General
Committee: David Gabai (chair), Paul Yang, Janos Kollar
Time: 05/06/2010, 13:00-14:45
Place: Fine 502, Professor Gabai¡¯s office
Special topic: Differential Geometry, Algebraic Topology
I arrived at Gabai¡¯s office a few minutes before one o¡¯clock. Gabai and Yang were already there, so we sat there
and waited for Kollar. During that time, Yang walked around Gabai¡¯s office and saw the posters hanged on the wall.
Kollar arrived at exactly one o¡¯clock. They asked me to choose which topic to talk about at first, and I knew that
they may prefer to start with general topic, so I said analysis.
Real Analysis:
Y: Do you know Fourier Transform?
Me: Yes, and I wrote down the definition for L^1 functions.
Y: What¡¯s the property of it?
Me: It¡¯s bounded.
(I also wanted to say it is continuous and decay to zero at infinity, but he stopped me.)
Y: What about L^2 functions?
Me: If f belongs to L^1 intersect L^2, then there is Plancherel equality, so we can extend Fourier transform to L^2.
Y: What can you say about the Fourier transform of a compacted supported function?
Me: It can¡¯t be compacted support.
(In fact, it¡¯s an entire function)
Then Yang said he was happy with the real analysis part, and I was a little surprised since it was so short. Then we
switched to the complex analysis part.
Complex Analysis:
Y: What is analytic function?
Y: What¡¯s the property of this function?
Me: Cauchy¡¯s theorem for non-homologous curve in the region where the function is defined.
G: What¡¯s the orientation of curves if you do the integration on the boundary of an annulus?
Y: Proof of Cauchy¡¯s theorem?
Y: Other properties of analytic function?
Me: Power series expansion, and Cauchy¡¯s equation.
Y: Maximum value principle?
Y: What can you say about an entire function defined on the plane?
Me: Weierstrass factorization theorem.
Y: What can you say if an analytic function growth as a polynomial?
By that time, I was so stupid that I stuck there for a few minutes and got the answer by some hint.
This is the Complex Analysis part. They did not require any details, but only the idea of the proof.
Algebra:
K: What is normal subgroup? Can you get some natural map from a normal subgroup?
K: What topological objects can the original group, normal subgroup and quotient group relate to?
Me: A topological space, normal covering space and deck transformation group.
G: Is the action of deck transformation on the normal covering space by left or by right?
Me: I did not know the answer, but I can try to figure out the answer here.
I studied the lifting of curves to the covering space, and found that it was right action. But it seemed that Dave was
not so satisfied with it, and he asked me to do a specific example: figure eight, its universal covering space and the
action of deck transformation group. I did it, and took some time to figure out what the deck transformation group does
to the picture. Dave said he thought the action should be left action, but, anyway, I got the right answer in this specific
example.
After long digression, we came back to algebra again.
K: Do you know the Ext functor of abelian group?
K: Do you know where does it appear?
Me: All I know about it is in algebraic topology, and I stated the universal coefficient theorem.
K: What is Ext(Z_m,Z_n)?
G: What is Ext(Z_m,Z)?
K: Can you get some long exact sequence from a short exact sequence of abelian group and another abelian group?
I tried to give the whole proof, since I was not quite sure about the accurate answer, and wanted to figure it out in the
process of proof. But Kollar just wanted the answer. Luckily, I got the right answer.
Then Gabai gave me a Dodecahedron.
G: What is the symmetric group of this guy?
Me: It is A_5.
G: It is the orientation preserving symmetric group or the whole symmetric group?
I thought about it for one minute and found it is the orientation preserving symmetric group.
G: Does it have any normal subgroup? Can you get the answer by this Dodecahedron?
I watched the Dodecahedron for maybe five minutes, but could not get the answer. After the Differential Geometry part, we
came back and Gabai supposed there was a 2\pi/5 rotation in the normal subgroup, in this case I could prove it.
The algebraic part is really weird, since all the things here are related with topology and there is nothing about some Galois Theory.
Differential Geometry:
Y: Define Gauss curvature and mean curvature.
Me: Defined Gauss curvature by Gauss map, defined mean curvature by principle curvature.
Y: What can you say about a surface in R^3 with positive curvature?
Me: It must be locally convex, and I guess it should be globally convex.
But I could not prove it. I struggled here for a few minutes, and then Yang switched to another question.
Y: Do you know Gauss-Bonnet Theorem?
Y: Gauss-Bonnet theorem on a triangle.
I made a mistake on the sign of the angles, but it seemed that they did not realize it.
Y: Gauss-Bonnet theorem for complete non-compact surface?
Me: Cohn-Vossen theorem.
Y: Gauss-Bonnet theorem for higher dimensional manifolds?
Y: In four dimensional case, what is the name of the form you integrate?
Me: I don¡¯t know.
Y: Use this theorem, what can you say about the quotient of S^4?
Me: S^4 does not have an orientable quotient. Since the deck transformation group has order at most two, and there is not a fixed
point free orientation preserving isometry on S^4.
Y: Why?
I guess he wanted me to talk about Synge Theorem, but I used Lefschetz fixed point theorem. So we switched to algebraic topology,
and I was very happy with it.
Almost everything in the Differential Geometry are about surfaces in R^3, and in fact, I only reviewed very little of this part, so
I felt a little awkward in this part. But Yang is really a nice questioner.
Algebraic Topology:
G: What is Lefschetz fixed point theorem?
G: Hopf fibration?
G: \pi_3(S^2)?
G: Euler characteristic of a compact three manifold with boundary?
G: Poincare duality, non-orientable case?
G: What is the homology group of the complement of a twenty components link in S^3?
Me: Alexander duality.
G: How to prove it?
Me: Deduce it by Lefschetz duality
Dave was so nice that he did not ask me to prove Lefschetz duality, so in some sense I was cheating.
G: Give an example of contractible one manifold.
Me: R^1.
G: Any others?
Me: (I was confused here.) Do you mean manifold with boundary?
G: No.
Me: Do you suppose the manifold has a countable topological basis?
G: Yes, so there is another condition.
Me: Hausdorff!!
So I constructed the one dimensional non-Hausdorff manifold. (It is the point, and sometimes it is really difficult follow the idea
of the questioner.)
G: Can it be the quotient of the plane?
Me: It can be the quotient of some foliation.
I struggled for three minutes to construct the foliation, but I failed. Since I have not learned foliation for more than one year, but
it seemed that Dave did not mind. I guess maybe he is happy with I have mentioned foliation.
G: Construct a contractible four manifold.
Me: Trivial example: D^4. Then I drew a two components link on the blackboard with linking number one, one component is unknot, and the
link is not Hopf link. It gives us the Kirby diagram of a nontrivial contractible four manifold.
G: Why?
Then we talked a little about Hurewicz theorem and Whitehead theorem.
That¡¯s all. They asked me to wait outside of the office, and two minutes later, the committees came out and said congratulations to me.