Anshul Adve's Generals Committee: Peter Sarnak [S] (chair), Alexandru Ionescu [I], Ian Zemke [Z] Special topics: harmonic analysis, analytic number theory April 19, 2022, 3:00-5:45 PM, Sarnak's office (Disclaimer: I've probably forgotten several questions, hints, and fun comments from the exam. The order of the questions is certainly not quite correct. Most importantly, any mistakes allegedly made by the committee are almost certainly a transcription error on my part.) I arrived at [S]'s office at 3. There was some brief discussion about Covid - several of [Z]'s students had missed an exam due to sickness, and [S] complimented [I] on his decision to double mask. I was then asked to choose the order of my topics. I suggested complex, real, algebra, harmonic, and finally number theory. This choice was approved because [I] had to leave early, so they wanted harmonic analysis to be the first special topic. Complex Analysis ---------------- [S] What did you read? A combination of Ahlfors and Stein–Shakarchi. [S] OK good, so you've looked at Ahlfors. State the Riemann mapping theorem. (Stated). How unique is the map? You can choose which point gets sent to zero by applying a Mobius transformation, and then it's unique up to rotations. [S] Write explicitly what you mean. (Done). What are the automorphisms of the disc? I wrote down a rotation times a Blaschke factor. [S] What group do these transformations form? I didn't know. After some fumbling at the blackboard, [S] told me the answer: [S] Have you heard of the group SU(1,1)? (No). What's the automorphism group of the upper half-plane? PSL_2(R) (in hindsight I should have picked up on the fact that Aut(D) and Aut(H) are conjugate by the Cayley transform). [S] How about the automorphisms of the Riemann sphere? Mobius transformations, PSL_2(C). [S] Do you know anything about Riemann surfaces? (A bit). What are your favorite Riemann surfaces? I quickly decided I didn't want to be asked much about Riemann surfaces, so I said: "the simply connected ones!" [S] List them. (Listed). Can you prove uniformization? Uh, yeah, at least I think I can sketch it (I wasn't pressed for details, which is probably fortunate). [S] How can you distinguish the three simply connected Riemann surfaces? The sphere is compact, the disc admits a Green's function, and the plane satisfies neither of these two properties. [S] Draw a nice open subset of C, say bounded with smooth boundary. What's a Green's function on this domain? (Defined). How would you construct one? For subsets of the plane, you can subtract off the fundamental solution to reduce to solving the Dirichlet problem. Then you can use, e.g., Perron's method. [S] What's Perron's method? (Described). How do you know the supremum you wrote is actually a harmonic function? I gave the standard argument from Ahlfors. It took a few minutes of thinking aloud and a leading comment from [S] for me to figure out all the details, but the committee was very patient. [S] I'm satisfied with complex. (Turning to [I] and [Z]) What do you think? [I] Wait a minute. Why are there enough subharmonic functions to force the Perron solution to obey the desired boundary conditions? When [S] asked me to draw this domain, he said I could assume smooth boundary, so I'll use this assumption now. [S] Yeah I only care about smooth boundary, but [I] can change the rules! [I] No I don't mind, continue. I showed how to construct barriers by means of a picture, and [I] was happy. [I] We still haven't asked him anything outside of conformal mapping! [S] Oh right. What's the order of an entire function? (Defined). How are entire functions of finite order determined by their zeros? (Stated Hadamard factorization). OK let's move on to real. Real Analysis: -------------- [I] What are L^p spaces? (Defined). How can you control a product of functions in L^p? (Holder). What's the dual of L^p? (Stated for 1 \leq p < \infty). Do you need a condition on the underlying measure space to make this work? (\sigma-finite). Prove it. I began the proof, but [I] wanted me to skip ahead. [I] What tool do you finally use to construct the representing function in L^{p'}? (Radon–Nikodym). Where does the proof fail for p = \infty? (Explained). Can you find an element of the dual of L^{\infty} which is not in L^1? (Subsequential limit of L^1 functions escaping to infinity). In which topology is this limit taken? (Weak*). [S] Do you know what the dual of L^{\infty} is? Finitely additive measures ... ? [S] But on which \sigma-algebra? This is important. Oh, I don't know. [S] The Lebesgue \sigma-algebra. (Not Borel?) No, Lebesgue! Before I could respond, [S] turned to [I] and said: [S] You'll test him on harmonic analysis later, right? So we don't have to ask the usual questions? [I] seemed to agree, so [S] continued: Great, we can talk about nasty stuff then! ([S] said this with zeal. Thankfully, I understood that "nasty" just meant "pathological and possibly requiring the axiom of choice," not necessarily "difficult"). [S] Have you heard of the Ruziewicz problem? (No). It's to show that the only rotation-invariant, finitely additive measure on the Lebesgue sets of S^n for n \geq 2 is Lebesgue measure, up to scaling. This was proven by Margulis and Sullivan for n \geq 4 by showing that a certain subgroup of SO(n+1) has property (T), and it was proven by Drinfeld for n = 2,3 by establishing a spectral gap. (Wow!) In this context, is the distinction between Lebesgue and Borel important? [S] Yes! This problem is open for finitely additive measures defined only on the Borel sets. In fact, the Borel case is X's favorite problem! (Sadly, I've forgotten since the exam who X is). By the way, why do you think the situation for n = 1 is different from n > 1? Maybe because SO(2) is amenable, but SO(3) is not? [S] nodded encouragingly. And why isn't SO(3) amenable? It contains a copy of the free group on two generators. [S] Can you show that? I can't give any details, but I think you can do it with the ping-pong lemma. [S] Yeah that's right. Do you know the Banach–Tarski paradox? It's something like, you can decompose a ball in finitely many pieces and reassemble them ... [S] Yeah, you can reassemble them with rigid motions to get two copies of the ball. If you allow a countable exceptional set, then this was already shown for the sphere by Hausdorff! The exceptional set here is the set of points fixed by at least one nontrivial element of a chosen free subgroup of SO(3). [S] OK, now, state Hahn–Banach. (Stated). [I] Does a version of this hold in a more general setting? (Locally convex spaces). [S] Do you know why this is named after Hahn and Banach? It wasn't a joint paper. No I don't. [S] Banach developed it to extend Lebesgue measure to a finitely additive measure on all subsets of R^d. I've forgotten what Hahn's contribution was. Speaking of, why can't Lebesgue measure be extended countably additivitely to all subsets of R^d? I defined the Vitali set and gave the standard argument. [S] asked [I] if he had any more questions. [I] State the Baire category theorem. (Stated). Give some applications. Open mapping, closed graph, and uniform boundedness in functional analysis. [I] Can you use Baire to show that there are continuous functions which are nowhere differentiable? Yes – if there aren't, then you can write C([0,1]) as a countable union of nowhere dense sets. I drew a picture to illustrate what the sets should be, and then added a sawtooth-looking thing to demonstrate that they are indeed nowhere dense. This satisfied [I]. [S] (musing) What other nasty stuff can we ask about? Do you know Krein-Milman? (Stated). I'm sure [S] had an application in mind for this, but [I] was starting to get nervous about the time, so we switched to algebra. Algebra: -------- [Z] OK, I prepared a list of questions .... What's a projective module? I was just beginning to answer when [S] kindly interjected: [S] Hold on a moment! Let's start simple. What's a module? (Defined). OK, go on. [Z] So what's a projective module? (Defined in terms of exact sequences splitting). That's correct, but it's usually a theorem that that's equivalent to the standard definition. Do you know what the standard definition is? (I guess not). Draw the following diagram: you have a surjection from N to M, and P maps to M. Then P is projective if and only if there's always a lift P to N. Have you seen this? Yeah, I have, but I wouldn't have remembered if you hadn't reminded me. [Z] That's fine. Do you know any other characterizations of projective modules? You can direct sum them with something to get a free module. [Z] Right, they're direct summands of free modules. Yeah, "direct summand!" That's the phrase I was looking for. [Z] So how do you prove that? I proved one direction of the equivalence, and was stopped before giving the other direction. [Z] What's a projective resolution? (Defined). What can you say if you have two projective resolutions? I was out of my depth, but I made a tentative guess and looked over to [Z] for confirmation: You can find a map from one to the other? [Z] nodded. Can you prove that? I managed with some hand holding. [Z] Good. Do you know what Tor(M,N) is? Uh, I've seen the definition but don't know anything more. I think you take a projective resolution of M, tensor with N, and take homology? [Z] Almost, you also drop the last term. (Oh, right). Let's try an example. Let k be a field, and view k as a module over the polynomial ring k[x]. Then what's Tor^{k[x]}(k,k)? This was pretty much a disaster. I found a projective resolution, but couldn't work out what happens when you tensor with k. [Z] had to walk me through every step from then on. Given my cluelessness, he explained things very kindly throughout the exam. [Z] It turns out that Tor^{k[x]}(k,k) has a natural algebra structure. Can you see how? (Uh, no). There's no way you can think of to combine symbols together and get a multiplication? I gave two suggestions, which, from the look on [Z]'s face, clearly made no sense. [Z] Hmmm, it seems like the rest of my questions might be a waste of time .... [S] (pitying me) Maybe we should go back to more standard stuff. What did you study? Group theory, Galois theory, representation theory, and finitely generated modules over a PID. [S] OK, should we do group theory or Galois theory? [I] I've never seen a qual which doesn't do any Galois theory. [S] Let's do that then. Do you like finite fields or number fields? (Finite fields). Tell me about extensions of finite fields. (Gave standard facts). What's the algebraic closure of a finite field? I accidentally wrote it as a limit rather than a colimit (lesson: don't unnecessarily use notation you're not used to), but they didn't bother correcting me. I guess it was clear what I meant. [S] Give an example of a non-abelian Galois extension, say over Q. Splitting field of x^3-2. [S] Do you know any simple non-abelian groups? (A_5). What's the significance of A_5 in Galois theory? S_n and A_n aren't solvable for n \geq 5. [S] Why do we care about that? We can find Galois extensions which are not radical extensions. [S] OK but what does this mean in terms of polynomials? There are polynomials of degree 5 whose roots can't be expressed in radicals. [S] Can you give an example? Take an irreducible polynomial with three real roots and two strictly complex roots. [S] What's the Galois group of such a polynomial? (S_5). Why? View the Galois group as a permutation group acting on the roots. Then it contains a transposition (namely complex conjugation). By Cauchy's theorem, it also contains an element of order 5, which must be a 5-cycle. Any transposition and any 5-cycle generate S_5 - more generally this is true in S_p for p prime (I saw this argument in Keith Conrad's expository note "Galois groups as permutation groups"). [S] You said S_5 isn't solvable because A_5 is simple. How do you prove that? I first said you could look at the sizes of conjugacy classes and see that no subset of them adds up to a possible size of a normal subgroup. [S] Does this use the Sylow theorems? I was pretty sure the answer was no, but I hadn't done this computation in a while, so I dodged the question by saying: "Never mind. Instead, you can check by hand that A_5 is simple by playing around with conjugation by 3-cycles." [S] didn't mind this rather vague answer. [S] Do you know any other simple groups? (PSL_n(F_q)). Why is that simple? I said I couldn't give the details, but I mumbled something including the phrases "Iwasawa criterion" and "doubly transitive action on lines." [S] Yeah that works. I'm done with algebra. Ian, do you want to ask anything else? [Z] Nope. [S] Let's take a five minute break then. While drinking water outside [S]'s office, I overheard [S] saying to [Z]: You probably think homological algebra is the most important tool in mathematics, and that everyone ought to know it! [Z] didn't dispute this. Harmonic Analysis: ------------------ [I] What did you learn? Calderon–Zygmund, Littlewood–Paley, pseudodifferential operators, and oscillatory integrals. [I] Let's start with oscillatory integrals. What's stationary phase? I began to write, and [I] interrupted: [I] Just start with the quadratic phase e^{\lambda ix^2} in one dimension. What do you get? I wrote the asymptotic. [I] What's the constant in the first term? (Answered). How do you prove the asymptotic? I was indecisive about which proof to give. Then I panicked because I was just standing there saying nothing, and I rushed into the wrong choice: I began giving a proof which is a little messier than necessary in this simple case. [I] Why did you make that change of variable? It's counterproductive. It's a nice normalization so that the phase starts oscillating when |x| \gtrsim 1 .... Oh I see, you want me to just do it directly. Starting over from the beginning, use Parseval and then Taylor expand the phase. [I] How do the coefficients in the asymptotic depend on the derivatives of the cutoff? (Described using Fourier inversion). You can also see this without using Fourier analysis. How? Starting from the beginning again, localize to |x| \lesssim \lambda^{-0.1} (say) using dyadic decomposition and nonstationary phase, Taylor expand the cutoff, and use dyadic decomposition and nonstationary phase once more to estimate \int_{|x| \lesssim \lambda^{-0.1}} x^j e^{\lambda ix^2} dx. [I] What's the contribution of the integral you just wrote down? \lambda^{-(j+1)/2} for j even, 0 for j odd. [I] Alright. What if the original stationary phase integral is just over an interval? Boundary terms arising from integration by parts contribute to the asymptotic. [I] Tell me about stationary phase in higher dimensions. I wrote the standard result for a phase with nondegenerate critical points. [I] What's the constant in the first term now? I gave the constant up to a phase. This was all [I] wanted. [I] Why should you expect that? I drew an ellipse where the phase is stationary. [I] That's enough about oscillatory integrals. What's a Calderon–Zygmund kernel? You can just do the translation-invariant case. (Defined). Give a sufficient condition for the Fourier transform of the kernel to be in L^{\infty}. The integral of the kernel against an L^{\infty}-normalized bump function adapted to the ball of radius R is uniformly bounded in R. [S] Where did you read this stuff by the way? Stein's book and Tao's old course notes. [I] There are many sources for this – it's very standard. [S] Why do we always ask standard questions? We've gone soft with qualifying exams lately. (It hadn't felt too easy so far, but OK.) Were they harder in the past? [S] Yes! Stanford used to be known for having a terrifying exam. You know that problem book by Polya and Szego? (Yeah). Those were all Stanford qual problems. In fact, Hormander didn't believe it could be as hard as people said, so once when he came to visit, he asked to try the real analysis portion. He took the exam home with him that evening. But when he came back the next day, he was red in the face. He couldn't answer all the questions! When was this, roughly? [S] When I was a faculty member at Stanford. And how was it when you were a student there? [S] Even worse! [I] Let's get back to harmonic analysis. What's a classical example of a Calderon–Zygmund operator? (Hilbert transform). What's the kernel, as a distribution? (Described). Tell me if the following are Calderon–Zygmund kernels: \langle x \rangle^{-1}? (No). |x|^{-1/2}? (No). A Dirac delta at 0? (Yes). Give an application of Calderon–Zygmund theory. Once again, I was indecisive about which application to give. I settled on the one I thought would lead to the fewest followup questions: L^p control on second derivatives of solutions of the Poisson equation. In hindsight, though, I'm pretty sure [I] wanted me to say something people use every day, like Littlewood–Paley, or: [I] How about something involving Sobolev spaces? Sure. I gave the physical and Fourier space definitions of Sobolev norms and showed they're equivalent. [I] OK what else ... can you state the Cotlar–Stein lemma? [S] Wow, this takes me back. I haven't thought about this for years! I stated it, but forgot at first to include bounds on both T_iT_j* and T_i*T_j. [I] Are you missing a condition? Oh, of course! I corrected myself. This was the last harmonic analysis question. [I] had to leave, so [S] and [I] went outside briefly to discuss if my performance had been satisfactory. When [S] returned, we began the final topic. Analytic Number Theory: ----------------------- [S] What did you read? Both Davenport books recommended by [S], as well as Tao's course notes. [S] It always comes back to Tao with you, doesn't it! By the way Ian, you should feel free to chime in. [Z] That's alright, I don't do much number theory. [S] Come on, ask him anything you've ever wanted to know about L-functions. You don't often get an opportunity to watch a guy under stress answer questions at your whim! [Z] OK, what do you like about L-functions? [S] (before I could answer) I once saw someone get so inspired after one conversation that they changed their field of math. I want to see this happen to Ian today! I wrote Riemann's explicit formula, and described how analytic continuation and crude growth bounds on L-functions sort of magically give asymptotics for counts of arithmetic interest. I wasn't sure how else to motivate L-functions, but I'm sure [Z] had seen this all before, so he was understandably unmoved. [S] Tell me about sums of three primes. (To [Z]) This will force him to synthesize everything he's learned. (As you'll see below, [S] pushed for very few details, so this didn't actually cover much of what [S] had asked me to prepare. I wasn't complaining, though.) I wrote the relevant exponential sum S_X(\theta). To motivate the major/minor arc decomposition for [Z], I drew a graph of |S_X|, and drew on the graph where the major arcs are. [Z] Why does the graph have spikes where you've drawn them? Because the primes are far from uniformly distributed mod q for small q. [S] How large is the union of the major arcs? X^{-1} times a power of log X. I explained that the spikes in the graph have width roughly X^{-1}, and that this is consistent with the expected main term being of size X^2 in the asymptotic for weak Goldbach. [S] Why do we expect the main term to have size X^2? (Standard heuristic). Since we're running out of time, I won't make you analyze the major arcs. How do you estimate the contribution of the minor arcs? We want to control the L^3 norm of S_X on the minor arcs. By Parseval and Holder, we win if we can get a nontrivial L^{\infty} bound for S_X on the minor arcs. To do this, I described the Type I/II decomposition, and [S] stopped me as soon as I said "bilinear forms." He only wanted a broad outline (probably because we were short on time). [S] Let's do something less standard. If we tried to prove Goldbach or twin primes in exactly the same way, then Parseval would have to be replaced by an L^1 upper bound on S_X. To show that this strategy won't work, can you give an L^1 lower bound on S_X? You can get a lower bound of X^{1/2} by examining the 1/X-neighborhood of the rationals of denominator at most X^{1/2} (thanks Mayank for showing this to me!). [S] Good, so you've thought about this before? Well, you asked it on a previous Generals. [S] Oh, alright. Actually, Paul Cohen and I once put this on the Stanford qual (with a hint). There was a grad student who thought he'd proven Goldbach, so we added this to the exam to show him he was wrong. We were forbidden from writing the quals after that! [S] Here's a similar question: can you give an L^1 lower bound on \sum_{n < X} e(n^2\theta)? Interpolate with L^2 and L^4. [S] Don't tell me – I've asked this in previous Generals too? Sorry, yeah (on like every single one!). [S] Fine. Have you heard of the Littlewood conjecture? (No). Write the following exponential sum: T_X(\theta) = \sum_{j < X} e(n_j\theta), where the n_j's are an increasing sequence of integers. The conjecture is that the L^1 norm of T_X is minimized (up to constant factors) when the n_j form an arithmetic progression. My advisor (i.e., Cohen) showed that the L^1 norm of T_X at least goes to infinity with X. This is the first thing that made him famous, before he proved the independence of the continuum hypothesis. [S] In the arithmetic progression case, can you tell me what the L^1 norm is? Actually just take n_j = j. I mumbled "it should grow like log," and I drew a graph of T_X to convince myself this was correct. Unfortunately, this just confused me, because I only drew the portion of the graph where |\theta| \lesssim 1/X. [S] You should know this, it's the Dirichlet kernel! Right (that was embarrassing). I drew the rest of the graph with oscillations of amplitude 1/|\theta|, and concluded that the L^1 norm should be log X. It was now getting late, and [S] asked me to wait outside so he and [Z] could deliberate. After a couple minutes, they invited me back in and told me I passed! [Z] then left, and [S] asked me about my post-Generals plans – "you're not going to drink yourself off the face of the earth, are you?" We continued talking for a few minutes in [S]'s office and then went down the elevator together. [S] accidentally came all the way to the sub-basement with me. From his expression of curiosity, I'm not sure he'd been this far down in Fine before. Anyway, I proceeded to the first-year offices, and [S] went back up to leave.