Daniel Vitek's generals
Date and time:October 23, 2015 9:30 AM
Committee: Peter Ozsvath (chair), Gang Tian, Fabio Pusateri
Special topics: algebraic topology, symplectic geometry
*Complex Analysis*
P: State Cauchy's integral formula? How do you prove it?
V: OK, despite several computational errors.
P: What is Liouville's theorem?
V: OK.
P: How do you deduce it from Cauchy's formula?
V: We want the Cauchy inequalities, hence the Cauchy formula for arbitrary
derivatives. We just have to differentiate under the integral sign.
P: How do you use it to prove the fundamental theorem of algebra?
V: OK.
P: Talk about the Fourier transform, thought of as a complex function.
What conditions on f do we need to ensure that this is holomorphic?
V: OK, I'm going to think about it as a real function first. We get nice
decay of the coefficients from smoothness of our original function, via
integration by parts. If we want it to be holomorphic, perhaps we can
differentiate under the integral sign? I didn't really answer the question.
P: What about just convergence in the upper half plane?
V: OK, we just end up multiplying by an exponential, so we'll need our
function to decay faster than any exponential, and then possibly some.
This answer was also not fully satisfactory.
*Real Analysis*
T: Define L^p spaces.
V: OK.
T: When do we have inclusions of an L^p space into an L^q space?
V: I write down the exactly wrong inclusion for compact spaces. I
eventually fix it, giving an example of a function that's in L^1[0,1] but
not L^infinity[0,1]. I then explain heuristically why none of this works
over non-compact spaces.
O: Why do we like L^p spaces?
V: They're Banach spaces. I can show they're complete, too! (The
committee ignored that statement and did not ask me to show that L^p spaces
are complete.)
T: How does the set of compactly-supported smooth functions sit inside an
L^p space?
V: I claim the inclusion is dense. I struggle to prove this, saying several
silly things.
O: Do you know any operators on L^1?
V: Not any interesting ones...
O: Do you know what convolution is?
V: Oh, now I know an interesting operator on L^1! I write down the
formula, and claim some smoothness properties. I try and prove the
denseness result above using convolution, screwing up several times. I
repeatedly assert that the convolution of an L^1 function and a compactly
supported smooth function is compactly supported; I usually caught this
error. I finally show that the convolution of an L^1 function and a
compactly supported smooth function is smooth (differentiate under the
integral sign again!). I write down the appropriate approximations to the
delta-distribution, but I did not know Hardy-Littlewood maximal theory so I
was unable to bound the L^1-distance of an L^1 function f and its
convolution with an approximate delta-distribution, which was what I needed
to finish.
*Symplectic Geometry*
T: Define a symplectic manifold.
V: OK.
T: Give an example of a manifold that is not symplectic.
V: S^4, and the standard argument.
T: What do symplectic manifolds look like locally?
V: R^2n with the standard form, by Darboux's theorem.
T: If I have two symplectic forms that are cohomologous, what can I say?
V: Oh, we can use the Moser trick! I proceed to jump all over the place,
and am carefully prodded to write down the hypotheses, at which point I
realize that we need a symplectomorphism induced by an isotopy to start the
Moser trick.
T: Okay, now prove Moser's theorem.
V: I write down the Lie derivative, use Cartan's formula, and stare at the
result for way too long before realizing that I've actually written down
what I need, given that the symplectic form is non-degenerate.
T: Define an almost complex structure.
V: OK.
T: What conditions do we want on the interaction between almost complex
structures and symplectic forms?
V: Compatible and tamed.
T: What can you say about the space of compatible almost complex structures
for a fixed symplectic form?
V: It's path-connected and contractible! (It's not contractible.) I
struggle mightily in proving these statements, especially the one that's
false. The only possibly useful thing I say is that the space of metrics
induced by tame almost complex structures is convex, but that's not helpful
in this context.
T: Talk about the construction of Lagrangian Floer homology. How do we set
everything up: moduli spaces, differential, etc. What can go wrong?
What's an example of a space where everything works?
V: OK, with minor screw-ups such as forgetting to quotient the moduli
spaces by translation. For the nice space question, I first suggested
T*S^1. I was told that this was noncompact and thus not a good idea, even
after I mentioned we could weight differentials by symplectic area. Tian
suggested complex tori, and I mentioned how pi_2 vanishing precludes sphere
bubbling. (Note that Tian and I had previously agreed that Lagrangian
Floer homology was part of my curriculum - it wasn't out of the blue.)
*Algebraic Topology*
(Tian had to leave for a meeting during this section.)
O: What are the homology groups of a compact oriented manifold?
V: (snarkily) If it's connected, we have the standard H_0, H_n, finitely
generated, and Poincare duality.
O: What does Poincare duality actually give you?
V: I mention the perfect pairing.
O: What's the homology of a simply connected 4-manifold?
V: H_1 and H_2 vanish by standard PD + UCT arguments. I mentioned the
intersection form.
O: What are examples of a 4-manifold with H_2 = 0, Z, Z+Z?
V: S^4, CP^2, and S^2 x S^2.
O: What's your favorite theorem in algebraic topology?
V: (ambitiously) A simply connected 3-manifold is homotopy equivalent to
S^3.
O: Prove it.
V: OK. I struggle a bit with the right way to pass from homology
isomorphisms to relative Hurewicz, which I need in order to apply
Whitehead's theorem and get the homotopy equivalence.
O: What space should go in the third column of the long exact sequence?
V: The mapping cylinder. Then everything is OK.
O: Suppose that G is a finite group acting freely on CP^2. What can you
say?
V: I start to talk about what a group element must do on homology. (I was
not thinking of the Lefschetz fixed-point theorem, unfortunately.)
O: (cutting me off) What do you know about Euler characteristic? What must
the orbit space be?
V: It's multiplicative under coverings. The orbit space must be a
manifold. x(CP^2) = 3, so we have G = 1 or G = Z/3. And we have the
permuting-the-coordinates action of Z/3 on CP^2. (Note: this action is not
free. There is no free action of Z/3 on CP^2, by Lefschetz. Nobody
commented on this fact.)
*Algebra*
(Tian came back right around now, just in time for the last two minutes of
the exam.)
O: Define a PID.
V: OK.
O: What's an example of a PID?
V: (unambitiously) Z. I think Z[x] might be ... (I was thinking of
polynomial GCDs, i.e. Q[x].)
O: (quickly) Why is Z[x] not a PID?
V: (quickly) Because (2, 1+x) is not a principal ideal. (I don't know why
I said 1+x instead of x, which gives a simpler example.)
They sent me out of the room and came out after a couple minutes to
congratulate me. I mention that I now know to differentiate under the
integral sign.
Duration: 2.5 hours. (It felt nowhere near as long.)