May 19 2pm-5pm (2017)
Committee: Shouwu Zhang (Chair) Christopher Skinner Javier Gómez-Serrano
Below is what I can recall after my memory decayed quite a bit, so very likely there are things missing or inaccurate.
(Before we started, Chris noticed my bottle and commented: “That’s real mathematician. She has coffee!” “No it’s just water.” ”Too bad…”)
Complex Analysis
———————————————————————————————————————————
G: (State and) prove Rouché’s theorem.
(I wanted to prove using argument principal but forgot what to do. I eventually did it this way with hints from Javier. Along the way I offered to prove argument principal. Shouwu noticed a simpler proof using maximum principal before I figured out mine. With the comment “so Shouwu passed his general”, they asked me to finish mine.)
G: (State and) prove Cauchy Integral Formula
(I gave the standard argument)
G: Do you like integrals?
(I don’t know… Go ahead.)
G: We are going to compute the integral $I(\xi)=\int _{\R} e^{-ix\xi}/{x^2+1}dx$, for now let’s assume $\xi >0$.
(I did contour integration)
(Not done yet — a very smooth transition to Real Analysis)
Real Analysis
———————————————————————————————————————————
G: Now if you know $I(\xi)$ for both $\xi >0$ and $\xi <0$, what can you say about $I(0)$
(I is continuous at 0.)
G: If you replace $1/(x^2+1)$ by some other $f(x)$, what can you say?
(Though it seems clear what this sentence means now that I’m typing it, I did not understand what he was asking then, so they had to make it very explicit…)
Z&S: Write down $I(\xi)=\int _{\R} e^{-ix\xi}f(x)dx$. What is it?
(Fourier transform? (Up to some scaling))
Z&S&G: Yes!! So you know it’s Fourier transform. What do you need $f$ to be in order that it’s Fourier Transform is continuous?
($L^1$.)
G: How would you show it?
(I was stupid enough to write down the definition for differentiability when I said “continuity”, they pointed it out before I messed up. After I wrote down the correct definition it became clear that DCT will give the result.)
G: We’ve talked about $L^1$, so what’s $L^p$ space?
(I wrote down $L^p(\R)=\{f:\int _{\R} |f|^p dx < \infty \}$)
S: So you are saying it’s a collection of functions?
(Yes… Ah no! Mod out the equivalence relation defined by almost everywhere equality.)
G: What’s the norm?
(p-th root of that integral.)
G: Let $1\leq p < r < q < \infty$, if $f \in L^p$ and $f\in L^q$, what can you say about $L^r$?
($f\in L^r$ because roughly speaking $f\in L^q$ takes care of the part where $f$ is big whereas $f\in L^p$ takes care of the tail. Maybe that’s too rough.)
G: Let’s just prove it.
(With hints I did the Hölder’s inequality proof.)
S: But I still like your idea just now, you can make it rigorous by dividing the domain into where $|f|\geq 1$ and where $|f|<1$.
(Yes!)
G: Do we stop here for analysis or do we have time for one more integral? It may be related to number theory.
S&Z: Yeah let’s do it.
G: Let $b>1$. Show that $\int _0^1 (x^{b-1}/(1-x)) \log (1/x) dx =\sum_{j=0}^{\infty} 1/(b+j)^2$.
(Expand $1/(1-x)$ as a power series and do integration by parts. Before the integration by parts they told me it became a test of whether you can teach calculus… Chris and Shouwu tried to ask me something like how would you evaluate the right-hand-side, I didn’t get what they were looking for.)
Algebra
————————————————————————————————————————————
S: Can you state the structure theorem of finitely generated modules over a PID.
(I stated it.)
S: Say we have a finite dimensional vector space $V$ over a field and we have a linear map $T: V\rightarrow V$.
(We want to apply the structure theorem to $k[T]$ and get rational canonical form?)
S: Yeah that’s a good idea. Explain it.
(I did it, messed up a little with the rows and columns of the rational canonical form, and figured out what to do after going through the proof.)
S: So how many conjugacy classes does $\GL _2 (\F_p)$ have?
($p-1+p(p-1=p^2-1)$. But I struggled way more than I should on this question)
S: So how many irreducible linear representations, up to equivalence, does $\GL _2 (\F_p)$ have?
(The answer is the same as the previous question. I confessed that I did not revise representation theory when I was preparing for the exam and was only able to say why the number of irreducible linear representations cannot exceed the number of conjugacy classes of the group.)
Z: How can you generalize the structure theorem to noetherian domains.
(I explained that for any finitely generated module $M$ over a noetherian domain $R$, there is a surjection from $R^n$ to $M$, whose kernel $K$ is also finitely generated, thus there is some surjection from $R^m$ to $K$. Shouwu then asked me what would happen for Dedekind domains and suggested that I should consider localizations.)
Algebraic Geometry
——————————————————————————————————————————
(This section was a complete disaster on my part — for most of the 40-50 min in this session, I had no idea what was going on, merely wrote down what my committee said and did small bite-sized problems along the way. Since I hardly knew what I was doing and a few days has passed after the exam, I remember very little of what I was asked.
I only remember it was about $\Spec Z_p[[T]]$ and $\Spec Z[T]$. Shouwu told me after the exam that they wanted to test my understanding of Hartshorne Chapter II by applying them to these situations. So if you are fluent with the scheme language, you will certainly do better than I did.
I spent most of my time preparing for my advisor’s favourite question on classification of curves and things that could possibly arise along the way, but they did not ask anything along those lines. Perhaps because I have done part of the problem in Shouwu’s office a few weeks before the exam, he believed I would be fine with this so decided to ask something else.)
Algebraic Number Theory
———————————————————————————————————————————
S: How would you find the integral points on $y^2=x^3-13$?
(Write the equation as $x^3=(y+\sqrt{-13})(y-\sqrt{-13})$. Show that $(y+\sqrt{-13})$ must be the cube of some principal ideal in $Z[\sqrt{-13}]$ and solve the problem. This was done with hints from my committee.)
S: What’s your favourite theorem of algebraic number theory?
(Perhaps because 2h50min has passed and I was already getting silly and miserable, my committee decided to save me? I said I liked the main theorems of class field theory and wrote down the local one before they stopped me to ask questions)
Z: How do you construct $K^{ab}$.
(I knew $\Gal (K^{ab}/K)$ is the projective limit of $\Gal(L/K)$ over finite abelian extensions L/K, but I didn’t know anything about the proof of local class field theory. They did not pursue, but after the general Shouwu said I should learn this.)
S: What does the inertia correspond to under this correspondence.
(The units in $O_K$. I briefly explained why.)
S: I like it that you wrote elements of $K ^*$ as $u\pi ^n$. Let’s say $K=\Q_p$, what are its quadratic extensions?
(It boils down to find $\Q_p ^*/{\Q_p ^*}^2$, which is represented by $\{1, u, p, up\}$, so quadratic extensions of $Q_p$ are $Q_p[\sqrt {u}]$, $Q_p[\sqrt {p}]$, and $Q_p[\sqrt {up}]$. I forgot to say $u$ should be a quadratic nonresidue mod p, but they didn’t mind.)
S: That is true for $p>2$.
(Yes. I should have mentioned it.)
———————————————————————————————————————————
I went out of the room waiting for my fate. After a minute or so they told me I have passed.
Comments:
— It seemed impossible for me to pass the generals a month before I took it and I had a break-down two weeks before the exam feeling I knew nothing. But the real exam, though far from being great, went fine. You will be OK, though it’s probably hard to believe it the moment you are reading past transcripts. I have tried to include my silliness in this transcript and very likely I was even sillier in the exam than what’s revealed here.
— Talk to your committee. I could have had a better start — Rouché’s theorem was used (but not proved) in my practice general with Javier and I’m sure he asked me to prove it just as a warm-up, but it was my fault not reading the proof carefully after the practice general. Talking to Chris and Shouwu helped me make progress in studying my special topics, and probably saved me a little when I did not do well. My committee had been really helpful both before and during the exam, which I’m really grateful for.
— I am also very grateful to fellow grad students in the math department for their help and support.