Committee: John Pardon (P, chair), Peter Ozsvath (O), Yakov Shlapentokh-Rothman (S)
Date/Time: 1/07/19, 2 - 5 PM
Topics: Differential Topology, Symplectic topology
Differential Topology
O: What are characteristic classes?
Me: They're cohomology classes in the classifying space...alternatively (abelian group) homomorphisms from K(X) to H*(X) (this second statement needs more elaboration to actually be true…)?
O: Give me an example of a characteristic class.
Me: The Stiefel-Whitney classes! Proceed to outline a couple of the definitions (Steenrod squares, obstruction theory, O mentions a definition of w_1 as maps from loops to Z/2 depending on whether the bundle is twisted over the loop or not)
O: Here's another way to define Stiefel-Whitney classes.
(We proceed to work out the definition following from the application of Leray Hirsch to the cohomology of the projectivized bundle.)
O: What are the Stiefel-Whitney classes of RP^n?
Me: Let a be the generator of the first cohomology group, then the total class is (1+a)^n. Gave the proof in milnor stasheff.
O: Given an immersed complex curve of degree d in CP2, what is its genus?
Me: g = (d-1)(d-2)/2, comes from taking first Chern classes of the sequence TC -> TCP^2|_C -> N_C.
P: How would you generalize that to the case of an immersed singular curve?
Me: Each node adds two more self intersections, adjust the formula accordingly (genus goes down by k, k = number of nodes)
P: Given a continuous map of a surface of genus g to a surface of genus h, h greater than g, how can you show that it has degree zero?
Me: Let's look at the map on...homology?
P: Cohomology, what can you say about the map on H^1?
Me: ...oh, it has a kernel! So take an element in the kernel, take its cup square...
O: Cup square?
Me: Oops, not the cup square, that's zero. But take some other class that has nonzero cup product with this (exists by poincare duality) and then use the fact that it commutes with the homomorphism, producing an element in the kernel of the map on H^2, so degree is 0.
P/O: So now what happens if f is holomorphic?
Me: (after a reminder of what degree actually means) If f is holomorphic, all points in the preimage of a point are positively oriented, degree 0 implies this is empty so f has to be constant.
P: How do you show that the framing on a knot in S^3 is independent of the choice of bounding surface?
Me: No idea, glue the surfaces together and see what happens? (P takes me through a proof).
Symplectic Topology
P: Describe a Moser-type argument.
Me: Sure, here's the Darboux theorem.
P: What is the gromov compactification of the moduli space of curves?
Me: Can I do genus zero? Its the union of maps modelled on trees, one map per vertex, two adjacent vertices share a nodal point.
O: Careful, do we have any conditions on the constant maps?
Me: Right, the constant maps need at least 3 nodal/marked points to make the automorphism group finite. Also, for higher genus we have to start worrying about complex structures on the domain.
P: What can you say about holomorphic spheres in CP^2?
Me: For a generic compatible complex structure J, there is exactly one. Upper bound follows from positivity of intersection, lower bound follows by calculation for the standard structure and the fact that the moduli spaces are cobordant for generic J.
P: How do you show holomorphic curves for the standard J are algebraic?
Me: Look in an affine chart?
P: When are complex analytic submanifolds of a complex variety algebraic?
Me: No idea.
P: They have to be projective (not sure if this is exactly what he said?)
P: Describe the Hofer lemma/formation of bubbles.
Me: (Gives the standard argument)
P: Give me some examples of symplectic manifolds.
Me: Surfaces, Kahler manifolds, can make a lot with the Gompf fiber sum, cotangent bundles.
P: What does a Lagrangian in a cotangent bundle look like?
Me: The graph of a one-form is Lagrangian iff it is closed, so as a corollary any Lagrangian locally looks like the graph of a closed one-form.
O: Let's talk more about this gompf fiber sum.
(Long discussion with O about symplectic fiber sum. Highlights include an embarrassingly long effort to compute the fundamental group of a surface and then Gompfs argument for constructing large families of symplectic manifolds which are not Kahler.)
Real analysis:
S: Define Lp space
Me: Sure.
S: Given a finite measure space, are there any inclusions of Lp spaces?
Me: If q >= p, there is an inclusion Lq -> Lp. We can show this with Holders.
S: Prove Holders inequality
Me: Sure, assuming Youngs inequality. S mentions after I forget that Youngs inequality can be proved using convexity of log.
S: What is the limit of the Lp norms of a function? Define the essential supremum etc
Me: (Oops, I've never worked this out myself! I get tied up with this for a really long time again, eventually eke out a proof for finite measure spaces and handwave how I can generalize it, given the function in question decays at infinity. P also leads a discussion on whether the "inf" in the definition of essential supremum is realized.)
Algebra:
P: Describe all the subfields of Q[rt2, rt3].
Me: Sure, needed a hint for the last one because I forgot rt6 is in the field.
P: Which of these is Q[rt2 + rt3]?
Me: Its the whole field, I work it out using elementary methods but P leads me to do it using Galois theory too.
S: Take a smooth one parameter family of self adjoint operators A_s, what kind of regularity can you give their lowest eigenvalue? Show that this is sharp.
Me: (Another fumble. I write down the variational characterization of this eigenvalue but then S takes me through the rest of the proof pretty much. I kept trying to show it is continuous, so needed some guidance to the real conclusion that its Lipschitz continuous.)
Complex Analysis:
By now it was getting pretty late, so this was a short section.
P: Classify simply connected domains in C.
Me: C or the disk, consequence of Riemann mapping/Liouville.
P: Give me a holomorphic map on the disk that doesnt extend to the boundary.
Me: I write down \sum z^k/k, but am unsure and after some staring write \sum z^k.
P: What is liouvilles theorem? Prove it.
Me: Sure, slip up at first by estimating f instead of its derivative with Cauchy but then show f' = 0 so f is constant.
P: What are the biholomorphisms of the unit disk?
Me: Gave an explicit formula, justified by Schwarz reflection, which implies they are Mobius transformations which preserve the unit circle.
O: What are the biholomorphisms of an annulus?
Me: Handwave with Schwarz reflection again, generated by reflections and rotations.
O: That's a good enough answer for (looks at watch) 5 PM.
P: Could you step out into the hall so we can discuss?
(A minute or two later)
P: Congratulations, you passed.
Comments: My committee was pretty nice, and they asked interesting questions. At times I couldn't figure out what kind of answer they wanted, so I just said as much as I could until they were either satisfied or gave me a hint.