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The Italian prime minister is in the United States for the UN general assembly meeting, and he was in the San Francisco bay area on Sunday and Monday. (No, this is not the one who paid money to an underage prostitute and had she released from custody by falsely claiming she was the niece of Mubarak, it’s the new one.)

Those two days were busy, and he met Italian start-up founders in the area, went to a dinner at Stanford hosted by John Hennessy, presided the inauguration of a new Italian-language school, went to Twitter, Google, Facebook and Yahoo, and he met the local Italian research community.

For the last event, a few colleagues and I were asked to give a short presentation. Being not sure what to say to a prime minister, I asked a colleague who is the department chair at an Italian computer science department for some data on state funding of university research in computer science, and if there was a way to turn this data into a recommendation, and his response could be summarized as “we cannot be saved, there is no hope.” This might have made a good theme for a presentation, but instead I talked about the importance of fundamental research, and of working on ideas for which the technology is not ready yet, so that when the technology is ready the ideas are mature. Politicians are good at feigning attention when their mind is elsewhere, and he feigned it well.

Yesterday I was “interviewed” as part of the process to naturalize as an American citizen. Part of the interview is a test of the knowledge that an American is supposed to have. I liked to think that the officer would bring up a map of the world and ask me to point to France, then I would point to Brazil, and he would say, “great, now you are ready to be an American!” (Instead he asked how many US senators there are, when was the constitution written, and things like that.) The vibe was very different from any other interaction I have had with the American immigration system before; now it’s not any more “who are you, why are you stealing our jobs, and how do we know you are not a terrorist,” but it’s all “yay, you are going to be one us.”

A conjectural approaches to the Riemann hypothesis is to find a correspondence between the zeroes of the zeta function and the eigenvalues of a linear operator. This was first conceived by Hilbert and by Pólya in the 1910s. (See this correspondence with Odlyzko.) In the subsequent century, there have been a number of rigorous results relating the zeroes of zeta functions in other settings to eigenvalues of linear operators. There is also a connection between the distribution of known zeroes of the Riemann zeta function, as derived from explicit calculations, and the distribution of eigenvalues of the Gaussian unitary ensemble.

In a previous post we talked about the definition of the Ihara zeta function of a graph, and Ihara’s explicit formula for it, in terms of a determinant that is closely related to the characteristic polynomial of the adjacency matrix of the graph, so that the zeroes of the zeta function determine the eigenvalue of the graph. And if the zeta function of the graph satisfies a “Riemann hypothesis,” meaning that, after a change of variable, all zeroes and poles of the zeta function are in absolute value, then the graph is Ramanujan. Conversely, if the graph is Ramnujan then its Ihara zeta function must satisfy the Riemann hypothesis.

As I learned from notes by Pete Clark, the zeta functions of curves and varieties in finite fields also involve a determinant, and the “Riemann hypothesis” for such curves is a theorem (proved by Weil for curves and by Deligne for varieties), which says that (after an analogous change of variable) all zeroes and poles of the zeta function must be in absolute value, where is the size of the finite field. This means that one way to prove that a family of -regular graphs is Ramanujan is to construct for each graph in the family a variety over such that the determinant that comes up in the zeta function of is the “same” (up to terms that don’t affect the roots, and to the fact that if is a root for one characteristic polynomial, then has to be a root for the other) as the determinant that comes up in the zeta function of the variety . Clark shows that one can constructs such families of varieties (in fact, curves are enough) *for all the known algebraic constructions of Ramanujan graphs*, and one can use families of varieties for which the corresponding determinant is well understood to give new constructions. For example, for every , if is the number of distinct prime divisors of (which would be one if is a prime power) Clark gets a family of graphs with second eigenvalue . (The notes call the number of distinct prime divisors of , but it must be a typo.)

So spectral techniques underlie fundamental results in number theory and algebraic geometry, which then give us expander graphs. Sounds like something that we (theoretical computer scientists) should know more about.

The purpose of these notes is to explore a bit more the *statements* of these results, although we will not even begin to discuss their proofs.

One more reason why I find this material very interesting is that all the several applications of polynomials in computer science (to constructing error-correcting codes, secret sharing schemes, -wise independent generators, expanders of polylogarithmic degree, giving the codes used in PCP constructions, self-reducibility of the permanent, proving combinatorial bounds via the polynomial method, and so on), always eventually rely on three things:

- A multivariate polynomial restricted to a line can be seen as a univariate polynomial (and restricting to a -dimensional subspace gives a -variate polynomial); this means that results about multivariate polynomials can often be proved via a “randomized reduction” to the univariate case;
- A univariate polynomial of degree has at most roots, which follows from unique factorization
- Given desired roots we can always find an univariate polynomial of degree which has those roots, by defining it as .

This is enough to explain the remarkable pseudo-randomness of constructions that we get out of polynomials, and it is the set of principles that underlie much of what is below, except that we are going to restrict polynomials to the set of zeroes of another polynomial, instead of a line, and this is where things get much more complicated, and interesting.

Before getting started, however, I would like to work out a toy example (which is closely related to what goes on with the Ihara zeta function) showing how an expression that looks like the one defining zeta functions can be expressed as a characteristic polynomial of a linear operator, and how its roots (which are then the eigenvalues of the operator) help give bound to a counting problem, and how one can get such bounds directly from the trace of the operator.

If we have quantities that we want to bound, then the “zeta function” of the sequence is

For example, and this will be discussed in more detail below, if if a bivariate polynomial over , we may be interested in the number of solutions of the equation over , as well as the number of solutions over the extension . In this case, the zeta function of the curve defined by is precisely

and Weil proved that is a rational function, and that if are the zeroes and poles of , that is, the roots of the numerator and the denominator, then they are all at most in absolute value, and one has

How does one get an approximate formula for as in (2) from the zeroes of a function like (1), and what do determinants and traces have got to do with it?

Here is our toy example: suppose that we have a graph and that is the number of cycles of the graph of length . Here by “cycle of length ” we mean a sequence of vertices such that and, for , . So, for example, in a graph in which the vertices form a triangle, and are two distinct cycles of length 3, and is a cycle of length 2.

We could define the function

and hope that its zeroes have to do with the computation of , and that can be defined in terms of the characteristic polynomial.

Indeed, we have (remember, is a complex number, not a matrix):

and, if are the zeroes and poles of , then

How did we get these bounds? Well, we cheated because we already knew that , where are the eigenvalues of . This means that

Where we used . Now we see that the poles of are precisely the inverses of the eigenvalues of .

Now we are ready to look more closely at various zeta functions.

I would like to thank Ken Ribet and Jared Weinstein for the time they spent answering questions. Part of this post will follow, almost word for word, this post of Terry Tao. All the errors and misconceptions below, however, are my own original creation.

This year, perhaps because of a mistake, the winners of the Field Medals and the Nevanlinna prize were made public before the opening ceremony of the ICM.

Congratulations to my former colleague Maryam Mirzakhani for being the first Fields Medals winner from Iran, a nation that can certainly use some good news, and a nation that has always done well in identifying and nurturing talent in mathematics and related fields. She is also the first woman to receive this award in 78 years.

And congratulations to Subhash Khot for a very well deserved Nevanlinna prize, and one can read about his work in his own words, in my words, and about the latest impact of his work in the the words of Barak and Steurer.

The Simons foundations has excellent articles up about their work and the work of Artur Avila, Manjul Bhargava, and Martin Hairer, the other Fields Medal recipient. An unusual thing about Manjul Bhargava’s work is that one can actually understand the *statements* of some of his results.

The New York Times has a fascinating article according to which the Fields Medal got its current status because of Steve Smale and cold war paranoia. I don’t know if they are overstating their case, but it is a great story.

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