Konstantinos Varvarezos - General Examination
7 May, 2018 - 1:30 P.M.
Committee:
Zoltán Szabó (Chair)
Paul Yang
Vlad Vicol
Special Topics:
Algebraic Topology
Differential Geometry
Algebraic Topology:
Szabó: So you read most of Hatcher? That book is okay?
I say yes, and that I guess the Bott-Tu book would also be okay.
Sz: What is a CW complex?
I talk about gluing cells. I mention something about weak topology.
Sz: Finite case is clear, but what about infinitely many cells?
I mention how a subset is closed iff its intersection with each cell is closed.
Sz: What about conditions on the attaching maps?
I was uncertain. He mentioned how there could not only be cells of arbitrary high dimensions but also infinitely many cells in a given dimension. He says that the technical requirement is that the boundary must meet only finitely many cells. I suddenly remember that this is the "C" part of "CW".
Sz: Give a cell structure for RP^n.
Sz: Compute homology. Why are the boundary maps like that?
I mention something about the degree of the antipodal map depending on the parity of the dimension.
Sz: How about RP^2 x RP^2?
I start writing down a 3 x 3 grid of product cells and arrows.
Sz: There is a slight problem - d^2 is not zero - you need some signs.
I mention how there is also the Künneth formula.
Sz: What is it? Maybe for a field?
I write down the formula.
Sz: What about cohomology? What's the ring for RP^infinity with Z/2 coefficients? How to prove this?
I mention one could argue by embedding copies of RP^n and using Poincaré Duality, or one could use the spectral sequance.
Sz: What is Poincaré Duality? Do you remember proof?
I say not really and that I vaguely remember something about dual triangulations.
Sz: Let's compare S^2 x S^4 with CP^3 - compute homology.
Turns out they have the same. I mention that cohomology rings should be different.
Sz: What are they? What are the Poincaré duals of the generators for the case S^2 x S^4?
Sz: Okay what about homotopy groups?
There was a discussion about the first few homotopy groups of S^2 x S^4, then I wrote down the long exact sequence of the fibration for CP^3. Eventually, he asked me a bit about the cell structure of CP^n and about the stable homotopy groups of spheres. I mentioned Freudenthal Suspension Theorem, but I didn't quite remember the dimension range (I vaguely recalled 2n-1 as a bound). I guessed that pi_8(S^7) would perhaps be pi_3(S^2) but then I thought pi_4(S^3)=Z/2 made more sense, and that was the right one.
Differential Geometry:
Sz: Which book did you read?
I say I am familiar with Lee's book as well as do Carmo's
Yang: Given a surface, can it be locally realized as a subspace of R^3?
I am not sure, and I start mentioning that there are conditions on second fundamental form.
Y: There are two different questions - you're given an actual surface and want to see if it can be locally embedded in R^3 versus given an abstract metric and second fundamental form. So what about the first situation?
I doubtfully say I suppose it might be possible...
Y: There is the Gauss curvature
Okay...
Y: You'd say it's a hard problem?
I say it seems like it.
Y: What about the other situation, where you have an abstract second fundamental form.
I mention it must satisfy the Gauss-Codazzi equations. I begin to write the general form, but he says it's too complicated - just give the special case for the Gaussian curvature. After that I fumble about trying to recall the Codazzi equation. He asks just for an intuitive understanding, but I am unable to figure it out. He says that you basically want an integrability condition, and that it should be something like D_k H_ij = D_j H_ik (where H_ij are the coefficients of the second fundamental form in matrix form).
Y: Do you know Gauss-Bonnet?
I write down the formula and mention there is a boundary term in the case of surface with boundary.
Y: Can you give a proof?
I mention that it will follow from the formula for a triangle along with a triangulation. For that formula, I first mention the lemma about the total winding angle of a polygon being 2p, and then began proving the formula - at some point in the middle of some calculation, he says it's okay, and so I just finish by saying what the calculation will give, and sketch the rest.
Y: We know there are generalizations for higher even dimensions, but what about odd?
I say I'm not aware of such results, and that the Euler characteristic wouldn't be of much help.
Y: Some 3-manifolds are the boundary of 4-manifolds. Can all of them be realized this way?
Ah! I say that certainly topologically, and after considering it for a few moments I agree that they all can.
Y: So can you guess what the invariant would be for the boundary term.
I ask if he's asking for an analogue of the "geodesic curvature" which appeared in the Gauss-Bonnet, but I can't come up with anything. He mentions it should involve the second fundamental form. Eventually he says that it's a 3 x 3 matrix and that there are many invariants, so it's okay.
Y: Back to the 2-dimensional case - what about complete, not necessarily compact?
I mention the Cohn-Vossen inequality. He says that there should be some extra finiteness condition, otherwise one could conceivably have metrics on R^2 with unbounded (curved) area.
Y: What about globally embedding a surface in R^3? Can you find a conformal map?
I initially was thinking about isometric, so I was saying that a flat torus cannot be embedded. Then when he mentioned conformal, I said wouldn't it still be flat? He asked if I knew about the Clifford torus, which I did not. He said in S^3, considered as the unit sphere in C^2, taking fixed radii, one gets a flat torus. I say okay and so since the sphere is conformally equivalent to R^3, that works.
Y: So now what do you think?
I say maybe you can embed them all conformally.
Y: Good guess - it turns out to be true, although quite hard.
Real Analysis:
Vicol: Define L^p space.
I write down the integral pth power must be finite.
V: Is that a definition or a norm?
I say that the norm needs a pth root - p between 1 and infinity. In the infinite case, it's essentially bounded functions, and the norm is the essential supremum.
V: Also they shouldn't be too crazy.
Okay...what do you mean?
V: You wrote dµ.
Well that's for an abstract measure.
V: It should be measureable.
Oh yes, of course! Otherwise the integral doesn't really make sense...
V: Are the L^p spaces contained in each other?
Not in general, but in some situations...
V: Okay. Given p