Committee: Fernando Coda Marques (Chair), Mihalis Dafermos, Gabriele Di Cerbo
Fernando asked me which topic I wanted to start with. I decided on algebra first.
Algebra
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[G]: State the Sylow theorems
I did
[G]: Use this to classify groups of order pq where p and q are different primes.
I wrote some stuff about how there’s a normal Sylow q subgroup where q > p, and so it splits as a semi direct product, I mumbled something about the divisibility condition you get to tell whether or not the semidirect product is direct or not.
[G]: What do you know about Galois extensions
I defined a few equivalent definitions of a Galois extension
[G]: If you take a Galois extension of a Galois extension is the whole extension Galois.
I stumbled through this one, I said you could try to take a tower of extensions of degree two to show it can’t be, Di Cerbo eventually suggested I look at adjoining the fourth root of 2, then it was obvious that x^4 - 2 has imaginary roots, so the extension can’t be Galois.
[G]: What’s you’re favorite proof of the fundamental theorem of algebra
I gave the Galois theory proof, Di Cerbo asked me to explain why it was my favorite, and I said I like how it explicitly only uses that you have square roots over C, so somehow what defines C gives you the whole fundamental theorem.
After this we moved on to complex analysis.
Complex Analysis
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[M]: Prove the Schwarz lemma
I can never remember the statement, but it’s somehow easy to figure out what the statement should be just from trying to prove it
[M]: What applications does this have?
I started talking about the Riemann mapping theorem, but Mihalis directed me to think about what I needed to prove that, so I realized he wanted me to classify the automorphisms of the disk, which I did.
[M]: Prove Riemann mapping
I did, I had just gone through the proof in Stein and Shakarchi the other day so I went through that one, shout out to Ryan for grilling me on that.
Then we moved on to Real analysis
Real Analysis
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[M]: State Fatou’s lemma
He didn’t ask me to prove it once I stated it.
[M]: Prove the monotone convergence theorem
I stumbled through this. I wrote everything I needed to show it down and it took Mihalis telling me I’d already done it to realize I had.
[M]: State and prove the dominated convergence theorem
I totally forgot how you do this, and at some point he asked me to look at the bounded convergence theorem and use it and I made it through eventually.
Mihalis suggested we move on but Fernando wanted to ask me a few more questions.
[F]: What do you know about regularity of analytic functions
I wasn’t sure what exactly he wanted me to show, eventually I wrote down what the radius of convergence was, and that analytic functions were smooth and derived the radius of convergence. This led to a digression back into complex analysis where he had me show that complex differentiability implied a function is analytic.
After this we moved on to PDE
PDE
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[M]: Tell me about the Cauchy-Kowaleski theorem
I stated the theorem and explained how you would prove it.
[M]: Why doesn’t this tell you all you want to know about PDE?
He wanted me to talk about illposedness of problems. Eventually I was prodded to realize that if you assign non-analytic data on the plane for the Laplace equation we have a compatibility issue.
[M]: What is Holmgren’s uniqueness theorem?
I explained what it was. I wasn’t asked to prove it. Mihalis then asked me a sequence of questions about the wave equation and it required a lot of drawing. I don’t really remember much about what happened in this part, I was very hungry. I remember it wasn’t very hard in retrospect, but I was fumbling a lot with visualizing things.
At this point we moved on to geometry.
Differential Geometry
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[F]: What do you know about manifolds admitting a Ricci Positive metric
I said the first Betti number vanishes if the manifold is compact and proved it with the Bochner formula and saying something about Hodge theory. At some point I fumbled with showing a term vanished and I needed to be told I’d already written what I wanted to show.
[F]: What if Ricci is just nonnegative
I said the same proof gives you a Betti bound from above by the dimension of manifold.
[F]: How would you prove it with minimal hypersurfaces
I fumbled majorly here but in retrospect everything was pretty reasonable, the stability inequality with the Ricci lower bound shows you that there are no stable minimal hypersurfaces, so H_{n-1}(M) = 0.
[F]: What else can you say about Ricci bounds.
I wrote Bonnet-Myers and proved it, this launched into a discussion about a bunch of theorems you prove using the index form, we talked about volume comparison, too, at this point it was kind of all a haze for me.
They popped out in a breakout room for a couple minutes and came back and told me I passed.
I was very nervous and definitely fumbled a lot, in retrospect all the questions were very reasonable but you shouldn’t necessarily expect it all to come back to you when you’re doing it live. It also helps that the committee was very nice.