Noah Stevenson's generals

Exam date, start time, location: 18 April 2022, 9:30a, Fine Hall 705
Duration: approximately 110 minutes
Special topics: 'Harmonic analysis' and 'fluid dynamics'
Examiners: Alexandru Ionescu [AI] (Chair), Peter Constantin [PC], and Lue Pan [LP]

I arrived about 10 minutes early to AI's office, along with PC. I was told to take this time to relax. PC and AI chatted about their young grandchildren and children, respectively. The conversation then moved to a discussion about campus construction's daily disruptive dynamite blasting (personally I had never heard the blasting). AI then said I hope that LP shows up, because I am unable to examine you in algebra. PC agreed. LP then arrived exactly on time.
I was asked which topic I would like to do first. My selection was real analysis (my though process was that I should choose what I feel most comfortable with first, as I was feeling quite nervous).

~Real Analysis~

[AI] Define the Lebesgue measure on the real line.
I gave the definition of the Lebesgue outer measure and then proceeded to say that a set E is measurable if for every epsilon positive there exists an open set U containing E for which the outer measure of their difference is at most epsilon. This led to a discussion about Carathéodory's criterion and the metric measure criterion.
[PC] What is your favorite convergence theorem (in measure theory)?
I said Vitali's convergence theorem. Before I stated what this result said, the committee said they did not know this result by this name. I was then told that they expected me to state one of the monotone, dominated, or Fatou's convergence theorems.
[AI] What is Fatou's lemma.
Answered.
[PC] I want a sequence of functions on [0,1] that are bounded in L^1, converge pointwise everywhere to zero, and do not converge in L1 to zero. Also make a picture we (the committee) cannot read equations.
My example's graph looked was apparently too rectangular for PC and he asked, "What, you don't like triangles?". I changed it to a more triangular example which was no longer satisfying the required conditions. The committee though this was humorous.
[AI] I want a sequence of functions on [0,1] that converge pointwise nowhere, but converge to zero in L1.
Gave an example.
[AI] Define convergence in measure.
Defined.
[AI] Give me a sequence of functions that converge in measure but not in L1.
Gave.
[PC] What's the Radon Nikodym theorem?
I stated the theorem and mentioned that I did not know how to spell 'Nikodym'. PC said that this is understandable as there is an unexpected 'y'. AI then got out what appeared to be Stein and Shakarchi's functional analysis to look up the either the correct spelling of 'Nikodym' or the correct statement of the theorem. This took the committee's whole attention away from me and I had to talk about Radon Nikodym over their murmurs. 
[PC] Why do you need sigma-finiteness as a hypothesis in Radon-Nikodym? Give me a counter example when we don't have it.
He hinted me towards thinking about Hausdorff measures.
[PC] Give me a classical application of Radon-Nikodym.
I said if you supplement it with Lebesgue differentiation theorem you could give a proof of the fundamental theorem of calculus for absolutely continuous functions and the almost everywhere differentiability of monotone functions. PC was not amused with this answer. I then mentioned that Radon-Nikodym is used in finding the duals of Lebesgue spaces. This was the answer PC was looking for.
[PC] Apply the Radon Nikodym theorem with the Dirac mass and Lebesgue measure for both combinations.
Applied.
[AI] What is the Baire Category Theorem?
Gave.
[PC] Let's talk about applications of Baire Category theorem to Fourier series.
I said I have not seen this one, but I mentioned that one can show that functions that are differentiable somewhere are meager in continuous functions. This comment seemed to have 
[AI] What are some other applications of Baire Category theorem?
I said Banach-Steinhaus and the open mapping theorem.
[PC] State the open mapping theorem.
Stated.
[PC] Do you need completeness in both the domain and codomain?
I said yes. This was incorrect. PC explained how you can relax the hypothesis of completeness in the domain.
At this point the committee had commented that more than thirty minutes had passed. They decided to change topics. They told me I am doing complex analysis next, since they thought as soon as I would be asked algebra questions I would become hopelessly confused.

~Complex Analysis~

[AI] Talk about analytic functions.
I listed six equivalent definitions.
[PC] Say something more about Goursat's theorem.
Said more.
[AI] Suppose you have a real valued function on the real line, how do you make it the real part of the boundary value of a holomorphic function in the lower half plane?
Described how to do this via convolution with the Poisson kernel and taking harmonic conjugates.
[AI] Can you do it a different way, maybe using the Fourier transform?
I said I think this is related to the Hilbert transform and proceeded to fumble for a few minutes trying to work out the details. At this point I found it difficult to problem solve in front of my audience and they had to help me through it.
[PC] Say you have a sequence of holomorphic functions converging uniformly on compact sets, what more can you say?
Promoted the convergence via the Cauchy integral formula. I mentioned that you can also relax the hypothesis of uniform convergence on compact subsets to convergence locally in L^1.
[AI] State Riemann mapping theorem.
Stated. I asked if they wanted me to prove it. They said no, it would take too long.
[AI] Why not all of the complex plane?
Liouville's theorem prevents this.
[AI] How do you prove Liouville?
Proved (verbally).
[PC] Should we talk about boundary values for Riemann mappings?
The committee decided not to not let me answer this and to change topics instead. They seemed to be either in a rush or bored. We then moved to algebra.

~Algebra~

Before LP asked the first question, AI and PC both got out their phones and proceeded not listen to the algebra portion of the exam.
[LP] Can you find a 2x2 matrix that is symmetric, nonzero, and nilpotent?
I said no, because you can diagonalize symmetric matrices. LP corrected me and emphasized that he said 'symmetric' and not 'self-adjoint'. I then realized that I needed to find a 2x2 symmetric matrix with complex entries and zero eigenvalues. This determines a system of equations for the coefficients, I found all the examples.
[LP] Same question, but I insist that the matrix have real entries.
I said you cannot find such a matrix because of the spectral theorem.
[LP] State the definition of a Galois extension.
Gave.
[LP] Give me an equivalent definition.
Gave.
[LP] What is the automorphism group of an extension of finite fields? Why is it a Galois group?
Answered. I stumbled a bit in proving that the Frobenius automorphism is the generator for the Galois group of a finite field over its prime subfield.
[LP] What can you say about the multiplicative group of a finite field?
It is cyclic, which is a consequence of the structure theorem for finitely generated modules over a PID.
[LP] Can you use the aforementioned fact to prove that the Frobenius automorphism is the generator for the Galois group of a finite field over its prime subfield?
I was given lots of hints from LP to show this.
[LP] How are the structure theorem for finitely generated abelian groups and Jordan normal form for matrices related?
The structure theorem for finitely generated modules over a PID.
[LP] Derive the Jordan normal form.
Derived.
[LP] Why are the integers and polynomials with coefficients in a field examples of PIDs?
Division algorithm.
[LP] Is the ring of polynomials in two indeterminates with coefficients in a field a PID?
Found an ideal that is not generated by a single element.
[LP] Why is the aforementioned ring nice anyway?
I said it was Noetherian and was about to talk about Hilbert's basis theorem, but was stopped. LP wanted to hear that it was a UFD.
This concluded the algebra section. At this point AI and PC looked up from their phones and commented that they didn't understand any of that, but were impressed. AI then said we ought to do Harmonic analysis next, as we should save fluids - the main topic - for last. I then corrected AI and said I thought harmonic analysis was my main topic. He replied that they are both my main topics.

~Harmonic Analysis~

[AI] Define a Calderón-Zygmund kernel/operator. Talk about cancellation conditions.
Defined, but didn't include a cancelation condition in the definition. Rather I said that if we knew the associated operator were bounded on some Lebesgue space, then we'd know that it would obey a weak type 1,1 estimate. Then I gave two different necessary and sufficient conditions for L2 boundedness in the translation invariant case.
[AI] Is x \mapsto |x|^{-1} a Calderón-Zygmund kernel on the real line?
Answered.
[PC] How do you make the aforementioned function agree with a tempered distribution away from the origin?
Described.
[AI] Is the Dirac mass a Calderón-Zygmund kernel?
Yes.
[AI] Is a Gaussian a Calderón-Zygmund kernel?
Yes. At this point I thought he was trying to illustrate that my original definition of a Calderón-Zygmund kernel was incorrect, so we had a brief discussion on what the right one should be. I don't recall reaching an agreement.
[AI] Is x \mapsto |x|^{-1/2} a Calderón-Zygmund kernel on the real line?
No.
[AI] Does the Hilbert transform have higher dimensional analogs?
Yes.
[Al] Give some more examples of Calderón-Zygmund kernels.
I proceeded to explain how the Mikhlin-Hörmander multiplier theorem gives you a bunch of examples of Calderón-Zygmund kernels. Sadly, I was not asked to prove this result.
[PC] What are some applications of Calderón-Zygmund theory?
Fourier space characterizations of your favorite function spaces.
[AI] Talk about oscillatory integrals. Let's begin in 1-dimension.
I talked about decay and asymptotic developments.
[AI] Be more precise on the asymptotic developments? Tell me which terms vanish.
Was precise.
[AI] Now look at integrating the Fresnel phase against a smooth (but not necessarily compactly supported) amplitude on the interval [-1,1]. Derive the asymptotic developments for each part that contributes to the decay.
Derived. The committee was only interested in a sketch and I was not permitted to say how to calculate the coefficients in the asymptotic development.
[AI] Let's move on to higher dimensions. Consider the case of nondegenerate isolated critical points. What're the asymptotic developments.
Gave a rough idea, they didn't want proofs.
[PC] What's the dependence of the first term in the asymptotic development on the Hessian of the phase?
Derived.
This concluded the harmonic analysis portion of the exam.

~Fluid Dynamics~

PC decided to begin the 'fluid dynamics' section by asking some PDE questions. This worried me, as I had not prepared much PDE outside of what is used in fluids.
[PC] Talk about the wave equation in three dimensions. How do you set up the Cauchy problem?
Set up. I mentioned that I know of the Huygens principle and decay, but I didn't know how to prove it. I then reiterated that I was expecting fluids as a special topic.
[PC] What PDE do you know then?
I said I know the L^2 and L^p theory for elliptic equations and some parabolic.
[PC] Okay I will ask you an elliptic question. Say you have a bounded domain U with smooth boundary and a bounded sequence {f_n}_n of essentially bounded functions in U that are converging uniformly to a function on compact subsets of U. Now consider the sequence {u_n}_n of solutions to the PDE -\Delta u_n=f_n in U and u_n=0 on the boundary of U. Show that this sequence also converges uniformly on compact subsets of U.
Showed.
[PC] What books did you read for fluids?
I replied bits and pieces of: Constantin-Foias, Majda-Bertozzi, Boyer-Fabrie, Marchioro-Pulvirenti.
[PC] Talk about some nice properties of the Euler equations in two dimensions.
Talked. My pronunciation of 'Biot-Savart' was corrected by PC. I apologized and he told me not to worry as it is French.
[PC] What's the vorticity formulation of Euler?
Given.
[PC] Why is there not a vortex stretching term in two dimensions?
Explained.
[PC] Say that X is a passive scalar moved by a solution to the Euler equations. Show that the derivative of X in the direction of the vorticity is conserved along the flow.
I attempted brute force calculations here. The committee did not like this and told me to stop prematurely. PC told me to stare at the Euler equations and notice that it says exactly that the material derivative and the derivative along the vorticity commute.
[PC] Show that vortex lines are mapped to vortex lines under the flow of a solution to the Euler equations using the aforementioned vanishing commutator.
At this point I had lost a lot of confidence in myself and was unable to answer the question. We just moved on.
[PC] Use the previous passive scalar computation to show that two dimensional flows stay two dimensional and do not have vortex stretching.
Before I could even think, PC told me how to do it. Choose the passive scalar to be the vertical coordinate.
[PC] Consider a sequence of smooth initial vorticities for 2D Euler in all of R^2 that are converging to a Dirac mass. Describe what happens.
I said that they would hopefully converge to the point vortex stationary solution. I then proceeded to hand-wave and explanation.
[PC] If the initial smooth vorticity to 2D Euler is quantified close to a Dirac mass, for how long during the flow does the vorticity stay localized near a Dirac mass.
I had no idea how to answer this one and I could not think clearly after botching the answers of the previous several questions. PC quickly said what the answer should be and then we just moved on. PC also commented that he was trying to make me confused and think on the spot as he was not interested in listening to me answer questions from memory.
[PC] What is Beale-Kato-Majda?
Gave. The committee was not interested in a proof.
[PC] Give a blow-up criterion for Navier-Stokes.
Gave.
The committee then decided that it was time to conclude the exam. I was asked to go out in the hall. Maybe 30 seconds later I was called back into AI's office and was congratulated and told I had passed. I then shook the hands of the committee, while explaining that I was not proud of how I handled my fluids section. I was told not to worry.