Trajan Hammonds' Generals
Committee : Peter Sarnak , Chris Skinner , and Aleksandr Logunov
Special topics : analytic number theory and representation theory of compact Lie groups
May 11, 2021, 12pm-3:15pm via zoom
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Comments : The following write-up is just what I remember, but there were definitely
way more hints in the actual exam. Also the answers I give here may not be 100% precise.
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REAL ANALYSIS :
Sarnak : Ok, Aleksandr why don't you start
Logunov : Does the sum of 1/nlogn diverge or converge?
- I mistakenly said it converges because I flipped some inequality. After receiving
some strange looks I remembered to just do the integral test and showed it grows like loglog(n)
Sarnak : Ok write down sum of n^{-s}, do you know what this function is? (-___-)
- Yeah, it's the zeta function
Sarnak : What's the derivative
- I wrote down \zeta'(s) = -\sum log(n)/n^s
Sarnak : Now consider \sum a(n)/n^s,call it F(s). Do you know a complex analytic way to
express \sum_{n \leq x} a(n)?
- I wrote down the Perron formula and I remember to include the ' on the summation
since the last term should be multiplied by 1/2 if it's an integer, but he told me to
just get rid of this.
Logunov : Does \sum_{p \leq x} 1/p diverge or converge?
- It grows like log log x. You can show it via partial summation of Von Mangoldt.
(finally i started to put things together)
Logunov : Can you define a measurable set and give an example of a nonmeasurable set.
- I gave the Caratheodory criterion and gave a proof that a set of positive measure
contains a non-measurable set by doing the equivalence relation where x ~ y iff x-y in Q
and then deriving a contradiction
Logunov : But how do you know A has positive measure? Can you give an example of a
typical measurable set
- At this point I realized (or maybe Skinner pointed to it) he wanted me to define Borel sets
and the Lebesgue measure which I did
Logunov : is there a relationship between Borel sets and Lebesgue measure
- I said that Lebesgue is the completion
Logunov : what does that mean
- subsets of a set w/ measure zero should also be in the sigma algebra
Sarnak : Can you define the Fourier transform?
- I defined it on L^1
Sarnak : What properties does it have
- Continuous and goes to zero. Started to do the proof that it's continuous and got to "dominated convergence theorem"
before he cut me off and told me to just prove Riemann Lesbegue, which I did by approximating with step functions.
Sarnak: Can f and \hat{f} both have compact support.
- I started naively writing things down because I completely forgot to review this, until Sarnak suggested I define the Fourier
transform over the complex numbers and then we smoothly transitioned to complex.
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COMPLEX ANALYSIS:
We continued with the fourier transform problem. Sarnak asked me about convergence and then he asked me
to define the order of an entire function and then he walked me through the bound.
Skinner : What's the Riemann mapping theorem?
- I stated it, forgetting to say that \Omega is simply connected
Skinner : Can you do this for an annulus
- Realized my mistake and corrected to \Omega being simply connected.
Skinner : Ok prove it
- I went through the proof. At some point I realized I was just writing down the proof from memory and wasn't explaining anything
which I mentioned out loud and Sarnak suggested I talk through it. So I explained some of the intuition as I was writing it out.
(at this point Skinner or Sarnak made some remark about whether this was the proof in Ahlfors or Stein-Shakarchi and I told them
it's from my class in undergrad, but now that I check it's identical to the proof in S-S).
- At some point in the proof Sarnak asked me to prove Montel's theorem which I did.
Sarnak : Draw an annulus. Now draw a *blobbier* annulus. Can you map these two biholomorphically?
- I started talking about the Dirichlet problem and going through the explanation in Ahlfors but I pointed out the parts that
I didn't understand perfectly and Sarnak explained some of it
(at this point Sarnak started talking about stronger proofs where you can map multiply connected
domains to circles not just slits)
Sarnak : Ok let's step back a bit then. Do you know how Riemann tried to prove the mapping theorem
- I talked about how you start with a mapping and suppose F(z_0) = 0, then you can write F(z) = (z-z_0)e^g(z)
and showed it amounted to solving Dirichlet problem with boundary value -\log|z-z_0|. I then said you can construct
the biholomorphism from this harmonic function
Sarnak : What do you need to do that?
- That your domain is simply connected.
Logunov : If you have a planar graph you can treat each point as a disk, and if there is an edge between two points then you make
those discs tangent. Draw a triangle with a vertex in the middle and connect all the vertices. Then this maps to three circles which are
mutually tangent with a fourth tiny circle in the middle tangent to all three.
- (At this point I wasn't sure what the question was but Sarnak interrupted)
Sarnak : Ah, this is my favorite problem!
- (I think the problem is to show that there is a unique circle tangent to the three bigger ones)
Sarnak : What are the automorphisms of the disk?
- I wrote down Blaschke factors. (At some point before this there might have been some discussion about whether the maps in Riemann mapping
and I said only if you fix a point to send to zero). There was some discussion about the automorphism of the upper half plane and what group
it is. Anyways the problem reduced to mapping two of the disks to straight lines with a circle in the middle tangent to the two lines, and then
finding the circle tangent to all three by drawing a circle tangent to the two lines and the circle in between. (I'm forgetting a lot of details)
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ALGEBRA
Skinner : If G is a finite group when is Aut(G) cyclic?
- He basically walked me through the whole proof of this which touched on a number of ideas
Skinner : Show if G/Z(G) is cyclic then G is abelian
-Figure out the proof after about 2 minutes of struggling before I realized to just write down a generator and take the commutator of two elements.
Skinner : Can you classify groups of order 132?
- I started talking about Sylow's theorems and Sarnak remarked about how my pronounciation was wrong. Anyways eventually I figured out that there
was a normal subgroup and eventually they stopped me cuz apparently Skinner just wanted to know whether such a group is simple
Sarnak : When can you solve e^A = B for complex matrices A and B?
- Reduced it to jordan normal form, wrote down ttaylor expansion of logarithm and showed that the matrix with lambdas above the diagonal is nilpotent
to conclude that the taylor expansion converges
Skinner : How do you get Jordan canonical form?
- Talked about fundamental theorem of finitely generated modules over a PID. Typical questions about what's the module, what's the PID. eventually
Sarnak interrupted to ask
Sarnak : Why is C[x] a PID?
- I said cuz C is a field
Sarnak : Ok but why is it, *really*
- You can do a division algorithm
Sarnak : Great!
- I continued showing how to get Jordan normal form and they cut me off when I started writing the basis and said we should take a *5*
minute break
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We ended first half at 1:35pm
Halftime Comments : At this point I was pretty flustered because I thought the first half did not go very well. I spent so much time reviewing involved proofs
and big theorems that I forgot to review "simpler" stuff and was slow to answer some basics. There were a lot of things I knew that weren't
asked about at all like galois theory, rep theory of finite groups, functional analysis, and close to all of the non-conformal-mapping part of complex
analysis.
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ANALYTIC NUMBER THEORY :
Sarnak : So I assume you read both Davenports?
- Yes, that's right
Sarnak : Ok so perhaps I'll ask you about the classical stuff and Chris will ask about circle method since that was his original area of expertise
- Sounds good
Sarnak : State Dirichlet's theorem
- I joked that he has many theorems but began writing the asymptotic for the number of primes a mod q . Sarnak pointed out that this isn't
quite what Dirichlet proved so then I wrote down that \sum_{p \equiv a mod q} 1/p^s diverges as s-> 1
Sarnak : Good, and what does this boil down to?
- Showing L(1,\chi) \neq 0
Sarnak : OK but before we go there, what's \chi?
- I defined the Dirichlet character and he started asking about the structure of (Z/nZ)* and the dual group of a group G and whether that was isomorphic
to G itself. Eventually I realized he just wanted me to write the sum above as a sum over characters in the dual group, i.e. \sum_{p \equiv a mod q} 1/p^s
= 1/|G^|\sum_{\chi \in G^} \chi(a)\log L(s,\bar{\chi}) + O(1)
- I went on to explain how using this you could argue that L(1,\chi) couldn't possibly vanish for a complex character, since the LHS of the above equation
is nonnegative but the RHS would tend to -\infty as s->1 since L(1,\chi) and L(1,\bar{\chi}) would both vanish which will more than undo the pole of \zeta
(given by the principal character). So L(1,\chi) could only possibly vanish for real chi. (Sarnak made some remark about how Dirichlet invented this character
theory in order to prove the theorem for primes in APs)
Sarnak : Ok, so what about real characters?
-I stated the Dirichlet class number formula. I couldn't remember where to put the \pi and then Skinner asked me what \epsilon was. I said the fundamental unit
and talked a bit about Pell's equation and units in quadratic number fields. Sarnak asked me if I knew Dirichlet's unit theorem. I said I was familiar
with it and just said a couple words about it. Mainly he wanted to know whether I knew about the behavior of
units in more general number fields.
Sarnak : So this shows L(1,\chi_d) >> 1/\sqrt{d}. Can you give a better bound?
- I wrote down Siegel's theorem L(1,\chi_d) >> d^{-\epsilon} and remarked that the constant was ineffective. A conversation ensued about why it is ineffective.
Sarnak : Do you know any other theorems that are ineffective?
- I mentioned Siegel-Walfisz and Sarnak started talking about Roth's theorem
Sarnak : Ok before we go on, what can you do for complex characters? Why is the bound different for real characters?
- I talked about how you can do the same cosine trick (3-4-1 inequality), but then when you consider L'/L(\sigma + 2it, \chi^2) you have a \chi^2 character,
which is the principal character when \chi is real. So then if the imaginary part is small, you are getting too close to the pole of \zeta(s). Anyways, for
the complex characters you'll get a zero free region of shape 1-1/log(q(|t|+2)), assuming t is not too small.
Sarnak : Ok so what's the general approach for Siegel's theorem?
-I wrote down F(s) = \zeta(s)L(s,\chi_1)L(s,\chi_2)L(s,\chi_1\chi_2) and you then case on whether L(s,\chi_1)
has an exceptional zero \beta, and from that deduce a lower bound for L(s,\chi_2) using L(1,\chi_1) << log q_1, L(1,\chi_1\chi_2) << log q_1q_2
Sarnak : Ok so go through the proof and we can point out exactly where it becomes ineffective
- I wrote the same F(s) as above and noted that it is the Dedekind zeta function of a biquadratic number field. Then because the coefficients are
nonnegative we can consider the smoothed integral \int_{2-i\infty}^{2+i\infty} F(s+\beta)x^s/(s(s+1)(s+2)(s+3)(s+4)) ds...-
Sarnak : Wow! Where did you learn this?
- I said it was Goldfeld's proof and Skinner and Sarnak both said it's their favorite proof and the one they like to use in class. Anyways, they stopped
me there after I explained the ineffectivity.
Sarnak : Ok Chris why don't you ask him about some circle method stuff
Skinner : OK let's talk about Vinogradov's theorem. Tell me about it
- I talked about how to count solutions using orthogonality and splitting integral into major and minor arcs. I mentioned the singular series and we talked
a bit about it and I mentioned wanting to look at obstructions to solutions p-adically. Then talked about the singular integral. Then had a discussion about
the minor arcs and bounding with a combination of sup norm and L^2 norm via Parseval
Sarnak : How do you bound the sup
- I talked about using Vaughan's identity and then a discussion about power savings ensued and Sarnak asked me to consider the following sums
\sum_{n \leq x} \Lambda(n)e(n\sqrt{2}), \sum_{n \leq x} \Lambda(n)\chi(n)e(n\sqrt{2}), \sum_{n \leq x} \mu(n)e(n\sqrt{2})\chi(n), the last one was
"just for fun" as he put it. We discussed philosophy of square root cancellation
Sarnak : Ok great, so what about just \sum_{n \leq X} e(n\alpha), how do you bound that?
- I said you can use Weyl's inequality
Sarnak : Ok, but what about something simpler
- I then realized it's a geometric series and I wrote << min{X, \norm{\alpha}^{-1}} and Sarnak remarked how it was good I
remembered the ^{-1} for the norm
Skinner : Ok, so how about counting number of solutions in primes to the equation x_1 - x_2 = 2
- I wrote out the same deal but this time Sarnak interjected asking that I write S_X(\alpha) for the exponential sum instead of S(\alpha) because the
lack of the X dependence made him "uncomfortable". This time the expected main term is around size N. But if you try to bound on the minor arcs
using the same methods you get something that's larger than the main term. That's if you try to bound it just by the L^2 norm. If you try to pull out
a sup and do L^1 norm, even if you assumed L^1 norm was around \sqrt{N} on average, you would need better than square root cancellation for the sup which
is impossible.
Sarnak : Can you show the L^1 norm is at least lower bounded by X^{1/2 - \epsilon}
- I think you would need to do the circle method to show this.?
Sarnak : No no, there is an elementary way. Ok write down S_X(\alpha) = \sum_{n \leq X} e(n^2\alpha). What's the L^2 norm?
- (at this point I realized he wasn't talking about \sum_{n \leq X} \Lambda(n)e(n\alpha) and getting the L^1 norm of the function above
is indeed easier.) I said \sqrt{X}
Sarnak : OK what about the L^4 norm?
- I wrote down that we want to count solutions to {u_1^2 - u_2^2 = v_1^2 - v_2^2} and we can use the divisor bound
Sarnak : Ok so how might I get a lower bound for the L^1 norm
- I wrote down Holder's inequality by writing |S_X(\alpha)|^2 = |S_X(\alpha)|^{4/3}|S_X(\alpha)|^{2/3} with Holder conjugates
p = 3, q = 3/2.
Sarnak : WOW you're a genius!
- (actually no but I must've practiced this problem at least half a dozen times to get the exponents right)
Skinner : Ok how might I count solutions to a_1x_1^k + \dots + a_sx_s^k = b
- Same set up for the third time, integrating S_X(\alpha)^s e(-b\alpha)\,d\alpha where this time S_X(\alpha) = \sum_{x \leq X} e(\alpha x^k). At
some point I asked if I can make all the a_i = 1 since it was getting annoying to write. I talked about the major arcs, minor arcs, shape of main term.
Skinner: How large should s be?
- I said we can take s \geq 2^k+1 and then talked about Hua's lemma and how to prove it.
Skinner : How do you estimate sup |S_X(\alpha)| on the minor arcs?
- I talked about Weyl's inequality when |\alpha - a/q| < 1/q^2. The shape X^{1+\epsilon}(1/q + 1/X + q/X^k)^{2^{1-k}} and since you're on the minor arcs
q > X^\delta
- I don't remember how we transitioned but basically Skinner asked me about Weyl differencing and I explained it and remarked how I could never
remember how many factors of (2X) you get because it "disappears" in the base case (as you prove Weyl differencing via induction)
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REPRESENTATION THEORY OF COMPACT LIE GROUPS :
(For this section, it was mostly a conversation. In the sense that for half of it I wasn't even writing anything down, so the record will be less detailed
as I don't remember what exactly was said. Though I guess this is what people had to do before they had saved zoom whiteboard notes to reference...)
Sarnak : What did you read?
- Adams and a couple chapters of Fulton-Harris for the representations
Sarnak : Ok so let's say G is a compact topological group. What measure can I put on it and what properties does it have
- Talked about Haar measure, left and right invariance
Sarnak : So how can I get a nontrivial finite dimensional rep of G
- I said to look at L^2(G) and consider a convolution operator T_\phi f = \int_G \phi(g^{-1}h)f(h)dh. Then show it's compact and self-adjoint and then you
can use the spectral theorem to get finite dimensional eigenspaces. And these will give you the reps. (Throughout I was asked why the operator is compact
and I said it suffices to show it's compact in L^\infty(G) and from there you can use Arzela Ascoli.
Sarnak : Why are these representations
- Because they are G-invariant subspaces. (Also had some remarks about whether there is an infinite dimensional zero eigenspace
Skinner : Can you define the exponential map
- I wrote down one parameter subgroup and then the differential equation and wrote exp(x) = \varphi_x(1)
Skinner : Where is x?
- It's in the Lie algebra and i wrote x \in gl_n, but they reminded me that this is just some arbitrary Lie group G. Then they asked me what
the Lie algebra is in terms of G and I said it's the tangent space at the identity, and so then I realized they wanted me to say x is a tangent vector.
Sarnak : Ok, I want to get to the Weyl character formula because that's the heart of all this. Let's do an example, have you studied SU(3)
- Yeah a bit
Sarnak : Ok maybe let's start with SU(2)
- No no, SU(3) is fine (I had practiced SU(3) half a dozen times on the chalkboards in Fine so I really wanted to do SU(3))
Sarnak : OK write out the Lie algebra explicitly
- Wrote out the Lie algebra. Then I wrote down the torus
Sarnak : Is this just any torus, is it unique?
- I said that it's the maximal torus and that they're all conjugate. (By this point I had transitioned to auto-pilot)
- I wrote out the weights, roots, Weyl group, brief explanation why its finite, calculated it, calculated Weyl element (and Sarnak
remarked that he had never heard that name for the half sum of the positive roots). Then I talked about weights and drew a very
bad hexagon diagram. Before I could get to the representations Sarnak moved on
Sarnak : Ok can you define a character?
- I just wrote that its the trace of \rho(g)
Sarnak : Ok and describe the character formula
- I said that it allows you to express the character of an irreducible representation in terms of its highest weight via a fraction of
these alternating sums of exponentials (which he said are basically trigonometric polynomials, which might be more clear to see from the
SU(2) example oops)
Sarnak : Why do we only care about the characters restriction to tori
- I again mentioned because every element of G is conjugate to an element of the torus and characters are constant on conjugacy classes
Sarnak : Ok what about SL_n(C), how about finite dimensional reps in here?
- I talked about the unitary trick.
I'm hazy on the details but some more discussion followed about L^2(SL_n(C)) and how I need to learn this later and maybe some more stuff about SL_n(C)
Sarnak : Ok how about SL_2(C) does it have any finite dim unitary reps?
- First he made me define unitary. And then I gave the argument by noting that (1 x; 0 1 ) would map to a unitary matrix which would be diagonalizable. so
the only hope is to map to the identity
Some other things that came up in this section but I don't remember where :
Why all reps occur in L^2(G)
Why you can assume irreps on compact G are unitary
A bit about the proof of maximal tori (and some comment about the whole reason Adams wrote the book on Lie groups was to give the topological proof)
Surprisingly nothing about the integration formula
Some discussion about the dimension formula and verifying whether representations are irreducible
The Peter-Weyl theorem (at some point Sarnak talked about how Weyl's student Fritz Peter went into finance after proving this. I made some joke about
not realizing Peter was someone's last name and Sarnak laughed)
Skinner asked me about the exceptional group G_2 at the end since I had a recent paper about it with one of his former students and
Sarnak joked that the experts probably consider G_2 a classical group by now.
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they kicked me out for 5 minutes. even though i did not like how the first half went I was pretty happy with the second half so I felt not too nervous.
I returned to the zoom room and they told me I passed!
Final Comments:
- In my experience the best resource to use is people! I learned a lot from talking to people. Talking to people is great because you get
different perspectives and ideas and sometimes you just can't really get a hold of something on your own. But here's a list of resources I used
- Complex : Stein Shakarchi and Ahlfors
- Real and Algebra : mainly just tried to go over material from undergrad courses. I think my favorite book for real though is either
Stein Shakarchi or Folland, and Reed-Simon is great for functional. Dummit and Foote has everything for algebra.
- Analytic Number Theory : the two Davenports, the new book by Koukoulopoulos, Steven J Miller's notes on circle method, Wooley's notes
on arithmetic harmonic analysis, Joni Teravainen's blog, Noam Elkies 229x notes
- Lie Groups : I started with a couple read throughs of Adams and then tried Bump, both Knapp's and Fulton-Harris but eventually I settled
for Wikipedia which actually has some nice articles about rep theory of Lie groups.
- And of course doing past generals problems is good practice. I tried to at least read all of the analytic nt/rep theory sections on the website.
- You will also learn something from the exam itself as there are lots of historical/mathematical comments and insights, many of which I have not
written here.
- By the end of it you will realize you learned quite a bit of math and will be happy about it.