Generals exam: Ben Lowe
May 18, 2018, 2:10 pm to 4ish pm
Examiners: Fernando Coda Marques [FCM], Peter Constantin [PC], Jonathan Hanselman [JH]
Special topics: Differential Geometry and PDE
This was [PC]’s 3rd exam of the day so I had to go down and get him in his office. Remember to bring water! I also brought a stress icosahedron with me to the exam hoping I would get asked about it but alas no..
Algebra
[JH]
-State Sylow’s theorems. How would you prove the first one? Said something about orbit stabilizer, have it act by left multiplication. What do you act on? Subsets of the group of size p^n (he had to tell me this) then you can compute the number of orbits and find the group as a stabilizer. Ok good.
-Ok so is a group of order 40 simple. 40=2^3 *5, there’s one sylow 5 subgroup, then started saying how it had to split as a semidirect product. What does simple mean? Oh no there aren’t any.
-Groups of order pq? q>p, you get a normal subgroup of order q. Order p? There’s a divisibility condition, wrote it. Then you want to look at how Z/pZ can act on Z/qZ, I said this could happen in a number of ways if you had the divisibility condition.
-Do you know about subgroups of free groups? Yes they’re all free. Why? Take the cover corresponding to it, it’s a graph, it’s a wedge of circles. Say it’s index n, how do you compute the rank? Thought about it for a little while. He said something like what’s a good invariant that behaves well with covers-- oh Euler characteristic, wrote down the equation.
-Count the number of linear automorphisms of the vector space (F_2)^4. You can use a normal form, do orbit stabilizer counting. I guess you can’t do Jordan normal form since it isn’t algebraically closed. Do you know of another one? Rational canonical form but I don’t really know what it is. That’s Ok so talk about the theorem you’re using to get the Jordan normal forms? Sure, stated it, said how you apply it to get JNF. Then the difference between the two is for Jordan normal form you break it into powers of primes, for rational canonical form you stack them.
-Do you know anything about the proof of the fundamental theorem of finitely generated modules over a PID? I said yeah a little. There’s an important lemma where you show a free submodule of a free module is free and can’t have rank bigger than the whole thing. Then the proof is an induction where you start off my looking at homomorphisms to the ground ring and choose a minimal one, and continue in that way.. So where are you using it’s a PID? Thought about it for a while didn’t know what to say. Then he told me for the step where you’re looking at homomorphisms to R, you use it there. Right the images are ideals which you can identify with elements of the ring since it’s a PID.
-Ok is there anything else to ask? Maybe talk about the main theorem of Galois theory, and illustrate in an example. Said it, wrote the polynomial x^5 -2, explained stumblingly why the galois group is the group of upper triangular matrices in GL_2(F_5) which have 1 as their lower right entry, esoteric meaning: the hyperbolic plane over F_5 (just thought that didn’t say it), applied the fundamental theorem of galois theory to the totally real subfield.
Complex Analysis
PC:
-Ok enough of this nonsense (referring to algebra.) What does it mean for a function R^2 -> R^2 to be complex differentiable?
-Prove Liouville’s theorem
-What is Rouche’s theorem how do you prove it.
-Uniformly bounded sequence of holomorphic functions what can you say
-State Riemann mapping theorem. Why not the whole complex plane?
FCM:
-Geometric meaning of complex differentiability.
-Prove Schur’s lemma ???? oh Schwarz lemma. Stated it wrong but eventually fixed it when he told me to do the proof. What could you do if f(0) is something else?
-Geometric meaning of conformal automorphisms of the disk? How would you construct the metric just knowing the symmetries? Started talking about metrics on Lie groups in general but then he suggested I do it more concretely, so I just wrote down a formula for it choosing a metric at the origin and pushing it forward.
-Geometric meaning of meromorphic function.
Real Analysis
[PC]
-Say you have functions bounded in L^2 what kind of compactness? L^1? Similar but you get a measure. What sort of condition to get a function? Being guided to say some form of uniform absolute continuity.
-Do you know about the Hilbert transform? Yes. Explain what it is. Started saying “the point of the Hilbert transform is-” no write it down, said basic properties. Another way of proving L^p boundedness besides Calderon Zygmund? I said that I guess people originally did this with complex analysis but I didn’t know about that. He said yeah you can use it to get L^4, and from there get everything.
PDE
[PC]
-Burger’s equation, derive the characteristics. Got myself all confused and had do write it down with p’s and z’s and x’s like in Evans, which prompted [PC] to say something like note to self strangle larry when I visit berkeley next week. The characteristics are lines that can run into each other.
-Asked to show you preserve L^p norms in Burger’s equation. Added a u_xx term to Burger’s equation and you get something where the L^2 norm decays. Did the computation and he told me that with 3 minutes more work you could see that you actually get smooth global solutions for this one. Cool. (In hindsight I think I was supposed to notice it was parabolic, oh well.)
-Ok let’s do elliptic. Start by writing it down in divergence form. Say you have H^1 sol’n how do you get to H^2? Explained about difference quotients.
-Now just solving Laplace equation, 0 boundary data, how do you show there’s a weak solution. Wrote down the bilinear form, Poincare inequality, it’s an inner product so you can represent bdd linear functionals.
-What if you have boundary data? Apply Laplacian to it if it extends and put it on the other side and you’re back to the previous. If it’s just continuous? You can do something else, like Perron’s method (he didn’t make me explain what this was.)
Differential Geometry
[FCM]
-Ok so let’s talk about examples with different curvature conditions. Positive scalar but not positive Ricci? Take a product with a small S^2. Or no, hesitated for a bit. Guided to say you can rule out positive ricci curvature if you have a first betti number, so take S^1 for the first factor.
-Ok so what about a manifold that has a positive ricci metric but not a positive sectional curvature metric? What kind of restrictions are there for positive sec? Even dimensions it’s very restricted, you have to be a sphere or an RP^{n} (this is untrue but I didn’t know yet) How would you prove that? Well for if it’s orientable and you have a nontrivial pi_1, then you can take a minimal geodesic and get a contradiction using the second variation formula. So that does the simple connectivity part. What about the rest. Thought about it. What about in dimension 4? (Oh no there’s CP^2 I was tripping.) But you still get simple connectivity of the oriented double cover. Eventually wrote down RP^2 \times RP^2.
- Fundamental group if sec>0? Started talking about it’s virtually perfect oh wait you have the theorem that says the diameter is bounded so it’s a finite group.
-What’s an Einstein metric? Said that the factor relating Ricci and the metric had to be constant. All dimensions? Not dimension 2, then after a little while, because you can have non-constant guassian curvature. Proof? I just said it’s a computation and you use the 2nd Bianchi formula.
-What’s an example of something without an Einstein metric? So in dimension 3 this is the same as sec constant. So S^1 \times S^2 should work. Gave a bad explanation of why there isn’t a nonpositively curved constant sec metric in terms of the linear growth of volumes of balls in the Cayley of Z. What are the constant sectional curvature manifolds? Oh R \times S^2 is different from R^3.
-Why can’t S^3 have a negative sectional curvature metric? Then there’d be no conjugate points, started talking about a bad solution looking at the homology of the path space. There’s probably a simpler way.. Can you look at the exponential map? Oh right that would have to be a cover.
-You’ve been using this fact that non-negative ricci curvature implies the first betti number vanishes. How do you prove it? Bochner formula. Tried to write it down, didn’t remember for a couple minutes that you had to look at the norm squared of the vector field. So say why this implies the fact. What is the name of the thing that lets you take harmonic representatives- Hodge theory.
It helped that I had thought about most of this stuff before, even if I didn’t remember right away.
Comments: I may not have remembered everything right and I definitely got more hints than what I wrote. I should say my committee members were highly pleasant and I actually sort of had fun, especially because I wasn’t forced to go into details, not that I could have. Doing a practice one really helped shout out to Akashdeep. I’m reminded of a quote which Vladimir Drinfeld said which is something like “mathematics is about saying the right words in the right order.”