Marco Antonio Sangiovanni Vincentelli’s Generals Committee: C. Skinner (chair) (Sk), P. Sarnak (Sa), Y. Shlapenthock-Rothman (S-R) Special topics: algebraic number theory and algebraic geometry June 19th, 2020, Zoom Generals ————— These are pandemic generals that took place through Zoom! I shared my screen on my iPad and answered the questions of the committee on the app Notability; it was very cozy and the exam lasted for 3h15: from 14:00 to 17:15 (with two 5/10 minutes breaks). I have kept the notes of the exam, so this report will be pretty accurate! ————— We start with Real Analysis. S-R: What are L^p spaces? When do we have L^q inside L^p when q \geq p? When is the opposite inclusion true? S-R: For concreteness, let’s consider L^p([0,1]). How can I approximate a function f in this space? (I said by C^{\infty} functions). How can you do that? (I said convolution by approximation of the identity) What are those? (Wrote down the definition but forgot the positivity, so S-R asked « are these all the conditions necessary? », and I quickly said « positive » and wrote \geq 0, S-R corrected me by saying non-negative; I said I was French-educated so I had a hard time with non-negativity and would say « strictly positive » for positive; the committee laughed saying « sure, makes sense » ) S-R: Can you give me an example of good kernels? (Made a drawing of a smooth function \phi with compact support strictly inside [0,1] and then wrote down n\phi(n x)). S-R: Good. Define the Fourier transform on L^1 of the real line. Sa: What properties does it have? (I said goes to zero at infinity). How do you prove that? (I said approximation by smooth functions + integration by parts). How regular is the Fourier transform of a function ? Sa: Is every function that goes to zero at infinity and is continuous a Fourier transform of something in L^1 (I said no by Banach isomorphim) What is THAT? (Gave the statement of the theorem) Ok so how does this imply non-surjectivity? (Wrote the inequality that would be verified should Fourier be surjective, then wrote down \hat f = convolution of the indicator of [-1,1] with indicator of [-n,n], corresponds to \sin(2\pix)\sin(2\pi nx)/(\pix)^2 and this has not bounded L^1 norm in the same way as one proves that the nth Dirichlet Kernel has L^1 norm approximately log(n)) Sa: Where is the applause emoji’s on this thing? And I received applause emojis from the committee (Sa actually applauded with his hands); we all laughed. S-R: Ok so now let’s do more interesting stuff. Let me give you f in L^2 of \R C^1. With L^2 norm equal to 1, and its FT supported on, say, a ball of radius n. Can you give a bound on the L^2 norm of f’? (I wrote down that the FT is an isometry then used that the FT of the derivative is multiplication by 2\pi \xi, but then I bounded \hat f by its infinity norm and then L^2 norm of \xi on the support of \hat f ) Wait is that a good idea? We know that the L^2 norm is 1! (« Oh yeah, true!, so I rather bounded \xi by n, and the reused the fact that the FT was an L^2 isometry) Sa: Is that inequality sharp? (I said I used an inequality only once, and wrote down that if there is equality then \hat f = 0, so we can’t have equality ) Ok, nice. but what if you want to « almost » have equality. (I thought for a bit without having a lot of ideas) Where does the support of \hat f must be? (I said immediately on the edges of [-n,n]) Good! Another small question. I give you an operator defined by \hat{T(f)} (\xi) = \hat{f}b(\xi). What is the norm of this guy? (I just said L^\infty, without any justification) Aha! Good! (To S-R) Do you have anymore questions? S-R: Last interesting question. I give you a Schwartz function on \R^2, with infinity norm \leq 1, the Fourier transform supported on B(0,R). Can we have an inequality of the type gradient of f in L^\infty norm \leq Constant \cdot R? (I said on the torus it seemed wrong, so I imagined it was wrong). No it’s actually true. Why in the torus is wrong (I babbled some nonsense) S-R: Let me give you a hint. Trivially you can write \hat f = \chi(x/R)\hatf where \chi is some cut off. Now what can you do? (Before I had a chance to do anything) Sa: Nooooo! No cut-off. Cut-offs are ugly when you don’t need them! Just write Fourier Inversion. (I did, differentiated under the integral and ended up with 2\pi R times the L^1 norm of \hat{f}. Didn’t know what to do next). S-R: This is why I liked my cut off! Sa: Yeah, Yeah. Cut-it off then! (Sk: was very amused) (I wrote f as convolution of fourier inverse of \chi(x/R) with f and then we basically did the computation together.) Sk: Ok let’s move on to complex! Remarks: As you can see, the committee was very nice and easy-going and tried to put me at ease (the applause gimmick was very nice). I thought the last question was pretty hard for a non-analyst, but it was nice that we basically did it together. —————— Complex Analysis: (From here and on S-R was silent for the rest of the exam) Sk: So Peter, do you want to start? Sa: Yeah, so what is the statement of the Riemann-Mapping theorem?(In the statement I wrote down f: \Omega \to \D a « biholomorphism ») Aha « biholomorphism », definitely French. (Everyone laughed, I didn’t really know that it wasn’t a thing in English...) Is the mapping unique? (I said you have the automorphism of the disk, wrote down a formula) What is this group? (I said conjugated to PSL_2(R)) Good, yes. Sa: Now draw a Blob. (Did it) We want to map it to the disk, how can I do it by solving the Dirichlet problem on the blob? (I gave Riemann’s original idea.) Sa: Okay, so now draw a blob with a hole in it. I want to map it to an annulus, how can I do it. (I said let’s imagine there’s no hole, so I can map the inside to a disk, and then I will have a hole in the disk. Then I stopped and thought for a while... then I said: actually just solve the Dirichlet problem) Aha! Good, so when are two annuli « biholomorphic » (he stressed the word, with a little laugh afterward). (I said that since we were solving Dirichlet problems I gave a proof using that and that biholomorphism preserve harmonicity). Sa: Oh, nice, but I had in mind another proof. Do you know any other? (With Schwarz Reflection?) Yes. (Reflect it and then if the radii are different it is not Lipschitz at $0$) Yes, or what are the « biholomorphisms » of \C? (z \mapsto az + b and so b=0 which also gives the result.) Sa: Very good so now, draw a blob and put three holes in it. And draw another one. When can I map one to the other? (I said that in Ahlfors it is proved we can map it to an annulus with arc- slits. And explained the proof). So how many parameters do you have? (I said at least the number of holes for the position of the slits + the outer circle, but did not know if the arc-lengths of the slits inside the annulus counted. And Sa said something yes because essentially the only freedom one has at the end are rotations). Sa: Ok, good, but the shortcoming of Ahlfors is that the slits are not very natural. You want to actually map it to a region bounded by circles only. How can you do that ? (I said I did not know). This is actually what Koebe proved. So what is the uniformization theorem? What is the cover of the previous blob with three holes? (Discussion about PSL_2(R), free and properly discontinuous actions...) Sa: Good, I am satisfied with your understanding of the Riemann mapping theorem! Should we move on to Algebra? Sk: Wait! I have one last question. Consider the meromorphic functions in some neighborhood of 0. What is the algebraic structure carried by this set? (I understood holomorphic so I said it is a DVR, because it’s essentially a formal series ring). Ok good but I said *meromorphic* (Oh right so it’s a field, with \Z-valued valuation). Is it algebraically closed? (I first said mistakingly it seems plausible since an essential singularity cannot appear) Hmm What about X^2 -z? (Backpedaled, and showed with the valuation that it had no roots in the field). Good, let’s move to algebra! — — — — — Algebra: Sk: So what is rational canonical form. (I explained it elementary à la française with « companion » matrices) Sa: What is that?? (Me: just a matrix associated to a polynomial P, aka cyclic endomorphisms in a cyclic base. And then explained the proof through structure theorem.) Sk: So using this theorem how many conjugacy classes does GL_2(\F_p) have? Sa: What is another characterization of this number? (Number of irreducible representations) What are those? (Well.. we have the one dimensional, such as the determinant) Sk: Right, what about non 1-dimensional? (We can induce from upper triangular matrices, (refrained very hard from saying Borel, I did not want to go there...)) Sa: What is induction? What is the dimension of the induced? Sk: So what do you induce from the upper triangular matrices ? (Induce from characters; but wrote down unfortunately B for the group of upper triangular matrices) Sa: Aha!! The choice of notation shows the you know more, than what you say. (Me: « I don’t want to go there!! ». Sa and Sk laughed with me) Sk: So what is the dimension of the induced. (The index G/B) Which is?? (Thought for a bit, but then got confused in my mind with the double cosets and said 2) Well.. Let’s think for a second at what G/B is (Me immediately noticing my mistake and internally thinking « idiot! » : it classifies flags on F_p^\oplus2! So it’s P^1_\F_p which has p+1 elements) Sa: Very good! I definitely cringed for a moment there, because G/B must be quite big; you essentially have PSL_2(F_p), modding the center. What can you say about PSL_2(F_p)? (Well it’s simple... then immediately after, if p is like not 2 or 3 !) I mean we are all analysts, so of course p is not 2 or 3. Sk: I beg to differ! (General giggles). Ensued an informal discussion about finite subgroups of GL_n(\C). [In the middle of this question Sk got mysteriously kicked out of the Zoom meeting, so I got to explain why I had said 2 for the index, and why the double coset was indeed of size 2, but I steered away from ever saying Bruhat, or Weyl subgroup ] Sa: Okay so what else can you say about the dimensions of the irreducible representations of a finite group G? (Sum of the dimension squared = order of G). How do you prove it ? (I just said: by considering the regular representation). Sa: Good. So if I give you a small finite group can you compute the table of characters? (« Yes, I hope so! » ) So take.... A_3 for instance. (Euuh, It’s cyclic.. ) Right A_4 then! (I wrote down the order of A_4 as |A_4|= two \times 3 \times 2 = 12 = 4 \times 3, general laughter ensued). (I started then by computing the commutator subgroup: it’s the Klein group, I showed it is « distinguished » first (Sa: « distinguished », so French!) by showing it’s the unique 2-Sylow. So we have three 1-dimensional representations and then I was going to look at the conjugacy classes) Sa: OkOK very good, I can see you can do it. Sa: Last question, I think I ask this to everyone. When can you solve exp(A) = B in complex matrices. (Did the standard argument) Sk: Excellent, let’s take a 10 minute break and then let’s do the special topics! — — — — — I drank some water during the break, and chatted with my roommate. Then went back in my room and noticed we just received an e-mail from the Dean saying that on-compus in-person teaching for graduate students was to resume in the Fall. Sa and Sk were surprised, and Sa joked about Princeton wanted to get rid of professors of his age. Another time everyone laughed and the we started with algebraic number theory. — — — — — Algebraic Number Theory: Sk: So what is a Dedekind domain? (I said Various equivalent definitions exist, my favorite is Noetherian, 1 dimensional, integral and integrally closed, ring, which is that which I gave). Sk: So can you give me a Dedekind domain that is not a PID? (I gave the classical example of \Z[\sqrt(-5)]). Sk: Take a module over a Dedekind domain. Since it’s not necessarily a PID, do you still have a structure theorem? (I said you can decompose a f.g. module in torsion \oplus projective, and that the projective part was a sum of ideals) Is the projective part unique written as this ideal sum ? (No, since if I, J are coprime I \oplus J is isomorphic as a module to A\oplus IJ, and this you can always assume, just by multiplying I or J by an element of the fraction field using the CRT. So I said we have the projective part characterized by rank + an element of the class group) Sa: Ok, Very good. What algebraic structure does the class group have? (Finite abelian group) How do you prove this ? (I said the standard way was through Minkowski’s « Geometry of numbers ») What’s the precise statement from which you deduce finiteness? (I gave the statement that says that any ideal contains an element with norm less than (constant depending only on the number field) \times (Norm of the ideal) ) Sa: So the fundamental tool for this is some kind of pigeonhole principle. What is it? (Take a lattice in R^n. And take a blob, but a nice one i.e. convex and symmetric (Sa: Aha good! I like this guy! [excited for my use of the word blob]), then if the volume of B is big enough with respect to the covolume of the lattice, you can find a non-zero point of the lattice in the blob) Ok good but where is the pigeonhole principle in this? (I explained one cannot fit the parts of B all into the fundamental domain of the lattice without overlapping) Sa: What other results use this fact that you know of? (Dirichlet’s unit theorem, and I merely stated it). Sk: Okay so let’s consider \Q(sqrt(-23)). How would you compute the class number? (I said one can either use Dedekind bijection with quadratic forms or just Minkowski bound + factorization of X^2 + 23 mod small primes, what would you like me to do?) Whatever you prefer! (Ok, so, I prefer Dedekind. I did Gauss’ algorithm and ended up with Cl(\Q[\sqrt(-23)]) = 3 ) Sk: Ok so now consider the 23rd cyclotomic extension. Can you show that the ring of integers is not principal ? (I said I was thinking class field theory) Yes good! (Then Q[\sqrt(-23)] is contained in \Q(\z_23) because of Gauss sums or by taking the square root of the discriminant. Then take H the Hilbert class field of Q[\sqrt(-23)] the composite of \Q(\z_23) with H gives a degree 3, unramified extension of Q(\z_23) (show that H and Q(\z_23) are disjoint) and by class field theory we’re done.) Sa: Aha! I found the applause emojis! (General laughter) Ok so we studied the class number of \Q(sqrt(-23)), do you know how to prove that there are finitely many fields of the form \Q(sqrt(- d)) that have class number one? (Yes, Siegel’s theorem + Dirichlet class number formula) What’s the problem with Siegel? (The constant is ineffective!) Good, so what goes wrong when you it’s real? (We have the regulator!) Sk: What is the regulator? (Gave the general definition) Ok but in the context of real quadratic? (Just log of the fundamental unit!) Sa: Ok good, so it could be that log of the fundamental unit carries all the « beef ». Let’s take an example where d = n^2 +1, square free by simplicity. What happens to the class number (There’s a trivial solution to Pell’s equation, then the regulator is of size less than log(n), so by Siegel the class number goes to infinity). Sk: Can you give a proof of quadratic reciprocity using Frobenius reciprocity? (Gave the standard proof inside the pth cyclotomic extension. I wrote p^* and Sa asked « What is that?? Another French trick? » No! Gauss’ notation for (-1)^{(p-1)/2}p. Sa: Aha! You think you can beat me at my own game? General laughter ensued) Sk: Ok very good. Do you know what the Hilbert symbol is?(Yes defined like Serre, (a,b) =1 iff the conic z^2 - ax^2 - by^2 has a solution on the base field. I said it corresponds to the 2-torsion of the Brauer group). Sk: What is the Brauer group? (I said many possible definitions. The fastest is the H^2 of K^\times.) So what is the two torsion in this? (I explained quaternionic algebras and the fact that they are split iff the norm does not represent 0 i.e. iff the conic does not have a point. Then I stopped thinking I was done, but then said « oops I guess this still doesn’t explain the 2- torsion... » ) Sk (laughing): Yes indeed! (So I explained the other definition of the Brauer group in terms of central simple algebra, and why A\otimes A^op is a matrix algebra, then showed that in this definition Q \otimes Q = M_4(K), so Q is two torsion. [I didn’t explain the H^1 of PGL_n and why the two definitions are the same]. Sk: Good! Do you know if there is any local to global principle? (I wrote down Brauer-Hasse- Noether exact sequence. And said that the injectivity shows that a central simple algebra is split iff it is locally split.) Sk: yes so now, last question. If I give you a quaternionic algebra. What can you tell me about the places where it is not split? (I said that there were an even number of them,) Why? (Euuh, not really sure anymore) Look at the exact sequence that you just wrote. (After staring at it for a few seconds: Oh right! Q splits in a quadratic extension, and the local invariant is then just 1/2, so since the sum of the local invariants must be an integer (i.e. 0 all to the right), there must be an even number of non-split ones) Sk: ok good! I am satisfied with Number theory. Let’s take a 5 minute break and then come back to do algebraic geometry. — — — — — Algebraic geometry. Sa: Can you tell me the statement of Riemann Roch for curves? (Gave it) What are the major ingredients of the proof? What does it say in general? (I gave Riemann’s inequality, and explained that it was an equality when the degree of the divisor is bigger than 2g-2) Sa: Ok, let’s work over the complex numbers and imagine I take D of degree g-1. Can anything happen? (I said if g is 1 it is just an elliptic curve so we can certainly have 0 or 1. 1 if it’s the canonical for instance and 0 if it’s just a divisor of the form (P) - (O), so a priori anything can happen...) Mmh ok, but can you be more precise for also general g? (I said I didn’t really know). Okay, that’s fine. This is what Riemann studied in great detail. He embedded the curve into its Jacobian that is an abelian variety, and then you can restrict the theta divisor to the curve and see what happens. Sk: If I give you an affine equation, can you compute the genus? (« Yes I think so! » ) Okay choose one that you want. Sa: Not a hyper elliptic one! (I chose y^3 = x^6 -1, and computed the genus by considering the double cover of the elliptic curve y^3 = x^3 -1). Sk: Ok, good. If I give you a scheme, how can I know if it is affine? (I said there was Serre’s criterion) Alright but if I say for instance X= projective space over A. How can I show that it is not affine? (Take the global sections!) Sk: Ok, what nice properties does the projective space have? ( proper!) What does it mean essentially? (The structure map is closed). Can you prove it ? (I gave a direct proof, instead of using the valuative criterion). Sk: Good. I have one last question. I give you a scheme over \Q. Is there a local to global principle to be able to tell if X is the projective space. Or rather say the projective line. (I was thinking and said if I can reduce then I know it’s number of F_q points for all q so its zeta function and i know that the genus is zero!) So ? (well then yes, it must be P^1... wait no! I don’t know it it has a \Q point! So we again have the Brauer group intervening and Brauer-Hasse- Noether!) Sa (with Applause emoji): Good! So one very last question. You talked about the Zeta function. Can you prove the RH for Elliptic curves ? (I did it through the Tate-module approach with the trace of the frobenius). Sk: Ok we’re done! let me put you in a break-out room so we can deliberate. [After a couple of minutes] Sk: Congratulations you passed! Sa: So you come from the Ecole Normale? (No! Ecole Polytechnique!) Aha, then I must think now the Ecole Polytechnique is better ! (General laughter). Then Sa told us a funny anecdote about his time at the ENS. Sk: Now you can go and binge-watch any show you want for a week! — — — — — As you can tell by this account the committee was extremely nice, they put me at ease right away and we had a lot of fun, mostly thanks to professor Sarnak’s jokes. It was a very nice experience overall, even though at the end I was pretty exhausted.