Do you know what is Picard big theorem? (Yes) Can you prove it? (No) Can you prove a weaker version of it?
-What can you say about the simply connected domains in the Riemann sphere?
-When is possible to extend a biholomorphism to the boundary of the domain? Can you extend it further?
-What is your favorite meromorphic function? (I don t have), choose one (conformal map of S^2), something more elaborated? (I can talk about gamma function, but I don t really like it, they laughted).
Real analysis
[G]: -Compare different kinds of convergence of functions.
[T]: -State monotone converge theorem and give examples where you can see that the hypothesis of the theorem are necessary.
-What do you know about fourier transform?
-Can you describe its eigenvalues? What about eigenfunctions? Compute the fourier transform of the Gaussian.
[G]: -Do you know any generalization of the fundamental theorem of calculus?
-If f is in L^1, what can you say about the integral of f between 0 and x (as a function of x).
Algebra:
[O]: -What is a PID? What is the fundamental theorem of finitely generated modules over a PID?
-Give a counterexample of this theorem if the ring is just a domain.
[G]: -State the main Galois Theorem. Give an example of a field extension which is not normal.
- State Sylow s theorem. What can you say about groups of order pq (p, q different primes).
[O]: -What is a simple group? Can you give examples? (alternating groups) Why are they simple? What about A_4?
-What can you say about the subgroup of a free group? (free) Prove it.
-What kind of fundamental group can you have in a complex manifold? (I mention something about Kahler groups and a theorem of Taubes for complex threefolds, I am glad they didn t ask how to prove them), now more concretely, fundamental group of a Riemann surface? Compute the fundamental group of CP^1 minus 3 points. ([T]: we should move to the special topic)
Symplectic Topology:
[O]: -Is there a complex manifold which is not symplectic? What about symplectic but not kahler?
[T]: -Why dimension of H^1(X,R) is even for a kahler manifold? Is there a serious reason for this?
[O]: Prove that the Kodaira-Thurston manifold is symplectic (long time talking about symplectic fibrations).
[O]: -When are two symplectic manifolds symplectomorphic? (mmm) Surfaces?
-What do you know about the fundamental group of symplectic manifolds? (again, long time talking about symplectic connected sum, and I proved Gompf s theorem).
Complex geometry:
[T]: -Prove Kodaira vanishing theorem.
-Give examples of Fano manifolds.
-Prove a formula for the canonical divisor of the blow up of a point in CP^n.
-Is this variety a Fano manifold?
-Compute H^2(X,TX) for X fano. How do you interpret this in terms of geometry?
Comments:
The exam was super fast at the beginning, as you can see the common topic questions are very straightforward, however I got confused in some moments (most of the times because I didn t understand what is the purpose of the question), the best choose, for sure, is to ask what kind of answer are they looking for. It s probable that I am the first person who has taken symplectic topology, but it was a good choose, the questions are relatively standard (nevertheless, I was supposed to study from a book that is focus on analysis, different kind of capacities, pseudoholomorphic curves, etc., but nothing about the things Peter asked me during the exam!), don t make my mistake, make sure you know what you are supposed to know!. The really painful part of the exam was Kodaira vanishing, I needed several hints to achieve the conclusion. The professors were very nice during the exam, I strongly recommend you to choose any of them to be in your committee (if they are close to your topics). As an anecdote, I remember professor Gunning said just after the exam something like now you should figure out how to forget this experience , the day after my exam I saw professor Gunning during tea time, he shouted: wow, you survived! , me and my friends laughed (you can guess what he meant, but he wasn t talking about the exam). Good luck! (and work with your classmates, you will end up learning a lot about their fields).
Thanks!,
Anibal