Victor Wang
Friday, April 27, 2018 from 1:30p to 4:45p (maybe?) in Sarnak's office (Fine 1101*)
Composed by: Peter Sarnak (chair), Shou-Wu Zhang, Tristan Buckmaster (third)
Special topics: Representation Theory of Compact** Lie Groups, Algebraic Geometry
Ft. Hermann Weyl & Bernhard Riemann
*This is a large number. Unfortunately, unlike MIT, Princeton doesn't (currently) have a pumpkin drop off the tallest building on campus. The pumpkin drop occurs next door on top of the much shorter Jadwin Hall instead.
**or rather "Complex" according to an official email, but my examiners didn't troll me accordingly
===
Meta comments/advice:
- As Wei Ho said on her generals page, "Take the right attitude towards the exam... try to actually learn some math and enjoy studying for it."
- Look through a few other pages (esp. those involving your examiners; for me those with S or Z helped the most) for enlightening problems, reading suggestions, and advice. Or more usefully, talk to your examiners for guidance (both w.r.t. exam and generally).
- Talking to people is very helpful for BS detection, appreciating alternative viewpoints, and learning to tell reasonably coherent/complete stories. And also for other things, of course...
- Circadian rhythm is a strange beast. Try to maintain a regular sleep schedule the week before the exam, or else you may run into technical difficulties the day before (sleeping on the Common Room couch to avoid missing the exam... but there was also luckily a free food talk at noon that day).
- As others have noted, the point of generals is to test your high-level understanding of definitions, proofs, and ideas. To robustify my understanding (in general, not just for the exam) I often find it helpful to spend quality time digesting common misconceptions and important-but-possibly-subtle examples and concepts (here are random examples that might be irrelevant to your interests: usually "measurable function" R\to R means f^{-1}(Borel) is Lebesgue; closed vs. non-closed linear subspaces; or left- vs. right- vs. bi- vs. Ad- invariant measures or metrics or inner products). Trivial-to-easy exercises are often good for this (but not all books have them). So is talking to people, and other interactive things like multiple choice questions, e.g. Terry Tao's MC quizzes or some I've compiled: https://www.expii.com/t/advanced-math-archive-10777 (still a work in progress).
- As others have also said, don't overstudy for generals, and remember that in an oral exam the phrasing of questions and hints will take you much further and faster than you could go alone in a vacuum. You'll probably forget all the details you don't actively use in the future, anyways.
===
(Disclaimer: I have likely forgotten many funny, embarrassing, interesting, and/or boring things from the exam. Sorry!)
NOTE 12/24/18: Most of the stuff below was written a couple days after the exam, but I got bored writing up some technical parts, where I only left sketchy notes (hopefully now revised to a legible state). The more thorough, polished parts should be more useful.
B and I awkwardly stood outside S's office a few minutes before the exam. Then the elevator sounded as S & Z arrived a few minutes late. I got a bit nervous before the exam, but the easygoing nature of my committee helped a lot. (I've heard similar sentiments from other grad students.) Before we all walked into S's office, S joked to Z that "B [a first time examiner] probably thought he had to prepare for the exam".
S asked what basic topic I wanted first; I said real. S let B start, but admonished B to keep things simple ("if we [S and Z] can't see how to solve them, something's wrong"). B had, indeed, prepared a list of questions and also brought a mysterious iPad (?).
REAL ANALYSIS
B: "Don't worry, I'll start with something easy..." Prove that two disjoint compact sets (in R^2 say) have positive distance between them.
[Drew K_1, K_2. I instinctively took neighborhoods of p\in K_1 disjoint from K_2, which may work with enough care, but being at the board in an exam, I got stuck. I eventually remembered / hit upon the concise way to think about this: look at d(x,y) as (x,y)\in K_1\times K_2 varies, which achieves a minimum by compactness.]
Inspired by my silly drawing of a neighborhood of p\in K_1...
S: Draw a region \Omega (in the plane). Can you solve the Dirichlet problem (f on boundary, \Delta u = f)?
[I first blurted out "Perron's subharmonic method... assuming the region is reasonably nice" (which I had recently read about in Ahlfors and heard about in Tao's recent blog post on Riemann-Roch).]
S: "[Yes, assume] nice boundary conditions and what not."
[But I also had a rather degenerate example in mind (0 on punctured disk boundary, 1 on center... but looking back, this doesn't really count as nice boundary, as it's not even codimension one) so I ended up backing down to assuming simply connected (even though it's true in much greater generality). S asked what fact I wanted to use: Riemann mapping theorem. We would come back to this in complex.]
S then briefly discussed Riemann's attempt minimizing Dirichlet energy integral \int |\nabla u|^2, and how Riemann implicitly assumed compactness of function spaces or similar (I guess he brought this up because of the first question).
[Accidentally wrote \Delta u at first, which must be missing the intuition/point of doing this... and Z quickly corrected me.]
S: "Do you know much about [method of] variations?"
[No, so we moved on.]
B & S:
Define Fourier transform on L^1
S: "L^1 of what???" (B: "L^1(R)")
[Defined it.]
S remarked something like "I see that 2pi: he's serious!"
S: "What can you say about \hat{f}... ?"
[why continuous? phase bounded]
"Yes, OK, so that's easy. Now what B asked [Riemann-Lebesgue]..."
DCT argument
lazily used smooth cpt supported fcns, integration by parts
(could also have done maybe characteristic functions of intervals)
"do you get decay" (no, b/c needed L^1 approx.)
at some point:
[non-]surjectivity of Fourier transform
brief comment about Royden-Fitzpatrick blah, and "it's in Rudin"
I can't prove but can do for Fourier series
S: "OK, do it for Fourier series."
use open mapping theorem
S: "OK, so to finish you need... do you know an example?" [Yes, Dirichlet kernel. (Wasn't asked to elaborate, but the point is D_N(x) is just an exponential sum, so its Fourier coefficients are all 0 or 1...) I had just learned this the previous day, since I'd managed to delay review of Fourier until then and B had suggested skimming through Rudin for concise general review.]
at some point, I was asked the following:
[maybe here after open mapping thm and comments on how functional analysis is not taught anymore... around the same time (maybe after the Fourier transform question), I was also asked what I had learned from undergrad Fourier analysis (with Larry Guth), and I said "everything was Schwartz" there]
"do you know about Fredholm operators"
I said "sort of".
"you better, because [of the connection to spectral theory]..."
in the end I was just asked to define compact operators
[image of bounded set is totally bounded (this is equivalent to usual when target space is Banach, i.e. complete)]
then a question about Hilbert Schmidt integral operators, explanation why compact
[I think my answer was along the lines of
https://math.stackexchange.com/questions/474503/compactness-of-hilbert-schmidt-operator
reducing to the case of separable kernel functions using, I guess, one of
https://math.stackexchange.com/questions/105451/orthonormal-basis-for-product-l2-space
or https://en.wikipedia.org/wiki/Tensor_product_of_Hilbert_spaces
but I don't remember how much detail was requested.
(in the case of separable kernels, the operator is compact since it has f.d., in fact one-dimensional, image)
S: "yes, that's one way to do it. that's correct"]
B asks if I want more real analysis or Fourier analysis; I say either, though S half-suggests real because Fourier may be included in representation theory later.
B (basically ends up being Fourier-like anyways): "Let f be a periodic function and \gamma an irrational number..." and then since words suck he goes up to board and writes down 1/N \sum f(\gamma n) \to \int_0^1 f(x) dx and asks me to prove it.
S & Z joke in the process: "... he is taking over... you should sit down as B presents"
[I thoughtlessly say {\gamma n} is equidistributed so by Weyl equidistribution theorem...]
Of course the point is I should outline the proof.
[I thoughtlessly say it's enough to prove for f characteristic function of an interval (having definition of equidistribution in mind rather than geometric series proof), then upon hearing "that's not very helpful here" realize that of course we want to test with f exponential; I instinctively write e^{itx} (implicitly t real) and correct to e^{imx} (implicitly m integer), possibly forgetting the 2\pi, blahblahblah I say "oscillate" and they prod me to clarify geometric series, and eventually address the boring m=0 case separately.]
Perhaps inspired by this...
S: Consider the equation f(x+\gamma) - f(x) = g(x) with f,g periodic. (Again, \gamma irrational.) What is a nececessary condition (on g) for this (to be solvable in f)?
[Integrate to get \int g = 0.]
S: What else can you say?
[Look at Fourier series f(x) = \sum a_n e(nx) and g(x) = \sum b_n e(nx); I wrote a_n = b_n / (e(n\gamma) - 1), but S leads me to note explicitly that b_0 = 0 which I didn't really think about.]
A brief back-and-forth with S on square-summability and smoothness assumptions ("g smooth assumed") ensues.
[I say if \gamma is really bad (I mention "computable" and "Liouville number" but S asks me what I mean by "computable", and we don't pursue it) then I think this could still blow up.]
But we end up just doing \gamma = sqrt(2) explicit example.
[Difference of squares lets you bound |\gamma - p/q| >= C/q^2. By instinct I thoughtlessly say "if it's nonzero" at some point... "can it be zero?"... "no", of course not.]
Z remarks that Euler solved this in general; S reacts and Z clarifies, "formally" (with power series).
S remarks that this equation is important in "KAM theory" (which I just Googled to check; sounds about right) to which B nods.
===
COMPLEX ANALYSIS
B: Liouville
wrote down Cauchy formula
units canceled out, so not useful
realized about same time as got hint to look at derivative instead
Note: I always forget the nicest proof (by the late Ed Nelson, I think from Princeton), which works for harmonic functions in any dimension: you just use the MVP to write f(a) and f(b) as the averages of f over large *solid* balls centered at a and b.
B: "Can you use this to prove the fundamental theorem of calculus?"
After some funny silence and an exchange I don't remember exactly, B corrected himself: "... of algebra!" (and either Z or S made a joke about the appropriateness of the question in the analysis section).
[I then gave the usual explanation looking at 1/P(z)... though I guess you only need MVP, not Liouville, for this.]
S: Define order of an entire function (so I erased "bounded" from the Liouville discussion).
[I tried to give the lim sup log log ... form of the definition (instead of the more intuitive exponential form) but hadn't slept well enough to do this properly, and they didn't correctly correct me. It didn't end up mattering so much.]
S: Estimate the number of zeros (in a disk) in terms of order.
- At some point we briefly discussed how the bound can only go in one way, as the general form of an entire function with no zeros is e^{g(z)}.
- Anyways, in answering this question, I stated Jensen. [If there were no zeros would get log |f(0)| = avg log |f(z)| over circle since log |f(z)| would be harmonic, but since we have zeros we need correction terms.]
- S: "You said log |f(z)| is harmonic if there are no zeros. If there are zeros, what kind of function is this?"
[I mumbled "Green's function?", even though I knew it's not quite. S: "It's subharmonic! You mentioned that earlier for a reason, right?" Well... this caught me off guard; somehow I hadn't thought about Jensen this way before.]
- Then a brief back-and-forth on how you get an actual estimate on number of zeros from this.
Next...
S: Is (s-1)\zeta holomorphic?
[I got confused by the sudden transition and said something incoherent I've forgotten; S: "If you don't know about \zeta we don't need to cover it..." and I then corrected myself: "Yes."]
S: What is the order of the zeta function? (OK, I guess one needs the previous question for this question to make sense.)
[1. For a lower bound, I mentioned the trivial zeros and then elaborated "sum of reciprocals" when S said Jensen bound doesn't have a (naive) converse.]
S: "OK, why not order 5, say?"
[I mentioned the functional equation, and that the gamma function has order 1. They didn't push for further details.]
S: "OK, let's go back to the Dirichlet problem."
- This led to a statement of Riemann mapping theorem. "Do you know how to prove it?" [Yes, and they didn't push for details.]
- "OK, now how do you solve it in the disk?" [Poisson.]
- At some point, maybe here, S might've asked why this solution is continuous / unique (or related to one of these). I think I gave an unsatisfactory answer specific to the plane (harmonic is real part of analytic), though they didn't mind as it is complex analysis.
- We briefly discuss the boundary mapping and its role in ensuring the validity of the conformal mapping solution. [I said something about properness ensuring the boundary goes to the boundary, but wasn't able to say off the top of my head how to justify that the image of the boundary doesn't go "back and forth". I don't remember the wording but S said we were getting into rather esoteric or technical details and we moved on.]
Of course, an S exam would be incomplete without the next two questions...
S: Describe the conformal mapping of doubly connected regions.
[Map to annulus. I gave more or less the proofs from Ahlfors. The point is that Ahlfors solved the Dirichlet problem to some level of generality, so we can use that to generalize Riemann mapping theorem to multiply-connected regions. Explicitly, if the boundary of \Omega is F_1 - F_2, consider the harmonic function \omega_i that's 1 on F_i and 0 on the other. Then (by considering the periods of the conjugate differentials *d\omega_i, measuring the obstruction to the existence of a *global* harmonic conjugate to \omega_i) the sum \sum \lambda_i \omega_i, for some real coefficients \lambda_i, is the real part of a *multi-valued* function with "well-defined exponential 2\pi i wrapping"; intuitively, think about universal cover of circles and annuli. Note that an easier similar proof also works for the ordinary Riemann mapping theorem.]
S: When are two annuli conformal? [Same ratio of radii...] Prove it.
- We discussed how I was planning to use the reflection principle, but I said "I guess I first need to show that the boundaries map to each other." In preparation I only saw how to do this by a stupid topological argument where you remove a circle to form two path-components to show, say, the inner boundary of one cannot map to both inner and outer boundary of other. (In particular, this could be painful for multiply-connected regions.)
- S said this is a technical issue we shouldn't worry about, and in fact the reflection principle (I guess some careful version of it) would give it for free. (Sorry if I misinterpreted what S said.) After Googling, it seems one could certainly pass from annuli to simply connected regions (for which Ahlfors discusses boundary extension issues in depth), either by cutting a radial slit, or by lifting to a map of universal covers.
- In any case, here are some details not mentioned on previous transcripts: S helped me notice the deep fact that the reflection of an annulus is an annulus (!) instead of my stupid just-do-it argument. Also, I cited Casorati-Weierstrass at the end for removing the point singularity, partially out of laziness, but S noted that the map on the disk is bounded, so we don't need to cite that. (Me in the process: "OK, I guess disk automorphisms are easier than plane automorphisms.")
===
ALGEBRA
Z: Define PID. State the classification of (f.g.) modules over PID.
Ring not a PID, with ideal that's not principal
[where was this asked? was it actually asked?
I thought I would do Z[sqrt5] or Z[sqrt17] at 2 as a fun example, or maybe k[x,y] or Z[x], but genuinely "Z[sqrt p_1p_2p_3]" was first example came to mind...
(S: "do you need 3 primes?")
"no, Z[sqrt p_1p_2]"
normally I'd say the example was in poor taste, but I guess brain was not a great expander graph in my mental state at the time]
Z: Why?
[I wrote down some ramification.]
Z: OK, so you're using Gauss' genus theory.
Z: Do you know what a Dedekind domain is? Define it.
Noetherian brief discussion with S.
Z: How many generators of an ideal are needed?
[Two. Start with any nonzero element x\in I, and then add an element y to lower the valuations v(x) to match the v(I), using CRT mod x: only finitely many primes are relevant.
During the exam I accidentally cited "weak approximation". They didn't object, but I think that refers to rational theory whereas we want to use integral strong approximation / CRT here; in other words, as Z said, I was using the structure theory of torsion modules over a Dedekind domain, a.k.a. CRT here.]
Z followed up with brief culture questions about torsion-free part in decomposition of modules over Dedekind domain.
(Do all of them need to be frac ideals rather than R? No, one is enough.)
(I'd seen most of this in a problem set from Poonen at MIT. Leo also told me about these qn's from Z just before the exam.)
Next, I forget the exact transition, but...
PID interpret as class number 1 in Dedekind domain (ring of ints)
gave examples of Z, Z[sqrt(-1)]
S joke: Can you construct infinitely many examples with class number 1? [No.]
"If you can do that, you get a PhD... and a Fields medal" (or something like that)
Z asked about sqrt(-5) (imaginary quadratic field), specifically compute class group/number.
They probably wanted me to say "Minkowski bound", but instead I just started mumbling something incoherent about its proof, getting confused about ideal representatives (it's an ideal, not an element, but the proof goes by constructing small elements which is why I got confused)
... so we moved on.
Z: what do you know about function fields?
say R[x,y]/(x^2+y^2-1), specifically about Pic
if \otimes C, then it's isomorphic to C[t,1/t] so a UFD, but...
for R, some geometric intuition (for Pic): drew stupid picture of line through real circle
curve intersects circle in even # points
S: let's come back to this later, should do basic things right now
like...
F_q^\times
use cyclotomic polynomials
simpler way using PID classification (over Z)
e^A = B over \C
B must be invertible
if so, solve by Jordan blocks
discussion about logarithm; I'd forgotten that power series literally terminates in our context
I mentioned Jordan form in the previous question, so:
When can A,B matrices over Q be simultaneously conjugated?
rational canonical form
stumbled hard on this, got very confused
k[X] is PID by "long division"
S: "that's the best answer you've given so far"
:(
A,B,C,D simultaneous conjugated thing
S: "there's a subtle algebraic variety [underlying this]... I only learned about this recently"
===
S asked which of the three topics I'd like to be asked a few further qn's in. I asked for clarification ("what topics?") and he listed "real, complex, topology, algebra" (in some order). Of course, I was intrigued by the fourth option. Hence the following surreal, embarrassing but funny, "extra credit" section.
S to Z: "Why don't we have topology? [I don't know.] I guess we came too late to change the system..."
TOPOLOGY
All S, in some order:
- What topology have you taken? [Homology, cohomology, some homotopy but not really...]
S: "OK, we can do homology, cohomology, ..." and proceeds to ask something more... fundamental!
- Compute the fundamental group of R^2 - three points.
- Actually, first define the fundamental group. ["pi_1(X,\ast) = ..."]
- What's another interpretation of pi_1 in terms of universal cover? ["Automorphisms of the cover over the base", i.e. deck transformations]
- Compute pi_1(R^2 - pt) [Deform to circle, so get pi_1(S^1) = Z], pi_1(R^2 - 2 or more given pts)... [I hadn't thought about algebraic topology in a long time... S: "Yes, if it weren't free you probably would've heard about it..."]
- Compute pi_1((closed) genus g surface)... [Mindlessly drew some stupid pictures of torii for g=1 and g=2 with stupid loops and 1-cell structure without thinking, accidentally stated the correct answer Z^2 with incorrect reasoning, forgot the answer and reasoning for general g as well as the S^1 x S^1 interpretation for g=1; S: "There's a relation..." (oops!). I was in a totally different brain zone from summer 2017 project in Grenoble about graph embedding algorithms (a la Hanani-Tutte theorem).]
- Something about multi-valued functions (possibly continuing from complex analysis or the pi_1(R^2 - pt) example or the universal cover interpretation of pi_1, don't remember) like log(z), sqrt(z), "how Riemann would've interpreted it", why anybody cares about universal cover [I thoughtlessly said "simply connected" before mentioning "uniformization [mumble universal property]" by which I really meant lifting property. I should've just given an example like little Picard...]
At times S asked if I had "thought much about [such and such] before" and Z joked that we were doing group theory.
===
We then took a 5-10 minute break. S half-suggested that I run up and down the stairs to fill my brain with oxygen, but I lazily took the elevator down to the Common Room with them and ate some fruit. I'm not sure how much time had elapsed by this point, but somehow it seemed that tea had already ended.
S, while walking and discussing the German (?) prelim system and real analysis books with B: "I know Royden, I've taught out of Royden's book. Who is Fitzpatrick?" (I'd mentioned that I'd read some of Royden-Fitzpatrick) and remarks "Rudin is too slick" (as to why he prefers Royden).
Z talks a little to Daniel Kriz about how Z won't be here next week (which also happens to be one reason why my exam was relatively early).
On the way back, B asks S some questions about PDE (I vaguely remember hearing about decoupling, and some remarks of the form "it's more of an art... decoupling only really helps [for now?] in special cases like sums of powers...").
===
ALGEBRAIC GEOMETRY
(Trigger warning: RR over arbitrary fields. I haven't had time to think through this carefully but from Googling it sounds like everything's supposed to be OK.)
EDIT 12/24/18: see any book on function fields, a good modern one being Stichtenoth's "Algebraic Function Fields and Codes"; this is the reference I used while learning, the summer after the exam, about elementary "polynomial / RR space method" proofs of RH for curves.
(Reflection: this section ended up being a sequence of questions about curves from genus 0 to 3, with genus 0 and 1 juiced up a bit at the beginning.)
Z: Let's go back to the R[x,y]/(x^2+y^2-1) thing earlier (specifically, about Picard group). Change R to F_p. [S, after leaving and re-entering the room: "OK, what now? Oh, I see you've changed R to F_p." I think he was secretly disappointed in some ways. Later in the exam he would say "AG is first over C, then over other fields..."]
has rational point, so birational to P^1
in fact, it's P^1_{F_p} - some deg 2 divisor (p\neq 2)
then can compute Pic by excision; get trivial Picard group if -1 is a square in the field (here F_p), else Z/2
(Note 12/24/18: I think I referred to the excised divisor as a "point" during the exam, but the divisor will split into 2 points if -1 is a square.)
They didn't ask for details on the excised divisor (either split, or a prime divisor of P^1), or equivalently, whether the points involved are deg 1 or 2. But IIRC that computation is not hard. (It's related to the denominator m^2+n^2 in the famous Pythagorean triple parameterization.)
(The problem over R, instead of F_p, was on a previous Z exam, and I wrote it up at https://www.expii.com/t/sheaves-algebraic-geometry-10798?type=problem&id=21010
I'd worked out this perspective in preparation after reading about the excision exact sequence in Hartshorne... but it can also be done more directly.)
Z: Change x^2 to x^3.
first, if E = \ol{C}, the excision exact sequence (excising the deg 1 point \infty \in E(F_p))
Z -> Pic(E) -> Pic(C) -> 0
splits via deg: Pic(E) -> Z, identifying Pic(C) with ker(deg) = Pic^0(E).
(at the time I got a little confused, since it is not literally just excision as in the previous problem; there is an extra step here with the deg splitting)
next, a textbook computation of Pic^0(E) (completely independent from previous part)
E(F_p) -> Pic^0(E) is injective by irrationality of E
for surjectivity, I got confused by F_p, but with some prodding, did it with RR (by Riemann's inequality, any divisor "in" Pic^0(E) + \infty is linearly/rationally equivalent to an effective degree 1 divisor of E, a.k.a. some point of E(F_p))
I had forgotten easy inductive proof of surjectivity E(k) \to Pic^0_k [EDIT 12/24/18: for alg. closed fields k], a.k.a. how to prove group law stuff, etc.
[EDIT 12/24/18: what's going on is that there are abstract methods without base change, vs. direct methods with base change as long as one proves that results w/ and w/o base change agree, which I guess is not literally true on level of Pic, but if one is careful working with actual divisors in, say, RR, everything should be OK... see e.g. Silverman AEC Chapter II. Algebraic Curves, Lemma 5.8.1 (Galois descent type stuff)]
S & Z (stepping back a bit):
define genus
(did geometric)
state RR, etc.
no Serre duality (Z wanted an algebraic proof, I think I also didn't give the definition of pairing wanted so we moved on)
S briefly asked what did Riemann prove what did Roch do [Riemann's inequality vs. full theorem]
Z moved on: what do you know about genus 2?
[hyerelliptic. "yes, branched double cover to P^1"]
Z: genus 3?
[if not hyperelliptic then smooth plane quartic
... got me to say canonical embedding.]
Z: why is it a quartic?
["Plucker formulas."
Z: "What?"
So I write down g = (d-1)(d-2)/2, which is apparently called the "degree-genus" formula.
(At some point during or near this question, I may have written down affine generators and relations for the sheaf of differentials; this was the classical way of thinking about genus, which ACGH proves using Riemann-Hurwitz, but there's a better modern answer; see below.)]
Z: conversely, are all canonically embedded?
I say some stupid things...
Z: "that's equivalent, but much harder"
so Z led me through a conormal exact seq computation
[note: we basically proved the https://en.wikipedia.org/wiki/Adjunction_formula
the reason this answer is better than degree-genus is that it gives both directions at once for "genus 3 is smooth plane quartic \iff canonically embedded"]
[note: I'm glad Z pushed me during exam to think about things more abstractly than before; however all of the above should be equivalent to concrete ideas in ACGH (basic Appendix on RR, not actual book) / Tao blog / Roy Smith post/notes:
https://mathoverflow.net/questions/253090/elementary-proof-of-riemann-roch-for-compact-riemann-surfaces
https://mathoverflow.net/a/253187/25123, which links to http://alpha.math.uga.edu/~roy/8320.pdf (apparently this concrete method is due to Brill & Noether)
https://terrytao.wordpress.com/2018/03/28/246c-notes-1-meromorphic-functions-on-riemann-surfaces-and-the-riemann-roch-theorem/]
then did dimension count of moduli space (still genus 3) with Z (using C -> P^2 counting)
I got confused with some notation about Aut P^2 blah stuff, but essentially we're just counting xyz monomials of degree 4 (up to scaling), then modding out by Aut(P^2) = PGL_3, to get [(4+3-1 choose 3-1) - 1] - (3^2 - 1) = 14 - 8 = 6.
Z: genus 4 (probably in Hartshorne, but I only really prepared up to genus 3)
some complete intersection and degree insight didn't quite understand (quartic/quadric and cubic? in P^3??)
also something about canonical embedding (2g-2 = 6)
S: apply RR with 2K_C to get 3g-3 (= dim M_g)
"Riemann knew this!"
Homework: think about this. After exam, read letter from Weil to E. Artin. "Learn from the masters." (Note 12/24/18: I procrastinated on this for a long time, lol. But it's nice, and even written in English rather than French!)
they asked if I knew about Abel-Jacobi, abelian varieties, Jacobian, etc. I didn't know much but mentioned Pic^0 C
S: "yes, that's related" but didn't ask much more about it
S asked if Z wanted to ask a few more questions, say about surfaces, but Z said he was satisfied (phew!).
===
REPRESENTATION THEORY OF COMPACT LIE GROUPS
S asks what I know (S: "representation theory of compact Lie groups... and Lie algebras?"). I say I've read Adams and some of Fulton-Harris; we start with "Adams stuff".
S: Construct a finite-dimensional representation of G (compact Lie group).
[Z: "do you mean faithful?" S: "let's start with any... we'll maybe later do faithful" the point being that the weaker result still captures all the fundamental ideas]
We end up discussing many of the foundational issues, e.g.:
- Haar measure, left- vs. right- vs. bi- invariant (use compactness of G for agreement of left and right; this uses the algebraic observation that \mu(S^{-1}) is right-invariant if \mu(S) is left-invariant, which I wasn't really aware of but also wasn't asked; I also haven't thought through whether/when one needs bi-invariance in Peter-Weyl theory).
- Hilbert-Schmidt operators on L^2(G) (infinite-dimensional as soon as G is infinite). I instinctively write out \int |k(x,y)|^2 < infty as one of the conditions, but S is quick to point out that this is subsumed by other assumptions (including continuity, as G is compact), so I erase that. (Note that k just needs to be left-invariant, not bi-invariant, though I wasn't asked about this.)
[S pointed out that we should really think of k as acting by convolution, given its left G-invariance.]
- Spectral theorem precise statement (not just V_\lambda finite-dimensional for \lambda nonzero, but decomposition into ker + V_\lambda's... I hesitate in completing the sum and Z is quick to correct me to "completed/topological sum").
We discuss proof
"variational argument" is keyword S wanted to hear
it's the one that generalizes from finite-dimensional to infinite-dimensional
Culture test:
- All irreps of G are finite-dimensional. And occur in L^2(G) ("regular representation") via V\otimes V^\ast \to L^2(G).
- Weyl's unitary trick. (S: "... it's much more than [a trick]!") S started asking about GL_n(C) but we switched to SL_n(C) (whose Lie algebra is the complexification of the Lie algebra of SU(n)) to avoid dealing with center (I haven't thought carefully about this). Looking back at Fulton-Harris, I suppose it's important that SU(n) is simply connected for Weyl's trick to work (that way you can exponentiate su(n) Lie algebra representations), but they certainly didn't ask about this (they seemed happy with general awareness).
- Sym^n C^2 for GL_2(C), interpretation via homogeneous 2-variable polynomials of degree n (maybe C^2 should be dual because polynomials are functions, but whatever).
- How do you prove complete reducibility for compact G? I said "averaging" (the main idea behind the proof of Maschke's theorem) and S was happy.
- Does complete reducibility always hold? I instinctively said "if semisimple?" (I guess I had Wedderburn theory in mind) but S quickly improved that to "reductive" (and not all groups are reductive).
S: Define maximal tori. [Embedding T = (R/Z)^r \to G with r maximal, called the rank of G... a possibly better abstract definition might've been compact connected abelian subgroup, together with the classification of such groups as tori.]
- What can you say about them? [Conjugate.]
- Can you outline a proof? [I asked if I could "cheat with Lie algebras".]
- S said OK, but also had an interesting comment on the role of topology in these results on tori. (Note: the topology is explicit in the Lefschetz and mapping degree proofs, and in the surjectivity of exp---see comment below on "cheating". And in the Lie algebras approach, I suppose the topology lies in the distinguished role of simply connected forms. Anyways, S noted how people have kept trying to improve the proofs in different ways.)
- [So I sketched a variational Lie algebra argument: take an Ad-invariant inner product B on \frak{g}. If \xi_1,\xi_2 are dense Lie algebra generators of T_1,T_2, then by compactness of G, the real value B(Ad_g \xi_1,\xi_2) is maximized at some g\in G, say g=1. Then by considering paths g = exp(t\eta) and differentiating in t, end up getting that \xi_1,\xi_2 commute, so by maximality T_1 = T_2. (This uses the fact that compact connected abelian groups are tori, but S was happy without inquiring about this or any other details.)]
- Note: In retrospect, this really was cheating. This proof doesn't show exp surjective, so we don't immediately get that G is covered by tori (which I suppose would be important for proving K(G) \to K(T)^W is injective, unless there's some way around it). A proof of exp surjective might need some differential geometry, perhaps using the fact that the Riemannian exponential map (w.r.t. a bi-invariant metric) agrees with the Lie-theoretic exponential map, see e.g. https://terrytao.wordpress.com/2011/06/25/two-small-facts-about-lie-groups/
Other comments from S after the exam:
- Conjugacy of maximal tori has to do with some simultaneous internal diagonazability... and if we're talking about a dense generator, can think about diagonalizing a single matrix in another basis or something. "It's a nontrivial fact."
- "Weyl didn't have a field, he just went and did this [great] thing."
Re: tori, at some point S stopped to ask about dense generators. (S to B: "we are constructing a great symphony", in reference to B's earlier question about Weyl equidistribution.)
- S asked me about the characterization (linearly independent over Q), and then brought up Kronecker's density theorem, which I had forgotten about (it certainly predates Weyl's equidistribution theorem).
- S also commented about density vs. equidistribution results, how the latter is often easier to prove or at least "usually one tries for that". (He mentioned that "Z is famous for an equidistribution theorem [Bogomolov]".)
S: Define characters, roots, integral lattice & weights ("this is something slightly annoying with Lie group approach compared to Lie algebra"), etc. "... this is what I really wanted you to know..." (maybe he phrased it as "define everything needed for WCF [Weyl character formula]" or something along those lines, I don't remember).
- I was writing L(G), L(T) for Lie algebras and S said "you can use the modern notation" to which I replied "OK... but my handwriting is bad".
- "OK I don't remember everything I need for WCF, so I'm going to work backwards." (Remembered to define highest weight at this point.)
- At some point S reminded me to define the Weyl group N(T)/T (Lie group definition), wasn't asked to explain why this is finite. (The reason is N(T)_1 is a (connected) subgroup containing and commuting element-wise with T, so N(T)_1 = T by maximality of T.)
- At some point: stopped while about to define alternating sum "A(-)" in WCF.
S: OK let's do an example, say SU(3).
- Have you worked out many examples? ["I've worked through at least SU(3)!"]
- Maximal torus, roots, weights, reminded to explicitly compute the Lie algebra.
- For WCF, computed FDWC (fundamental dual Weyl chamber) and \rho (half the sum of positive roots).
- Before computing anything with WCF, S stopped me and said he was "losing steam" at that point, and (in any case convinced I knew the stuff) wanted to end the exam. I think I was also going really slowly at that point; I'm not sure.
Practical tip: If you have to work out scarier examples than me, one way to keep things straight (when figuring out roots and weights) is the fact that weights mod roots (i.e. weight lattice mod lattice generated by roots) has index #Z(G) (center), e.g. \#Z(G) = n for G = SU(n). Thanks to Leo (preparing rep theory a few days before the exam) and Fulton-Harris for helping me realize this. Once you trace through definitions, the underlying reason is that Z(G) = \bigcap U_r where U_r = {t = e(x)\in T: \theta_r(t) = \theta_r(x)\in Z} (using t or x depending on preference for Lie groups or algebras).
===
S, after stopping me with SU(3): "... do you have any questions for us?" ["No." Wasn't expecting that. But in retrospect, there are some loose ends I could've asked about during the exam instead of tying them up on my own afterwards.]
They then kicked me out, and a minute or two later S told me I'd passed. The committee seemed reasonably happy, though I personally felt rather thoughtless throughout the exam. I was personally unsatisfied with my AG (and still am, I guess), but S and Z didn't seem to mind. Z preferred sticking to more algebraic things (S-style would have more complex, historical, and philosophical things, e.g. giving N equivalent definitions of genus, Riemann's approach to RR... which were cut off in my exam).
Z & B left, and S then talked to me a bit longer (until 5p or so), helping to orient my plans for the next 5 months or so... but enough stories for now. I have a high-stakes Princeton reenrollment form to fill.