Generals for Shilin Lai
Wednesday April 25 2018
14:00 -- 15:30

Committee: Chris Skinner (Chair), Shou-wu Zhang, Robert Gunning
Special topics: algebraic number theory, representation theory of compact Lie groups.

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Complex analysis:

G: What's the relation between holomorphic functions and conformal mapping?
- Locally around $z$ such that $f'(z)\neq 0$, $f$ defines a conformal mapping. 

G: What about the other direction?
- Conformal mappings are defined by holomorphic functions?

G [after a pause]: orientation-preserving ones.
- right...

G: In relation to conformal mapping, what is the most beautiful theorem?
- I stated the Riemann mapping theorem.

G: Why does $\mathbb{C}$ not work?
- Liouville's theorem.

G: What happens at the boundary of a conformal mapping from the upper half plane to a polygonal region?
- I don't know.

G: That's fine. What's your favourite transcendental function?
- If Peter Sarnak were present, the only correct answer is the Riemann zeta function, so I went with that.

G: What is its definition?
- I wrote down the series and the functional equation, then realized that I am missing the critical strip, so I mentioned Riemann's contour integral representation. He did not want the details.

G: What do you know about its zeroes?
- The trivial ones at negative even integers, and then we don't know.

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Real analysis:

G: What does it mean for a function to be measurable?
- Pre-image of measurable set is measurable.

G: Define the Lebesgue measure?
- I defined the outer measure and wrote down the Carathéodory condition. Gunning seems to like this definition.

G: Name some modes of convergence for functions on [0,1].
- Pointwise almost everywhere, $L^p$-convergence, convergence in measure.

G: What is the relation between $L^1$-convergence and a.e. pointwise convergence?
- Stumbled a bit on the counter-example for $L^1$-convergence implies a.e. pointwise convergence. Mentioned pointwise convergence and uniform integrability implies $L^1$-convergence.

G: What is a Hilbert space?
- A Banach space with an inner product [should also add that they induce the same topology, but they didn't seem to care].

G: Give an example of a Banach space which is not a Hilbert space.
- The sequence space $l^1$, because its dual is not separable. 

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Algebra:

G: What is an invariant of a linear map between two vector spaces?
- The characteristic polynomial?

Z: No, between two different vector spaces.
- Oh...[after an unreasonable number of seconds]...the rank. That completely determines the map.

Z: What do you mean by that?
- I drew the necessary commutative diagram.

Z: What is a PID?
- Every ideal is generated by a single element.

Z: What can you say about modules over PIDs?
- I wrote down the structure theorem for finitely generated modules.

Z: Do you know about Dedekind domains?
- Noetherian, integrally closed, and every non-zero prime ideal is maximal.

Z: What about finitely generated modules over a Dedekind domain?
- I mentioned that the localization at primes are DVRs, and a module is the intersection of all of its localizations.

Z: What are the invariants of a module over a Dedekind domain? Of course there is the torsion part, what about the torsion-free part?
- I guessed rank and discriminant, which is not what they are looking for and wrong. This started a long discussion during which I was led very slowly step-by-step through the complete classification by rank and ideal class (I probably should have reviewed the standard first algebraic number theory course, more on this later). 

Z: Give an example of a Dedekind domain which is not a PID.
- $\mathbb{Z}[\sqrt{-5}]$, because $(2,1+\sqrt{-5})$ is an ideal of normal 2, but there are no elements of norm 2.

S: What is its ideal class group?
- Z/2Z

S: How do you show that?
- Reduction theory of quadratic form [S: fine, I suppose that works here], or I can use Minkowski's bound.

Z: How about a function field example?
- Eventually, I said elliptic curve and wrote down $F_p[x,y]/(y^2-x^3-x)$.

S: Why is it a Dedekind domain?
- It's a regular curve, so it is normal. It's a curve, so all non-zero prime ideals have height 1. It's Noetherian.

Z: What is its ideal class group?
- I stated that the Pic^0 of an elliptic curve can be identified with its points, and then realized that I have an affine curve, so I tried to figure out the relation between that and its compactification, which took a while (getting the arrow direction wrong the first time). Eventually, with a lot of help, I wrote down that the Picard group of the ring is the Pic^0 of the elliptic curve.

Z: Let's do more algebraic geometry.
- I reminded them that my special topics do not contain algebraic geometry.

Z: That's fine, these are all part of group theory [- yes...]. Prove your claim that the Pic^0 of an elliptic curve is isomorphic to the points.
- I did the standard argument using Riemann-Roch. 

Z: How many points are there on your curve?
- I mentioned the Weil bound, so it's definitely non-empty if $p>5$.

S: In your case, half of the time you know exactly how many points there are.
- Because it has CM by Q(i), if $p$ is congruent to 3 mod 4, then the curve has $p+1$ points.

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Representation theory:

Z: State Weyl's integral formula.
- I wrote it down.

Z: Prove it.
- I started doing the usual proof. I got stuck half way through because I seem to be missing a few terms. Shou-wu reminded me that I should translate to the identity and stopped me from going further.

Z: Describe the finite dimensional representations of a compact Lie group.
- I mumbled about the highest weight theory.

Z: Write it down exactly.
S: Wait, first prove that there exists a non-trivial finite dimensional representation.
- I considered an integral operator on $L^2(G)$ and said that a generic kernel has an eigenspace which does not contain the constants. 

Z: $L^2(G)$ is a Hilbert space. Decompose it as a (G,G)-bimodule.
- $L^2(G)=\bigoplus_\pi \End(V_\pi)$.

Z: What is the map going from right to left?
- Given $\phi\in\End(V_\pi)$, it maps to $g\mapsto(\dim V_\pi)\Tr(\phi\circ\pi(g^{-1}))$ [I wrote this down without hesitation because I just worked out the detail the day before].

Z: How do you tell if a module is a highest weight module?
- Just look at the maximal torus action?

Z: There is a more geometrical way. What do you know about G/T?
- It's a manifold?

G: Does it have a complex structure?
- The roots pair up, so it is even dimensional...

Z: Do you know about complexification?
- Yes

Z: Do you know that G/B is a projective variety?
- Yes

Z: Are G/T and $G_C/B$ the same?
- They have the same dimension, so maybe...

Z: Write down the Bruhat decomposition [- I wrote it down]. No the other one, Iwasawa decomposition.
- I wrote it down, and then realized that G/T is the same as $G_C/B$.

Z: So G/T is a projective variety. [He then said something about highest weight modules corresponding to holomorphic line bundles on G/T because killed by positive roots are equivalent to the Cauchy-Riemann equations. I am not sure if I wrote that correctly. I just nodded and said I did not know a lot about complex geometry.]

Z: Give me some representations of SU(3).
- I gave the standard representation and its dual, then said everything occurs in their symmetric powers while mentioning their highest weights. Not much detail was given.

Z: Is SU(3) simply connected?
- Yes. I first showed that the stabilizer of a point of SU(3) acting on $\mathbb{C}^3$ is SU(2), then when I tried to write down the fibration, they reminded me that SU(3) only acts transitively on the sphere. But SU(2) is simply connected, so I do not need to know anything about $\pi_2(S^5)$ [Now that I have looked it up, it's obviously 0].

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Algebraic number theory

S: Let's start with local fields... of characteristic 0. How many extensions of a fixed degree are there?
- Finitely many. I gave the standard proof using Krasner's lemma.

S: Describe a natural filtration on the absolute Galois group of a local field.
- It has the upper ramification filtration [secretly hoping he does not ask for details because I will get them wrong].

S [surprised]: You mentioned ramification...
- Oh, there is the unramified part and the tamely ramified part [I think he was looking for this]. I then said the wild part is pro-p, and the upper ramification filtration jumps at every rational number.

S: What have you read about local class field theory?
- Serre's local fields, Serre's article in Cassels and Fröhlich, Lubin-Tate theory...

Z: Describe Lubin-Tate theory.
- I gave a brief description. They did not ask for proofs, recognizing that the details can get messy.

Z: Can you do similar things for global fields?
- No, we don't have the Jugendtraum yet. For Q, it's given by the cyclotomic extensions by  Kronecker-Weber. We also know things about imaginary quadratic field by CM theory.

Z: Do it for an imaginary quadratic field.
- I wrote down the ray class groups in terms of j and Weber function of division points.

Z: Do you know how to show that $e^{\pi\sqrt{163}}$ is close to an integer?
- Yes, I gave a talk on it [at the GSS] a couple of weeks ago [S: I don't even remember what I talked about last week. - It's easier if you only give one talk].
- I gave the explanation emphasizing $Q(\sqrt{-163})$ has class number 1.

S: Bonus question: do you know how to prove there are finitely many imaginary quadratic fields with class number 1?
- No, but I can try...[1 minute later]... No.

Z and S: [Heegner proved it by finding an algebraic equation satisfied by their associated singular moduli. One can then apply a Siegel-type result to conclude finiteness.]

S: Do you know the Brauer-Siegel theorem?
- No, but I have heard of it [I have not studied the analytic aspects].

Z: It's about comparing class number times regulator to the discriminant. [I showed signs of vaguely remembering something like that].
S: Is it effective?
- I heard it's not.

Z: Have you read Tate's thesis?
- Yes.

Z: Write down the Poisson summation formula.
- I was writing the version for Z and R when Shou-wu told me to write down the adelic version, which I did.

Z: Why is this also called the Riemann-Roch formula?
- If K is a function field, then this recovers the usual Riemann-Roch formula.

Z: Why is it hard to estimate the number of lattice points in a region?
- Because the Fourier transform of the indicator function decays too slowly?

Z: It's easy to give a lower bound [- Using the volume?]. Do you know Minkowski's convex body theorem?
- I stated that.

S: For a really long and narrow region, it may have many lattice points while having a small volume.
Z: Riemann's part in Riemann-Roch is to give the lower bound. That's like Minkowski's theorem. Roch gave the error term. The Fourier transform of the indicator function of a domain is like the indicator function of the dual domain.

S: Minkowski's theorem has applications to number theory.
- The finiteness of class number and Dirichlet's unit theorem.

Z: Another application is to prove there are finitely many number fields with a given discriminants.
S: I like to put that on my finals.
- I said the degree is bounded and could not go further. Chris asked me to consider the D=1 case. I knew the argument, but I could not reproduce it for many stupid reasons, in particular forgetting what Minkowski's theorem (for finding points in an ideal with small norms) says exactly, in particular forgetting why the discriminant shows up. This took a while... Eventually, I [rather, they] finished the proof.

S: What do you get when you combine Tate's thesis with class field theory?
- Huh?... Oh, meromorphic continuation and functional equation for Artin L-functions, for characters first, and then in general by Brauer's induction theorem.

S: What is the most important application?
- Chebotarev density theorem.

Z: Prove it. Don't go into details.
- First reduce to the abelian case, then take an appropriately weighted sum over the abelian Artin L-functions to isolate each element [I was very vague].

S: Do you know the Hasse-Minkowski theorem?
- I stated it.

S: Can you prove it?
- The n=3 case reduces to the Hasse norm theorem. The rest follows by clever quadratic form manipulations. They allowed me to only do the n=3 case. I did the cohomological proof of the Hasse norm theorem.

S: What is $H^1(K,\bar{K}^\times)$?
- Zero.

S: What are the elements of $H^1(K,O_{\bar{K}}^\times)$ which are locally trivial?
- I went to finite extensions to figure out what's going on. It became clear that applying Dirichlet unit theorem would be messy. Chris told me to consider embedding $O_{\bar{K}}^\times$ into $\bar{K}^\times$. I then identified the cokernel as a direct sum of copies of Q, one for each place of $\bar{K}$ [S: One should probably pass to the finite extensions. - I agreed and did not do it out of laziness]. Then the long exact sequence and Hilbert's theorem 90 shows that the answer is just the ideal class group.

S: One can look at this in terms of forms of O_K. [- That's neat.]

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Total time: 1h30min