Hung-Hsun (Hans) Yu's Qualifying Exam

Committee: Noga Alon (chair), Zeev Dvir, Peter Sarnak 

Special Topics: Probabilistic method, Discrete Geometry

2023/4/28 Friday in Noga's office, 14:00 -- 17:00

I arrived at Noga's office a bit earlier. We started to talk about how the office was freezingly cold, and Noga also said that he tried to request that they turn on the heater again but it did not work.
Zeev then showed up, and we were told that he had some allergy, which gave him a bit of hard time to speak up during the general.
Noga called Sarnak as he was a bit late, and we were told that he would arrive 10 minutes later.
Therefore we decided to start without him.

Usual process: Noga asked me to choose which of the general topics to start with. My plan was to start with Algebra so we can talk about Algebra forever, but I was then a bit hesitant as maybe I could just start with real or complex analysis so that I did not have to prove Riemann mapping theorem. When I was hesitating, Noga said that maybe we should start with Algebra as Sarnak would be the main one asking questions in the analysis part. 
I agreed so we started with Algebra.
I should comment before diving into the transcript that Algebra never really ends---it shows up in almost all the other topics for some reason.
I guess I should have been careful with what I wish for.

Algebra section: (asked by all of them)

[A]: Consider the finite field. What can you say about the multiplicative groups?

Me: Counting argument and brief remark that the fundamental theorem of finite abelian group is really not needed here. (That tends to be Sarnak's preferred proof, but he has not arrived yet :p )

[A]: What is the number of primitive roots?

Me: $\varphi(q-1)$.

[A]: Now say $q$ is a prime $p$.
Can you say anything about the smallest primitive roots?

I admitted that I have no idea, but I am fairly sure that there is some result like that (I recall reading something like that for some algorithm to work). 
I also said I could say something really dumb like $p-\phi(p-1)$. Noga said that he also had no idea, so he asked a simpler question.

[A]: The primitive element is necessarily a quadratic nonresidue (remark: when $p\neq 2$).
What about the smallest quadratic nonresidue?

After some silence, Noga decided to ask if it is possible to have all elements in $[\lfloor \sqrt{p}\rfloor]$ being a quadratic residue.

Me: Well then the subgroup generated by those guys all have to be quadratic residue. That sounds impossible. In particular, any number with its largest prime being at most $\sqrt{p}$ would have to be a quadratic residue.
Can I use prime number theorem?

Noga said that I could use prime number theorem, but I messed up really badly and somehow said $\sum p/q$ where $q$ runs over the prime up to $X$ is $p/\log\log X$.
This messes up the entire calculation but Noga told me that this way I can show that a constant portion of $[p-1]$ are in the subgroup generated by those guys.
He was happy about this and remarked that if $[\lfloor \sqrt{p}\rfloor]$ only consists of quadratic residue, then we have found a really large independent set in the Paley graph.
This will be revisited later.

[A]: What can you say about the ideals in the ring of polynomials over the integers?

Me: Its finitely generated by the Hilbert basis theorem.

Around this time, Sarnak arrived and Noga told him where we were at.
Meanwhile I set up the notation and sketched the proof of Hilbert basis theorem as Noga wanted to see it.

Me: By considering the ideal of leading coefficients, we can generate all polynomials of sufficiently large degree (this is wrong: see below).

[A]: Why?

Me: (Show the proof and realize that I made a wrong claim: I could show that this reduces the problem to generating polynomials of degree bounded by some number).

[A]: What is a solvable group.

Me: (Define it)

[A]: Now show $S_4$ is solvable.

Me: $S_4\supseteq A_4\supseteq K_4\supseteq \{e\}$.

[S]: What is $A_4/K_4$.

Me: (Almost said $S_3$ because $S_4/K_4\cong S_3$ appears way too often) $C_3$.

[A]: What about $S_5$? Is it solvable?

Me: No because $A_5$ is simple.

[A]: Do you know the proof of that, or $A_n$ in general?

Me: Yes and I also know that it is a bit tedious. (Remark: if they just ask for $A_5$ then it is easier: simply count the size of each conjugacy class and realize that there is no possible way to add them up to a nontrivial divisor of $60$.)

Noga was fine with that comment and Sarnak took over.

[S]: What about a simple group that is not $A_n$?

I guess I could have said $\ZZ/p$ but I thought Sarnak wanted $PSL(2,7)$. I made a comment that it has order $168$ according to wiki and Sarnak got amused and at the same time disliked that comment.

[S]: Don't learn from the net.

This I sort of disagree. The net is really useful for miscellaneous stuff. I learned about Hurwitz group after reading Riemann--Hurwitz from Vakil in one minute because Google is powerful. This is also something too miscellaneous that I think is too distracting to cover in a book. Anyways.

[S]: What do you use to study?

Me: Well I took Algebra in undergrad and we used Michael Artin's book.

Sarnak said that he would ask from that book then (which he did not).

[S]: What do you know about representation theory?

Me: Maschke's theorem?

[S]: State and proof.

Me: (State it over $\mathbb{C}$ and got interrupted to define all possible things you could have asked for definition in the statement)

Then I started the usual proof until the place where we take average to get a unitary representation. 
Sarnak then told me to stop and ask me if we need it to be algebraically closed.
I said probably because we need that eigenvector always exists (this is not true: somehow I thought ahead too much and was referring to Schur's lemma where you needed the field to be algebraically closed).
Sarnak commented that I was thinking ahead and said that the only thing we needed was that the characteristic of the field does not divide the order of the group.
I agreed and we moved on.

[S]: Can you show me an irreducible representation?

At that point it was unclear if he wanted a representation that works for all groups, so I said the trivial representation.

[S]: What about one that was not trivial?
Say the group is not trivial.

I asked if he wanted an explicit construction, and he said that he just wanted to show existence. 

Me: There should be another one as there are at least two conjugacy classes, which shows that there are two non-isomorphic irreducible representations.

[S]: Whoa you are using a lot there.

At this point it was a mix of Sarnak trying to ask me what I am trying to quote and me realizing that I could have just considered the regular representation and used Maschke's theorem.
The conversation became really divergent for a while, and eventually Sarnak asked me to define the regular representation.

Me: (define it)

[A]: Where does the dimension of the irreducible representations show up in the regular representation?

Me: (Write down $\chi_{\textup{reg}} = \sum_id_i\chi_i$.)

[A]: Can you give an upper bound on the dimension of an irreducible representation?

Me: From the formula we can get, either by directly looking at the direct sum of the representations or plugging in $1$ for the character, that $|G|=\sum_id_i^2$.
Therefore we have a trivial upper bound $\sqrt{|G|}$.

[S]: What can you say about the group if there are $|G|$ different irreducible representations?

Sarnak had been distracted by his phone up to this point for some reason, and he had to leave the room for a bit.
Noga took over and I answered it should be abelian by the consideration of number of conjugacy classes.
Sarnak came back and promised that he would now focus more.

[S]: Here is my favorite question (I know what is coming next).
Define the exponential of a matrix (I had the right guess).

Me: (define using Taylor series)

[S]: When can you solve $e^A=B$, given $B$?

I tried to argue that $B$ is invertible by looking at its Jordan form.
Sarnak said that there was a simpler way: what is its inverse?
I said $e^{-A}$ and he was happy (I guess I got too lazy and just `Jordan-form'-ed everything).

[S]: So you show that $B$ has to be invertible.
Now you need to show that if $B$ is invertible then you can find $A$.

Me: (Do this by zooming into a Jordan block and consider its log)

[S]: Why is it well-defined?

Me: This is a finite sum.

Then Sarnak asked me to prove Jordan canonical form. I forgot the exact wording but he asked me to not prove it by just messing around with the matrix. I got amused by this request and asked what it meant. He then told an interesting story of two great mathematicians (sorry I forgot whom) where one just told the other, why not mess around with the matrix first to simplify the proof?

Me: (Do the proof using generalized eigenspace)

[S]: Ah you are using the proof in Artin. Prove it in another way... do you know rational canonical form?

I generally do not like to overkill with powerful machineries while Sarnak definitely does not mind. I did this intentionally to see what comment he would give. I guess our tastes in math really differ and this shows up later too.

Me: Yes.

[S]: What is really the algebraic content behind the rational canonical form?

Me: Structural theorem of modules over PID (I almost said Smith's normal form but I decided to save some time and said the one canonical answer that Sarnak expected).

[S]: How do you apply it?

I started to define the $k[X]$-module and got immediately stopped as Sarnak thought I knew the entire thing.

Definitely more than 40 minutes have passed, and we are still deep in Algebra. Sarnak thought it would be a good idea for Zeev to ask some questions too. Zeev's questions were a bit unexpected, but he asked me those with the impression that I had to use this on my research.

[D]: Define the dimension and degree for a variety.

I am too used to thinking about it algebraically so my first instinct was to define it using the transcendental degree of the function field.
Zeev asked me to define it geometrically.

Me: Ah, its the maximum length of chain of irreducible closed, plus or minus $1$ (it's minus).

Zeev was satisfied so I moved on to define the degree of a variety using Hilbert's polynomial (which I guess can also be used as a definition of dimension).

[D]: What is Hilbert's polynomial?

I asked for clarification if Zeev wants to see an affine description or a projective description. Zeev said it should be something about homogeneous polynomials (so the projective description) so I did it. Then the same question: do it geometrically.

Me: Geometrically...??? Ah, take generic hyperplanes, consider their intersections with the variety and count the points with multiplicities (basically B\'ezout's).

At this point Sarnak got really excited (oh no) and thought I knew algebraic geometry (this is a huge misunderstanding). He asked me if I know Rieman--Hurwitz and I deliberately said no just to keep the general on track.

I thought the algebra part should have been over, but Sarnak carried on and asked me to consider $SO(3)$ and said a bit about its irreducible representation. I had zero clue and the only clever (though tautological) thing I could say is that the tautological representation seems irreducible to me. I also admitted that I knew nothing about representations for infinite groups. Sarnak said that we would come back to this in analysis... not a good sign...

Okay, so we did talk about algebra forever, and maybe this was not as enjoyable as I thought.

The three agreed that it was enough algebra, and Noga asked me which of the general topics to cover next. I said complex because I really wanted to shove real analysis away.

Complex analysis section: (mostly by Sarnak)

[S]: What do you study from for complext analysis?

Me: Ugh... friends? It's good to learn from friends :)

[A]: State Cauchy integral formula.

It became surprisingly chaotic. I wrote down
\[f(z_0)=\frac{1}{2\pi i}\int_{\gamma}\frac{f(z)}{z-z_0}\d z\]
(almost forgetting to put the $2\pi i$ there... and have forgot to put that when writing this transcript) and Noga and Sarnak both thought it was not Cauchy integral formula. At some point they said Cauchy formula so I thought maybe they meant the differentiation formula and modified the statement accordingly. Both of them were still not satisfied. Later I realized that Sarnak called the residue theorem Cauchy integral formula, and Noga called Goursat's theorem but for holomorphic function the Cauchy integral formula. I think both are nonstandard but I could be wrong.

Then we started with the Riemann mapping series! The questions are outlined here for completeness, but as other people with Sarnak in the committee already had lots of those in their transcripts, I would omit lots of details.

[S]: Draw a blob. Draw a disc. Are they conformally equivalent?

Me: Yes, by Riemann mapping.

[S]: Classify simply connected open subset of $\CC$ up to conformal equivalence.

Me: (Usual answer)

[S]: What are the conformal maps from one (bounded) simply connected region to another one? (Me: conjugate of automorphism groups of unit disk) What is it? (Me: easier to do it on upper half plane) How do you construct an explicit map between unit disk and upper half plane? (Me: (did it)) What is the automorphism group of the upper half plane then? (Me: $PSL(2,\mathbb{R})$).

[S]: Now if you puncture the two regions that you have on the board (both or simply connected and bounded), are they conformally equivalent? (Yes by transitivity) What if we dig out a hole? (Me: Not necessarily, showed that they can always be mapped to an annulus and that the ratio of the radii determines its equivalence class) How many parameters are there? (Me: one) Yes, one real parameter. 

Sarnak then proceeded to make remarks on how the proof I gave could not be generalized to triply or above connected regions, and asked if I went to the department colloquim on Wednesday (it was on moduli space and tropical geometries). I said I did, and he tried to draw similarities between $\mathcal{M}_g$ (which is $3g-3$ dimensional) and the multiply connected case.

[S]: What else do you know in complex analysis?

I did not really know how to answer that question, like which thing in complex analysis can I confidently say that I know? I guess I could have said something that Sarnak always asked, but maybe at that point it would become obvious that I intentionally studied Sarnak's transcripts (which I did). So I waited for Sarnak to suggest for topics, and apparently Sarnak did not like that. 

[S]: You must have read some complex analysis textbook, right?

I said I did take complex analysis in undergrad which followed Stein (I think) but I focused more on notes taken in class and not so much on the textbook. Sarnak really disliked this. He made some comments about how this is a graduate level exam and I am supposed to go through the entire book, and Ahlfors is a great source. 

He asked if I knew the things that he liked to ask (entire function, factorization) and I said yes, and he believed in me. We just talked a bit about Jensen's formula (he tried really hard to drag the word ``harmonic'' out of me as I sort of refused to use that language... it also feels a bit unnecessary for me) and then we were done with complex.

It was roughly 15:15, and Zeev said he had to go at four. Zeev said he thought it would only last for two hours, to which Sarnak replied it usually lasted for three hours (well, at least Sarnak's ones). We then decided to go over discrete geometry. Sarnak left for a bit and Noga asked if I wanted to take a break. I looked at my watch and was disappointed that cookies were not served yet, so I decided not to take a break.

Discrete geometry: (mostly Zeev)

[D]: State Szemer\'edi--Trotter.

Me: (State)

[D]: What proofs do you know?

Me: One with polynomial partitioning, and one with crossing number.

[D]: State the crossing number theorem.

This is something that I know I can derive immediately, so I did not try to memorize the statement. As a result, I oopsed a bit and state the wrong exponent. I said it could be wrong and Noga seemed to agree that it was wrong, so I suggested that I could go over the computation and find out the right bound.

Me: (State the weak crossing number inequality $\textup{cr}(G)\geq v-3e$, and then use the usual sampling argument to ``bootstrap'' it to $\Omega(v^3/e^2)$ if $e\geq 4v$.)

[D]: What about Szemer\'edi--Trotter over finite field?

Me: (State the one over $\FF_p$ with a small saving in the exponent)

[D]: What about general $\FF_q$?

I got confused a bit and thought that Zeev wanted me to state an analogous result but over $\FF_q$. I confessed that I did not know, and Zeev said it was in the handout so I got confused. Then I realized that he wanted me to say why something like that could not hold, and I said that the existence of subfield is a barrier. He was satisfied with that.

[D]: Can you go over the high level proof of that?

I only had time to mention that it was lots of additive combinatorics, and Sarnak interrupted and insisted that I gave a proof of Bourgain--Katz--Tao. I got really confused (partly because I forgot that Szemer\'edi--Trotter over finite field is Bourgain--Katz--Tao) as Sarnak said that it was a sum-product type result that is used in the proof (I think it probably should be the other way around). Noga and Zeev were also confused, and after all the confusion we decided to show some sum-product result over the real using Szemer\'edi--Trotter. Here I messed up a bit:

Me: I usually have to play around with the construction a bit to get the exact same thing, but the main idea is that you can construct a small set of lines and points to get $|A|^3$ incidences: $(a+b)c-ac=bc$. 

The right way of setting it up is to consider the set of lines $\{y=c(x-a):a,c\in A\}$ and points $\{(a+b,bc):a,b,c\in A\}$ to recover Elekes' result $\max(|A+A|,|A\cdot A|)\gtrsim |A|^{5/4}$. Because of how I wrote down the identity, I put the set of lines $\{y=cx-ac\}$ and bounded the cardinality by $|A|\cdot |A\cdot A|$ instead of $|A|^2$. I got $|A|^{7/6}$ because of this, but Noga and Zeev did not mind as they thought I got the right idea.

[D]: State Balog--Szemer\'edi--Gowers

Me: (State it)

[D]: On the high level, what is the proof?

Me: First, by considering popular differences, we can draw a bipartite graph between $A,B$ that is fairly dense where the edges correspond to a popular difference. Then we apply the path-of-length-$3$ lemma.

[D]: What does that say?

Me: (Roughly state the path-of-length-$3$ lemma)

Zeev was satisfied with it, and Noga made a brief comment on that the dependency of the constants are polynomial. Originally it was not polynomial when Balog and Szemer\'edi proved it, and Gowers showed how to get the polynomial factor using dependent random choice. I realized that it's probably why the names are not in alphabetical order.

They much declared that discrete geometry was done. They were discussing how to get Zeev updated in the final decision and quickly the entire discussion became completely in Hebrew. They promised that they did not say any bad thing for me, but I guess I would not be able to type this part down at all. Anyways, Zeev then left, and we returned to real analysis.

Real analysis (mostly Sarnak):

[S]: What do you study from? This time not your friends (he meant transcripts here) and wikipedia hopefully.

Me: Well I took a real analysis class in Taiwan several years ago and I went through the notes I took at that time.

He still disliked this answer and thought I should go through some textbook instead. He also made a comment that he realized all the people study his transcripts too well and he would ask some new question this time (oh no).

[S]: What do you know?

Again I do not even know where to start. Sarnak started to be annoyed and asked if I said this out of humbleness or I actually did not know anything. I did not know what would be a good answer to it, and after some thoughts Noga and I almost simultaneously said maybe measurability. I am then asked to define a measurable function. During the process, I am asked to define Lebesgue measurability.

Me: (Defined the outer measures and Lebesgue measurable sets)

[S]: What is the difference between Lebesgue measurable sets and Borel measurable sets?

Me: Measure zero sets.

[S]: Give an example of a function that is not measurable.

Me: An indicator function of any nonmeasurable sets.

[S]: Do those exist?

Me: Yes. (Proceed to give Vitali's construction.)

[S]: Whoa this is by Vitali? I didn't know!

I am not sure if he was saying this jokingly or he did not know. He made a comment that whoever taught the real analysis was intense as I knew this was by Vitali. Yes, he was.

[S]: Define Fourier transform for $L^1$-function.

Me: (Write the formula)

[S]: You probably know this question from the transcripts, but what is the image?

Me: $C_0(\mathbb{R})$.

[S]: Show it vanishes at infinity.

Me: (State Riemann--Lebesgue)

[S]: Well, just show it for compactly supported smooth function.

I started to do the $x+\pi/2\xi$ trick but got interrupted by Sarnak. He told me to show that their Fourier transform decay fast instead, and I realized that because of the smooth assumption I could show that it was Schwartz. Sarnak did not ask me to define Schwartz function and only asked me to do the trick with integration by parts.

[S]: Okay, I will still ask this, and you probably learned it from your friends too. Is Fourier transform surjective on $C_0(\mathbb{R})$?

Me: (Usual argument using Banach isomorphism theorem, throughout which Sarnak made sure I knew what Banach spaces are.)

[S]: Okay yeah I will have to switch up the problems. Now, here is a new problem (oh no x3). Actually before that, what is so special about $e^{-2\pi ix\xi}$? Why do we use that in the Fourier transform?

The question was pretty vague and I was not sure what he was at. He asked me what properties do those functions have, and I said something like orthogonality. He said I went a bit too far, and I was really confused. After a long time he said he was trying to go back to the Algebra section, to which I said that $e^{2\pi ix\xi}$ is a character on $\RR$. 

[S]: So, any character?

Me: I guess we definitely need some constraints like continuity.

[S]: Good, give an example that is not continuous.

I did the usual $\mathbb{Q}$-basis in $\mathbb{R}$ argument to get an additive function in $\mathbb{R}$ that is not continuous, and Sarnak was satisfied: ``This guy is sharp!'' I guess I really only know algebra.

Sarnak tried to push me to prove that $e^{2\pi ix\xi}$ are all the measurable characters on $\mathbb{R}$, to which I said that we probably also needed that the image lies in the unit circle. Sarnak asked me to prove this over finite abelian group (yes algebra here we go again).

Me: It has to be in the unit circle because all elements in $G$ have finite order.

Sarnak went on and comment that this is something we do not pay attention to in representations over finite groups, but it gets in our ways when the groups become infinite. Somehow I got away with not proving the statement he would ask me to prove.

[S]: Now let's go back to $SO(3)$ (oh no no no no). Can you come up with other irreducible representations?

Me: Ugh...... how about ``regular representations'' (though I had zero clue about what it would be for infinite groups)?

[S]: Great! So for infinite groups we usually consider the space $L^2(G)$. Do you know how to define this?

Me: Well I know Haar measure exists... but I know nearly nothing about it.

[S]: Define a $G$-action on $L^2(G)$.

Me: $\rho_g(f) = f\circ L_g$?

[S] Usually we do right action, but okay.

Sarnak asked if I know compact operators, and I said I knew nothing. He said it was basic Functional analysis (I guess but I have not learned about that at all). He then made a long comments about how compactness of $SO(3)$ plays an important role, using $SL_2(\mathbb{R})$ as a non-compact example where there are no finite dimensional nontrivial irreducible representations (I believe in him).

Anyways, the nightmare was over, and finally we moved on to the last topic.

Probabilistic method section (by Noga):

[A]: Can you say something about $\chi(G(n,1/2))$?

Me: It is asymptotically $2\log_2n$ (w.h.p,, but for convenience we dropped that really often).

[A]: What is the concentration window?

Me: From Azuma's we get an upper bound of $\sqrt{n}$.

[A]: What about lower bound?

I admitted that I did not know, and Noga said that lower bound is harder to show, and indeed we do have a matching lower bound.

[A]: What about $\chi(G(n,n^{-.9}))$? Do we have better concentration?

Me: By a clever application of Azuma's we can show a four-point concentration.

[A]: Not two-point? (Note: Noga got this result so this was probably why he asked)

Me: Ah yes you can prove something stronger, but then I do not know a proof.

[A]: And when do we have two-point concentration?

Me: Up to $n^{-1/2}$.

As the chromatic number is tightly connected to the independence number, we turn to the independence number.

[A]: What is the independence number of $G(n,1/2)$?

Me: Asymptotically $2\log_2 n$ too.

The conversation then became chaotic once more. Noga commented on that we had two-point concentration, but really mostly one. Sarnak suddenly interrupted and asked how was that possible. Noga tried to explain but Sarnak stopped him and ask me to explain instead. To be honest, I did not know how to respond because I thought the computation was really messy and there was no way that I could convince him. I tried to say something really high level and Sarnak was not convinced. Noga saw my utter confusion and suggested that I compute the expected number of $k$-independent sets, which I did. Noga asked me to do some computation to show that it dropped a lot when $k\sim 2\log_2n$. Noga then helped me in convincing Sarnak that this is the source of the phenomenon. Sarnak was finally satisfied.

We moved on and Noga was trying to think about a good problem to ask, and Sarnak interrupted and said that I should prove Ungar's theorem. Well, this is not probabilistic method anymore, so Noga quickly gave lots of hint and we went through that.

Finally, Noga decided to end the general with the Lovasz Local Lemma.

[A]: Do you know Lov\'asz Local Lemma?

Me: Yes.

[A]: Do you know the proof?

Me: Yes.

[A]: Okay. Now say we have a $d$-regular graph, and we color the graphs with two colors so that the number of red neighbors and the number of blue neighbors differ by not too much. How close can they be?

This shows up in a lemma in linear arboricity, so I immediately said $\sqrt{d\log n}$ and then corrected myself to $\sqrt{d\log d}$. Noga asked me if the $\log d$ was inside or outside the square root. I was leaning toward it being inside, but was not completely sure. I asked if I could show them the computation and Noga agreed to it. The usual application of the local lemma then gave $\sqrt{d\log d}$.

[A]: What about a lower bound?

I got stuck, and then Noga decided to get Sarnak's attention by throwing out the word Ramanujan graph.

[S]: Yes, show this for Ramanujan graph.

Noga quickly realized that it was not a good question to ask, but he asked me to do the computation and see what is wrong. Ramanujan graph is a really huge hint, so I started to set up to apply Courant--Fischer. Then I realized that I also needed a lower bobund on all eigenvalues.

[A]: So what graphs do you want to consider now?

Me: Paley graph!

Noga then tied this back to the question about quadratic non-residues hours ago. I think it is pretty nice that we start and end on the same topic!

It was 16:50 and Noga declared the end of the exam. I got out of the room and I supposed they talked about my ``wrong'' attitude towards learning math. Afterward Sarnak opened the door, congratulated me, and then said ``learn more math!'' and left. Well, in my defense, I didn't refuse to learn new math, but I just found learning other math way more interesting than... I guess memorizing the proof for Riemann mapping theorem and the non-surjectivity of Fourier transform. Anyways, I stuck around and Noga said that they talked a bit about it and he agreed that learning new was useful. 

Note: For probabilistic method we agreed to focus on the first half of Alon--Spencer. For discrete geometry we used Zeev's handout on incidence geometry.

Note2: Sarnak said that he would ask new problems as too many people just study from the transcripts...so if you are reading this because you have Sarnak on your committee... good luck ;)