Alexey Pozdnyakov’s Generals

Committee: Peter Sarnak [Sar] (chair), Will Sawin [Saw], Matija Bucic [B]
Special topics: analytic number theory, algebraic geometry.
May 9, 2025, 1:00-4:30 PM, Sarnak's office.

Sarnak tells me to erase the board and asks what I've read. I read Ahlfors for complex, Folland for real, and Dummit and Foote for algebra. For algebraic geometry, I read Hartshorne, but only the first half of Chapter III, and no surfaces. For analytic number theory, I read Davenport's Multiplicative Number Theory and Analytic Methods for Diophantine Equations. Sarnak asks me what I would like to start with. I say complex, and he invites Sawin and Bucic to ask questions.

Complex Analysis: 

[Saw] Do you know what an elliptic function is?

I say yes, doubly periodic and holomorphic? No meromorphic!

[Saw] What are doubly periodic holomorphic functions?

Constants by Liouville.

[Saw] What can you say about zeros of elliptic functions?

The zeros and poles cancel by the argument principle.

[Saw] What are doubly periodic meromorphic functions? Can you write one?

I wrote the Weierstrass P-function. I say that the sum converges since the terms are O(w^-3) and that it's clear from the definition that it's doubly periodic.

[Sar] What is this a function of? How does it transform?

I said a complex variable z and a lattice given by a point tau in the upper half plane. It’s doubly periodic in z, but I was unsure what to say about tau. I start to think outloud about automorphisms of lattices. I mention that certain lattices have extra symmetries, but Sarnak warns me that I am talking about something more subtle than he is looking for. He is happy once I say SL_2(Z). 

[Saw] You said it was clear from the definition. Why?

I started trying to show it and realized it was not clear. Sawin told me there is a small trick involved, and we moved on. 

[Sar] Why does it converge again?

I compare the series to a double integral and convince him that I actually knew that one.

[Sar] Draw a triply connected blob, draw another one, are they conformally equivalent?

I said probably not. He asks why, and I explain how to map triply connected regions to an annulus with a circular slit. He makes me define harmonic measures and explain how to build the conformal map out of them. When I mention the Dirichlet problem, he says that we'll get back to that. 

[Sar] How many parameters are there? What about doubly connected regions? Can we use complex parameters instead of real parameters?

I count the three real parameters for triply connected regions and explain that the annulus only depends on the ratio of the radii. I guess something about combining rotation and scaling with a complex parameter, but he stops me and says this isn’t in Alhfors. He then describes a deeper result due to Koebe.

[Sar] Explain the Dirichlet problem. How do you solve it?

I write down the Dirichlet problem. I accidentally wrote my triangle the wrong way, so they made me define the gradient and the Laplacian. I say that one way to solve it is with Perron’s method.

[Sar] Describe Perron's method.

I say that the idea is to find a maximal subharmonic function using Harnack's principle. He stops me when I start to write the set of functions to consider.

[Sar] I’m assuming you know Riemann’s proof and his mistake?

I said that he didn’t justify solving the Dirichlet problem, and jokingly, I say maybe it was trivial to him. Sarnak laughs and then pushes back, saying that Riemann is a genius and when he makes a mistake, there’s something subtle. 

[Sar] Do you know calculus of variations?

The last time I thought about that was in a mechanics class my freshman year.

[Sar] Write down \int |\del u|^2 du. 

Sarnak explained how you can show that if u minimizes this, then it must be harmonic. He then said Riemann's mistake was not justifying that the minimum is obtained in the space of functions he was considering. Since we're getting into real at this point, he asks the others if they have any more complex questions, and they shake their heads.

Real Analysis

[Sar] When is a minimum attained?

I say that something like a continuous function on a compact domain would be nice. We probably want some sort of compactness.

[Sar] What does Heine Borel say?

I say that in a nice space, closed and bounded is equivalent to compact. He says these guys are old, what kind of spaces were they working in? I say R^n. 

[Sar] Is the closed unit ball in C([0,1]) compact?

I say no, even though it's closed and bounded.

[Sar] Take a sequence of functions approaching the minimum. In what sense does it converge?

I guessed that we have some weak convergence, and he asks me what I mean by that. I write down some of the different topologies one can put on a Banach space and its dual. 

[Sar] When is the unit ball compact?

I know Banach-Aloaglu: the unit ball in X^* is compact in the weak^* topology.

[Sar] Do you know what a Sobolev space is?

I say that I can maybe guess the definition. I thought this would prevent him from asking about this further, but he then tells me to write ||f||_s = \sum_{n in Z} |\hat{f}|^2(|n|^2+1).

[Sar] Tell me about the Fourier transform. 

I define the Fourier transform from L^1(R) \to C_0(R).

[Sar] Why does it land in C_0? 

I say that you can pick your favorite density argument: use step functions and compute explicitly, use continuous functions and shift the phase, use C^1 functions and say smoothness corresponds to decay.

[Sar] Why does smoothness correspond to decay? 

You integrate by parts.

[Sar] What if your Fourier transform decays exponentially? When does that happen?

I wasn’t sure, so I said that maybe I could think about compactly supported Fourier transforms first.

[Sar] If the Fourier transform is compact supported, what can you say about your original function?

I say it’s not compactly supported. The committee laughs, and Sarnak says that's right, but he wants something else. Eventually, I say that it extends to a holomorphic function, and they ask me to show it is bounded in a strip. 

[Sar] Back to our original problem. Why does the Sobolev space embed into L^2?

After he says, “You must know this,” I say we can use Parseval.

[Sar] Suppose that you have a sequence of functions converging weakly. Show there's a convergence subsequence.

With the committee's help, I eventually realized that each Fourier coefficient has a convergent subsequence. I then spent some time working out a uniform bound on the tails of the Fourier series. Sarnak then asks what else I read about in Folland. I list a bunch of topics and theorems, but I don’t remember being asked anything about them. 

Algebra

[B] Do you know what a commutator subgroup is?

Yes, I defined it and explained why it's normal.

[Sar] Do you know what the abelianization is? 

Yes, but it took me a minute to remember the universal property. I decided it's the largest abelian quotient, and Sawin asks me what the smallest one is :)

[B] What is the commutator subgroup of S_n for n > 4?

I say that the only two options are A_n and S_n. They ask why, so I briefly describe a counting argument to show that A_5 is simple, and I say that it follows for A_n from A_5. I then realized that commutators are all even permutations. Bucic points out that I didn’t need to use the fact that A_n is simple, and we talked a bit about a more direct argument. 

[B] When can you solve exp(A) = B for matrices over complex numbers?

I say if and only if B is invertible, and prove it using Jordan normal form. Along the way, I'm asked why det(e^A) = e^tr(A).

[B] Suppose you have x_1,…,x_n \in R^d where all the pair-wise dot products have equal absolute value. What can you say about n?

I struggled with this problem for about 20 or 30 minutes. I tried for a while to rephrase the question into something about linear independence. I got stuck, so I drew the situation in R^2. Bucic points out that I can get n = 3 in this case, and he decides to make my life easier by removing the absolute values. Sarnak goes to the bathroom, and when he comes back, he says, “Has he written the matrix of inner products yet?”. I then write the matrix of inner products and continue to be stuck. At some point, Sarnak tries to give me another hint, but Sawin tells him to let me think and suggests that I am getting closer. Eventually, someone asks me what the rank of this matrix is, and I start to put it together: M = X^TX, and taking the rank yields n \leq d + 1. Bucic seems relieved, and Sarnak says that this is a great problem. They then discuss some recent results in dimensions 7 and 23, and the connection this has to sphere packing.

[Sar] Given your special topics, you should know some representation theory. Tell me about representations of finite groups. How can you find them?

I say that for small order groups, you can find the character table and get a lot of information about the representations this way. Sarnak asks what I mean, so I state Artin-Wedderburn and the consequences. 

[Sar] If I have a matrix representation that commutes with every other matrix, what can you say?

I was unsure and then repeated the question out loud and said, “Oh, Abelian, it's dimension 1”. We then took a five-minute break before moving on to the special topics.

[Analytic Number Theory]

[Sar] Can you bound L(1,\chi_d)?

Yes, I can do the class number formula, or Siegal’s theorem, or something elementary.

[Sar] Okay, but whichever you pick, I will ask for details

Okay, I’ll do Seigal. I write F(s) = \zeta(s)L(s,\chi_1)L(s,\chi_2)L(s,\chi_1\chi_2) and mention this is a Dedekind zeta function for a biquadratic extension.

[Sar] Can you prove that?

I explain how products of Dirichlet L-functions give Dedekind zeta functions. When I write Gal(Q(zeta_n)/Q), he asks what kind of fields I get this way. I said abelian extensions.

[Sar] Okay, go back to Seigal.

I start to write 1 \ll \frac{1}{2\pi i} \int, and I’m asked what’s so special about the product I chose. I said positivity is important, and he said why not the quadratic zeta function \zeta(s)L(s,\chi_1)? I struggle to answer, and he has me prove that L(s,\chi_1) is nonzero using it. At first, I argued by counting zeroes and poles, but Sawin points out that this requires the class number formula. Sarnak then shows me how to bound this directly by writing down the norm of an ideal in a quadratic extension. I was still unsure what was so special about the biquadratic zeta function.

[Sar] Okay, prove the prime number theorem.

The rest of the committee laughs, and I go to the other board and write the product that comes out of the 3-4-1 identity. Sarnak explains that the strength of the bound comes from getting more zeros than poles. He explains some history surrounding Seigal's theorem and why the biquadratic zeta function gives you such a strong bound. 

[Sar] Okay, let's talk about the other stuff. Solve a_1x_1 + a_2x_2 + a_3x_3 = b with x_i prime. 

I start to write the integral for the circle method. He asks if I can always solve this, and I say that there can be local obstructions. I continue, and he stops me when I write down what the major arcs should be.

[Sar] How do you win on the minor arcs?

I mentioned that you can get Vinogradov’s bound using Vaughan’s technique, and I mentioned type I vs type II sums. He asks what is really at the heart of this and emphasizes bilinear form inequalities.

[Sar] Why does this not work on the major arcs?

I start explaining how to get Vinogradov’s bound, and he says he believes I know how to do it, but intuitively, why does the method fail there? Eventually, he points out that the key thing is that e(n\alpha) is not multiplicative for \alpha irrational. 

[Sar] What goes wrong if you try to do twin primes, or Goldbach?

I say that you expect a main term of order N and explain that if you try to bound the minor arcs by an absolute value, then you can no longer beat the main term.

[Sar] What is a lower bound on the L^1 norm of \sum \Lambda(n)e(n\alpha)?

I say that you can look around the major arcs and get X^{1/2-\epsilon}. Before we get much further into this, he changes track and asks me to consider a different sum.

[Sar] How do you lower bound the L^1 norm of \sum e(n^2\alpha)?

I explained how to get an upper bound on the L^2 norm using orthogonality, an upper bound on the L^4 norm by counting solutions to u_1^2-u_2^2 = v_1^2-v_2^2 using the sum of divisors bound \sum_{n \leq X} d(n)^2 \ll X(\log X)^3, and then a lower bound on the L^1 norm by choosing a clever pair of Holder conjugates. 

[Sar] What can you say about two equations and four variables? Have you thought about that?

I haven't thought about this, but eventually I write a double integral. Sarnak describes some difficulties with the torus and some recent results.

Algebraic geometry

[Saw] Do you know a curve with genus 10?

I wrote y^2 = f(x) where f has degree 22 and distinct roots.

[Saw] Can you give me a plane curve of genus 10?

By the degree-genus formula, we just need a smooth, irreducible degree 5 plane curve.

[Saw] Why are these different?

I start to prove that a curve of genus >1 is either hyperelliptic or canonically embedded. I mess up the numbers and end up concluding that a canonically embedded curve is hyperelliptic. Sawin says it looks like I know what I’m doing, and this is not an arithmetic test, so we move on.

[Saw] When is a plane curve canonically embedded?

I work this out using the adjunction formula. 

[Saw] Talk about the line bundle \cal{O}_C(2), why is it very ample?

I first try to check the degree criteria, but he points out that this is much simpler than that. I say it's because it’s obtained by tensoring with \cal{O}_C(1), which is very ample. 

[Saw] Do you know what ample line bundles are good for?

I said that I asked someone I was studying with this same question a few days ago, and we didn’t come up with a satisfying answer other than sometimes it helps you determine that a line bundle is very ample. I also explain the correspondence between very ample line bundles and embeddings into P^n.

[Saw] If you view a hyperelliptic curve as a plane curve, can you compute its genus? Do you know how to find the genus of a singular curve?

I change charts so that the singularity is at the origin and begin to blow up the curve. After one blowup, I get something that still looks singular. Sawin says he believes I can do the algebra, but how does this get me the genus? I say that you can keep track of the multiplicity of the singularity at each blowup and subtract from the original arithmetic genus, which you can get using the degree-genus formula. 

Sawin then asks how far I got into Chapter III of Hartshorne. I said that I read Serre Duality, but I probably couldn't say much about it.

[Saw] Can you state Serre duality?

I gave Serre duality for projective Cohen-Macaulay schemes. 

[Saw] Do you know where the isomorphism comes from?

I said that Hartshorne does something with Ext sheaves, but I don’t know it very well. Instead, I described the cohomology of P^n and defined a perfect pairing in this specific case. I also mentioned that on a compact Riemann surface, this comes from integrating forms, and that for algebraic curves, one can define a residue map. He seemed satisfied with this and asked if Sarnak would like to ask anything. 

[Sar] How do you find elements of line bundles on a Riemann surface?

I said that in the analytic case, you can try working directly with their local trivializations. He mentioned the ddbar lemma and tied it back to the problem we discussed in the real analysis section. At this point, I was very fatigued, and Sawin says that the analytic side is an appendix of Hartshorne, so I probably haven’t seen this before. I had just taken a class on Riemman surfaces, but I was happy to move on to a different question. 

[Sar] Do you know about the moduli of curves? What is its dimension?

I wrote 3g-3, and Sarnak asks if I can prove it. I said that I can do it for g < 5.

[Sar] Okay, let’s go back to where we started this. Describe the moduli space of elliptic curves.

This was a nice way to finish. I used Riemann-Roch to get the Weierstrass equation, and then they stopped me when I defined the j-invariant. 

Sarnak tells me to leave the room and stay nearby. “Don’t leave the building, especially not in a projectile manner”. The committee deliberates for a few minutes, and I see Sawin and Bucic leave Sarnak’s office. After a few seconds, they say I passed and congratulate me. I go into Sarnak’s office, he shakes my hand, and tells me that I passed. “It was bumpy at times, especially with the more direct challenges, but you seem to know your theory and you prepared well.”