Nina's generals Committee: Sarnak, Katz, Nestoridi Special topics: analytic number theory and algebraic geometry November 2nd, 2020, zoom ----------------- 3 hours 30 minutes Sarnak asks what I want to start with, I say complex (tradition!). Complex Analysis ----------------- (S) Ok, so what did you read? I went through Ahlfors. Great! Comments about preferring it to Stein. (S): State the Riemann mapping theorem. Why does the subset have to be proper? (Liouville) (K): How do you map the upper half-plane? I construct the Cayley map, mention that there's a linear fractional transformation that takes any three points to any other three points, takes lines and circles to lines and circles, then write z-i/z+i. (S): Ok what proofs do you know? I know Koebe's rigorous proof, and I know Riemann's original idea. (S): Ok what's Riemann's proof I give Riemann's idea (S) So where was the mistake? I don't know if it's a mistake so much as it's unclear how to solve the Dirichlet problem! (S) Give the rigorous proof. Conversation ensues about who the proof is due to, I say Wikipedia has it wrong and that's probably why everyone keeps mentioning Caratheodory. I explain the proof is through normal families: define the family, show that it's non-empty, show that it's normal, take a subsequence for which the derivative approaches the supremum, show the limit function is in the family, do a transformation on the disc to see that the derivative maximizing function has to be onto (I gave some detail, but they seemed to want to move on). (N): can you prove the fundamental theorem of algebra using complex analysis? I can do it with Rouche... (N) what about Liuoville? Yeah, it suffices to show there's one root, just take 1/p(z), if there are no zeros it's entire and vanishes at infinity. (S): do you know the genus of a function? I define it via the Hadamard product. (S): What about some sort of growth rate? I define the order via log log(sup over x in dD, D disc of radius R of |f(x)|)/ log R. (S) What if we take sup inside the disc? Same thing by max. principle. (S) So what's the relationship between order and genus? Hadamard's theorem (g <= order <= g + 1). (S) How would you see something like this? I want something about zeros. Well, if the genus is g then 1/|rho|^{g + 1} converges... (S) No, something else. Jensen's formula! Proved and stated. (S) A-hah! Great. We can move on. (K) Actually, I have one last question. Suppose I have a meromorphic differential such that the integral of it is independent of the path. What can I say? Does it exist? it's exact? For example, take df... (K) Okay but there's some word here. Residues. I'm confused for no reason (K) Ok, suppose I take dz/(z + 1)... That doesn't work, for the same reason there's no single-valued log, there's a period. (S) So locally?... Take fdz. (I finally get it) Ooh! yes just the residue of f has to be 0 at a point. Real Analysis ----------------- (S) Ok, so what did you read? I mostly followed Folland and Terry Tao's notes (S and K smile when I mention the latter). (S) Evita, you want to ask something? (N) Ok so let's talk about Hilbert and Banach spaces... Suppose I have something converging weakly... Whoops, I don't even remember what that means. (S) Let's do it on R. (N) Ok, let's do it this way. Suppose for some function lim_{a \to 0^+} \int_a^1 f(x) dx exists. Must f be in L1? Well no, take 1/x ! (N) Ok ok exists and is finite. I'm blanking and eventually try to just say something. Well if they were positive we could use monotone convergence for f_n = f \chi_[a, 1]... So presumably the counterexample has to oscillate around 0 very fast. Eventually with (N)'s help I arrive to the idea of taking f with \int_[1/n, 1/(n-1)] f dx = (-1)^n 1/n. (S) Define Fourier transform on L1. Where does it land? It is clearly in L^\inf, it is also continuous and decays at infinity. (S) Why's that?? Riemann-Lebesgue, approximate by a smooth function, smoothness translates to decay on the Fourier side. (S) How do you approximate by a smooth function? Take a good convolution. (S) Ok ok now. We have a bounded operator from one space to another. What spaces are those btw? Complete, so Banach, with the L^inf norm on C0. (S) Right, we always have the infinity norm on C... Ok so... is this onto? It's a very difficult question! Not so difficult when it's been asked on previous generals! (S) WHo asks it??? Me?? :) So yes this is not the case by open mapping (Banach isomorphism). Take the family sin(2pi nx) sin(2pi x)/(pi ix)^2. The transform is \chi_[-1,1] * \chi_[-n,n], so we get a family with norm going to infinity that maps to something of bounded norm, so the inverse is not continuous (i.e. not bounded). (S) Ok, why does the L1 norm grow? Prove it. Actually don't prove it whatever. (K) What would Lebesgue say his best theorem is? Maybe Lebesgue differentiation? (K) Why? State it. I don't know, things always go wrong in real analysis and this is one of the few instances where everything goes right! K seems to like the answer. (K) Bochner used to say it was Dominated convergence! (S) According to Lebesgue or Bochner?? (S) Btw what is Lebesgue Differentiation on R? Fundamental theorem of calculus! Also, why do you need "almost everywhere" in the definition? Uuh you can take f(1) = 1; f(x) = 0 elsewhere. (S) Ok, what about something more interesting. Char. function of any measure 0 set. (S) Ok sure, obviously things can be wilder than that but ok. Nick, should we ask for the proof? Oh, I'm happy to give a proof! (S) Alright, let's do it! I give a proof. Once I get to the Hardy-Littlewood maximal function, Sarnak gets excited. Alright! By the way what kind of function is it? I didn't know and (stupidly) guessed L1 as well. (S) No-nono-noo ... Think they might have asked a few more questions, but I don't remember them. Algebra ----------------- (S) Evita, do you want to start? (N) Alright! As a warm-up, why does a p-group have a non-trivial center? Proof via class formula. (N) What's the action here? In the proof of the class formula? I just think about it in terms of the group splitting into conjugacy classes, but conjugation. (N) Ok great! Now show a group of order p^2 is abelian. (K) You can use representation theory to show this! I show it, mention you can use the same idea to show that if the Automorphism group is cyclic the group is abelian. (N) Ok, let's do rep theory. First, show that irred reps of abelian groups are 1-dimensional. Ok I always forget the name but there's Maschke's theorem and then there's... (N) Schur's lemma! Yes! Schur's lemma. State it. Applying to g we see that g acts by scaling so the dimension is 1. (S) More generally, what can you say about these representations? Well, they're called characters... The number of irreducible reps is the number of conjugacy classes... They form a group that is isomorphic to G... Noncanonically!!! (S) Prove that. (proved it) You keep saying non-canonically, what's a canonical isomorphism? (Double dual) (K) Is there a name associated with these sorts of hat operations? Pontryagin dual, but pretend like I didn't say it because I'm not ready to be asked more about this! (S) Ok so talk about the 1-dimensional representations of a group. (N) Those are representations of the abelianization (S) What's that?? I define everything, commutator subgroup, etc., Sarnak asks if it suffices to take just the commutators and makes a comment about his research and having to show that the commutator subgroup consists of just the commutators in some cases. (K) Ok, so how would you prove that a group of order p^2 is abelian using all this? I think for a moment; Katz asks me if I know another property of the dimensions of irreducible representations. I say yes, they divide the order of the group, and from there the idea is clear: each irreducible representation has dimension 1, p or p^2, and the sum of squares is p^2, hence (since there's the trivial representation) they're all 1-dimensional. (K) I learned recently that every element of SL_2(C) is a commutator! (K) gets a phone call and says "think about how you'd prove that" as he picks up. (K) Ok what about say Q(\sqrt{p_1}, ..., \sqrt{p_n}) for n primes p_i? Hmm... that's a degree 2^n extension. (K) why? We have to show that a prime is not in the field generated by the previous ones... I get very confused and there's some muddy discussion about fields generated by roots of unity. Eventually, Katz says this is related to ramification. K then asks me about finite field extensions and whether or not each such extension is generated by adjoining a root of unity. Eventually, this somehow turns into Sarnak asking (and me proving) that F_q^* is cyclic and we move on. Then Katz makes some comment about Galois extensions of Q_p and I just say I don't know anything about this. I feel embarrassed but they don't seem to mind! At some point (K) asks about a more general form of the structure theorem for abelian groups, so I state the structure theorem for f-g modules over PIDs and explain how this gives the rational canonical form and Jordan normal form. (N) Can you use this to show A and A^T are conjugate? I do it. ----------------- We take a 10-minute break. N and S are chatting about something. Eventually, Sarnak asks what I want to do first and I say I don't have a preference, so we do NT. Analytic Number Theory ----------------- (S) Define a Dirichlet L function. I define a character and then the L function. Where does it converge? Real part > 1. Where else? That's it for zeta, but for a primitive character, I can extend to real part > 0. Why? I do a partial summation argument, but also it's obvious because of the boundedness of the sum of chi(n). (K) and (S) both ask why L functions are important. I talk about the class number formula (write it down) and (K) asks for another important application. Before I answer: (S) Ok great, but what kinds of functions are these? Do they have an Euler product? me, cheekily: it wouldn't be called an L function if it didn't have one! (S) is happy with this because he always says this. Where are they defined? I talk about the functional equation. Sarnak asks what it is and I write it for the completed zeta function. How do you prove it? I say modularity of theta. How do you prove THAT? In this case, it's just Poisson summation. (S) Okay, what can we say about the orders of these functions? They all have order 1. We can see this for Gamma from the Hadamard product for sin, and such... Then a discussion ensues about the size of zeros. (S) Can it have finitely many zeros? No definitely not, it inherits the non-trivial zeros of zeta! (S) Why doesn't that have an infinite number of zeros? Because we can show it has (pi T)(log(pi T) - 1)+ O(log T) zeros with imaginary part up to T. (S) ah yes, winding number argument. (S) asks to estimate L(1/2 + it, \chi). I do a partial summation argument and get O(t). (S) So we can deduce the order from this, good. So now back to L(1, \chi): what do you know? I state Siegel. (S) Is it effective? No! (S) Why? And first, tell me what you mean by effective. Well, there are different schools of thought... (S laughs). Here I mean that you can't extract an explicit constant. The reason for ineffectivity is Siegel zeros. I can go through the proof! (S) Sure, do it, and point out where it becomes ineffective! I go through the proof; since I've been botching algebraic number theory questions so far, I joke about how zeta(s)L(s, \chi)L(s, \chi')L(s, \chi \chi') is just a random function that we define and not any function with a meaning. Eventually, I say it's the Dedekind zeta of a number field. (S) We should call it the Katz field! But what is the field? The biquadratic extension is given by the square roots of the conductors. I go through the proof with the power series coefficients and do everything rigorously, eventually, S gets bored and says okay so what do you get. I write down the final inequality and explain how you deal with Siegel 0s. Sarnak says great and makes a comment about Thue-Siegel-Roth being a similar balancing idea. (S) Okay, we should do something in the other Davenport! How do I solve \sum a_i x_i^r = b. I panic because I don't remember the details, but S only wants extremely broad strokes. I show what the integrand is, what the major/minor arcs are, and make some broad comments. I make some comments about squaring in Weyl, and Sarnak asks what you estimate in the end. I say you square k times and eventually get a geometric series, which makes Sarnak happy. (K) Ok but again what is another reason for the importance of L(1, \chi)? Oh, Dirichlet's prime number theorem of course! Stating it in terms of divergence of 1/p with p = a mod q. (K) But isn't there a stronger version? I say yes, the density is 1/phi(q), S talks about how actually that's not due to Dirichlet and the PNT comes much later than Dirichlet's proof. Katz starts asking about whether I know something roughly analogous for prime splitting. I say something incoherent but eventually produce "oh, yes, Chebotarev density". (S) If it looks like Chebotarev, quacks like Chebotarev... My brain is fried at this point. Algebraic Geometry ----------------- (K) Define the genus of a smooth projective geometrically connected curve in every way you can. I was expecting this exact question: H^0(omega), H^1(O_X), 1 - Hilbert Polynomial at 0; 1 - Euler char. (O_X), Riemann's inequality, through the degree of the canonical divisor. Discussion about equivalence/Serre duality. (K) Calculate the genus of y^2 = f(x) where f has distinct roots and is of degree n. Also, give a basis for the space of holomorphic differentials. For the genus, it's just Riemann-Hurwitz, and actually, you don't even need to check what happens at infinity because of a parity obstruction; you get 2g = n-1 or n-2. I write out the basis of differentials immediately (I practiced this the night before!) (K) Do the same for the Fermat curve. For the genus, I use genus-degree, mention it can be proved through Hilbert polynomial. For the basis of differentials, I write something almost correct (K) good enough, you clearly can do this in finite time. Then we move on to surfaces and I'm useless. Sarnak comes to my rescue: (S) Talk about the Riemann inequality more -- is there another part? (K) And who is Roch? I state it and say Riemann proved the inequality and Roch was his student and found the difference term. Sarnak asks me to define everything in the Statement, I do. S asks about what happens when you move the Riemann surface around, I don't know. At some point, I mention one of the reasons for the significance of RR is showing there is a meromorphic function to begin with. Eventually, K takes over again. (K) Ok let's look at SL(n); SO(n); GL(n)... Katz starts saying something about Lie groups. I'm completely brainless at this point. Katz asks about simply-connectedness, I get it wrong. He asks if these are rational; I explain why they are (the non-trivial thing is the Cayley transformation for SO(n)). (K) asks if I know any algebraic geometry over finite fields. I say I know the Weil conjectures but not much deeper than the statement. In retrospect, I should have said something about Hasse-Weil. K says alright, we can finish then. ----------------- Overall it was a much more pleasant experience than I imagined! The committee was very nice and they seemed to care more about what I did know than what I didn't. My biggest piece of advice is to not stress yourself out too much the days before the exam -- you'll do great!