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 --------------------------- 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... --------------------------- [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. --------------------------- [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. --------------------------- [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. --------------------------- [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. --------------------------- [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]]. --------------------------- 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!)