Matei Coiculescu's Generals
Exam date, start time, location: May 3rd, 2022, 3:30 PM, Fine Hall
1109
Duration: 2 hours 15 minutes
Special topics: the Navier-Stokes Equations, Riemannian Geometry
Examiners: Camillo De Lellis [DL] (Chair), Sergiu Klainerman [K],
and Jonathan Hanselman [H]
I was the last to arrive in [K]'s office, at around 3:27 PM. [DL]

asked me what topic I wanted to start with, and I chose Navier-
Stokes.

N.B. I certainly forgot a good number of questions that I was
asked.
Navier-Stokes
[DL] What is a Leray-Hopf Weak Solution?
[K] How do you get that a vector field in the energy space is
L^10/3 in space time.
First sobolev inequality, then Riesz-Thorin interpolation works.
[K] What about the pressure? How do you recover the pressure?
[DL] Sketch a proof of the the existence of Leray-Hopf weak
solutions
I begin by showing existence for a perturbation of the Stokes
system by the Galerkin Method. I need to use the cancellation
identity...
[DL] Prove the cancellation identity, in detail.
[DL] OK, let's go on to the actual equation, prove existence for
the actual equation.
Stated an appropriate regularized version of Navier-Stokes and
began.
[DL] Is weak convergence in the energy space of solutions to the
regularized version sufficient to get appropriate convergence of
the nonlinearity?
No, need strong L^2 convergence of the velocity.
[DL] How do you get strong L^2 convergence?
Aubin-Lions Lemma

[K] (to [DL]) Why is it called Aubin-Lions instead of Aubin-
Rellich (or maybe Rellich-Lions?).

[DL] Why can't you just use Rellich's Compactness theorem instead?
[DL] What did Leray prove about the epochs of regularity/ set of
singular times?
Stated the H^(1/2)(S) =0 result of Leray.
[DL] Sketch a proof of the result.
I began a proof similar to the one in Tai-Peng Tsai's book.
Needed to discuss mild solutions and the strong energy inequality.
[DL] Tell me about Weak-Strong uniqueness.

[DL] What are the two epsilon-regularity criteria in Caffarelli-
Kohn-Nirenberg?

[DL] What's the main partial regularity result in Caffarelli-Kohn-
Nirenberg?

[K] What good are Leray-Hopf solutions? Will they help you solve
the Clay problem? Physical meaning?
[K] What do you know about 2D Navier-Stokes, can you show global
regularity? Use the vorticity equation.
[K] Do you know any blow-up criteria?
Stated the Beale-Kato-Majda criterion.
Algebra
[H] Tell me about groups of order 35.
I instead stated and gave a sketch of the proof of the general
result about groups of order pq, with p, q primes p<q.
[H] State the Sylow Theorems
[H] Prove the first Sylow Theorem.
[H] Consider the polynomial x^3-x^2+x-1. Give me a 5x5 matrix over
C that has this polynomial as its minimal polynomial.
Used Jordan Canonical form.
[H] How about over Q? R?
Used Rational Canonical Form
[H] Where do these canonical forms come from?
Stated the fundamental structure theorem for finitely generated
modules over PIDs.
[H] Is the fact that the ring is a PID essential?
Yes, in the proof you need this to guarantee existence of maximal
ideals.
[H] Give me an example of an integral domain that is not a PID.
After a bit of thinking I stated that R= { continuous f:[0,1] ->
reals } is one such ring, with ideal I = {g | g(0)=0} an ideal
that is not principal. [DL] gave me hints to help me rigorously
prove that this is so.
Riemannian Geometry
[K] Favorite theorem from geometry?
Bonnet-Myers
[DL] State and prove it.
Stated and proved it, in detail.
[K] Do you remember the Lorentzian analogue from last semester's
course?
... no...
[K] It is the Penrose singularity theorem. ([K] then remarks to
[DL] that I don't appear to like Lorentzian geometry)
[K] Tell me about immersed hypersurfaces
Stated the 2nd fundamental form, Gauss map, equation of Gauss,
principal curvatures, Gaussian and Mean curvatures, all in
coordinate-free setting a la Do Carmo.
[K] But how can you do computations if you don't work in
coordinates! Write the Gauss equation in coordinates.
I wrote the equations in coordinates.
[K] How about the Codazzi equation?
[K] OK, do you remember your presentation from class last
semester? Tell me about the Sobolev inequalities on manifolds.
Stated the sobolev inequalities
[K] How do you use them to prove the isoperimetric inequality?

I had not thought about this in about 5 months... so I needed quite
a few hints here from both [DL] and [K].

The committee decided to forego an examination in complex or real
analysis, and told me to go out to the hallway. After a minute or
two, they opened the door, smiles on their faces, telling me I
passed. The test felt pretty intense, and I am happy I studied as
I much as I did. I only felt nervous at the start (anticipation of
the exam) and at the end (anticipation of the result), while I was
doing math I felt much more at ease, even when I needed hints.
Lastly, I want to thank Ben, Doug, and Dallas for their great
practice exams!