Lena Ji
June 9, 2017
2:00-4:30 p.m.
Committee: J\'{a}nos Koll\'{a}r (chair), Chris Skinner, Robert Gunning
Special topics: Algebraic geometry, algebraic number theory
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General comments: I was extremely scared at the beginning and my performance was not strong at all, but my committee was super friendly and gave lots of hints when I was stuck. You will be fine (and in Fine). Good luck!
Skinner, upon entering: “Don’t let him [Gunning] intimidate you!” But it was already too late...
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[Complex Analysis]
G: Let's start with the fun part! Say you have a function holomorphic in a region around the origin but not at the origin. What can you say about the function?
- It can have a removable singularity, a pole, or an essential singularity. Defined each, gave the example of e^{1/z} and explained why it's an essential singularity.
G: In the case of an essential singularity, can you say anything else about its behavior?
- Stated and sketched the proof of Casorati--Weierstrass.
I forgot what the question was but the answer was the Laurent series and I wrote out the coefficients.
G: What's the most beautiful theorem in complex -- no, in all of mathematics? Can you state it?
- Stated the Riemann mapping theorem.
G: If a sequence of holomorphic functions converges pointwise, when does it converge to a holomorphic function?
- Uniform convergence on compact subsets.
G: Can you prove it?
- Cauchy integral formula, use uniform convergence to exchange the limit and integral.
K: Is there any algebraic structure on the set of functions meromorphic on any neighborhood around the origin?
- It’s a field.
K: Is it algebraically closed?
- No -- it took a very long time and lots of hints (the quadratic formula..!) from my committee to actually write down an irreducible polynomial, C((t))[x]/(x^2-t). I vaguely said it seemed like it should be similar for degree n extensions.
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[Real Analysis]
G: What's a Lebesgue measurable set?
- I couldn’t remember so I just stated the Riesz representation theorem.........
G: What are some sets of measure 0 that seem big?
- Cantor set, Q?
G: Say you have a sequence of functions that converges pointwise to a limit. When do the integrals converge?
- Stated the monotone convergence and dominated convergence theorems.
G: Does L^2 convergence imply a.e. convergence?
- No, gave the standard example.
G: But it has a subsequence that converges almost everywhere. How would you go about showing this?
- Take a subsequence with ||f_{n_{i+1}}-f{n_i}|| < 1/2^i, show that f_{n_1}(x)+\sum_{i=1}^\infty(f_{n_{i+1}}(x)-f_{n_i}(x)) converges absolutely a.e.
G: What's the fundamental theorem of calculus?
- I failed to state the classical one so instead stated the Lebesgue versions...
G: What's the definition of absolutely continuous?
- Defined it.
G: Is every function that's differentiable a.e. absolutely continuous?
- No, the Cantor function has derivative 0 a.e.
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[Algebra]
S: What’s rational canonical form? How do you get it?
- Apply the structure theorem for finitely generated modules over PIDs to k[T].
S: Using this, how many conjugacy classes are there in GL_2(F_p)?
- p-1+p(p-1) = p^2-1.
Skinner started to say something about irreducible representations but I said I don't know any representation theory (of finite groups even!)
S: How many adjectives can you attach to the ring C[[t]]?
- Noetherian UFD, PID, DVR.
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[Algebraic Number Theory]
S: Ok so moving onto another DVR! Why are there finitely many extensions of Q_p of a given degree?
- Split it into the unramified and totally ramified parts. Unramified ones correspond to residue field extensions, and for totally ramified use Krasner’s lemma and compactness.
S: Does this work for equicharacteristic local fields?
- No, Krasner needs separable.
S: How many quadratic extensions are there of Q_p?
- This is the same as counting the size of Q_p^\times / (Q_p^\times)^2. Showed it for p not 2 using one version of Hensel's lemma. Stated the other version used for Q_2 but was stopped before going through the proof.
S: Compute the class group of Q[\sqrt{-13}]. How would you find the integer solutions to y^2=x^3-13?
- In Z[\sqrt{-13}] this factors as x^3=(y+\sqrt{-13})(y-\sqrt{-13}). Use unique factorization of ideals and the fact that the class number is 2 to say that y+\sqrt{-13}=(a+b\sqrt{-13})^3 for some a,b in Z.
S: What property of imaginary quadratic extensions lets you do this?
- The only units are 1 and -1.
S: How do you prove there are infinitely many units in a real quadratic extension?
- Drew the picture of the canonical embedding with the box getting shorter and wider.
S: What can you say about the density of primes that factor as the product of two distinct primes P_1 P_2 in K=Q[\sqrt[3]{2}]?
- One of the f(P_i | p) = 2, so the factorization of p in Q[\sqrt[3]{2}, \zeta_3] will be q_1 q_2 q_3 with f=2. The p that behave like this in Q[\sqrt[3]{2}, \zeta_3] are precisely the ones we are looking for (since p splits completely in K iff it splits completely in the normal closure, which I embarrassingly forgot), so by Chebotarev the density is 1/2.
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[Algebraic Geometry]
K: So how much of Hartshorne have you gone through?
- Not enough. [Face Screaming in Fear emoji]
K: Define an affine scheme.
- It's the set of prime ideals of a ring, the closed sets are V(I)...?
K: How do you tell if a scheme is affine?
- It has no cohomology, listed the 3 equivalent conditions in Hartshorne, sketched (i) implies (ii) but couldn’t do (iii) implies (i).
K: How else can you show that a scheme isn't affine? Why isn't P^n affine?
- H^n(P^n, O_{P^n}(-n-1))=k and computed the Cech complex for this.
K: But you can’t just use any cover for this right? You need the cover to have no higher cohomology, so somehow you’re using the other result that you couldn’t prove...
- It is a good point.
K: How else can you tell that P^n isn’t affine?
- Finally said that Spec(global sections) would be P^n again
K: Why doesn't P^n have any global sections other than k?
- Showed directly for P^1, and P^{n-1} is a closed subset of P^n.
K: What nice properties does P^n have? If you take the first projection X x P^n --> X, what can you say about it?
- Cl(X x P^n) = Cl(X) x Z?
K: Ok so that’s true, but what can you say about the image of a closed subset?
- I was confused and said something about the image of proper being proper before saying that P^n is proper over k.................
K: Can you prove that the structure morphism P^n --> Spec k is proper?
- I vaguely said something about the valuative criterion and the fact that you can check that every valuation ring of K(P^n)/k has a unique center, but couldn't say anything in more detail.
K: Ok so let’s move on. What can you say about the curve y^3=x^6+1?
- Non-reduced in characteristic 3, has three singular points in characteristic 2, smooth in other characteristics.
K: Assume the characteristic is not 2 or 3. Compute the genus.
- Riemann--Hurwitz, compute the blowup at infinity to see it’s unramified there.
K: What would a topologist see this, and how do you see this directly from the equation?
- Three connected components if you take out infinity. Factor the equation by taking a cube root of 1+1/x^6 in k[[1/x]].
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Then Koll\’{a}r asked me to leave. They talked for a few minutes and then asked me to come back in.
Gunning: “Congratulations! Now go write a thesis next week."
Koll\’{a}r: “Now you can relax for a bit. But I think you should still learn more Hartshorne..." (and I agreed!)