Is your job developing AI coding tools?

"Save for Maynard, a 37-year-old virtuoso who specializes in analytic number theory, for which he won the 2022 Fields Medal—math’s most prestigious award. In dedicated Friday afternoon thinking sessions, he returned to the problem again and again over the past decade, to no avail. At an American Mathematical Society meeting in 2020, he enlisted the help of Guth, who specializes in a technique known as harmonic analysis, which draws from ideas in physics for separating sounds into their constituent notes. Guth also sat with the problem for a few years. Just before giving up, he and Maynard hit a break. Borrowing tactics from their respective mathematical dialects and exchanging ideas late into the night over an email chain, they pulled some unorthodox moves to finally break Ingham’s bound."

This quote doesn't suggest that the only thing unorthodox about their approach was using some ideas from harmonic analysis. There's nothing remotely new about using harmonic analysis in number theory.

1. I would say the key idea in a first course in analytic number theory (and the key idea in Riemann's famous 1859 paper) is "harmonic analysis" (and this is no coincidence because Riemann was a pioneer in this area). See: https://old.reddit.com/r/math/comments/16bh3mi/what_is_the_b....

2. The hottest "big thing" in number theory right now is essentially "high dimensional" harmonic analysis on number fields https://en.wikipedia.org/wiki/Automorphic_form, https://en.wikipedia.org/wiki/Langlands_program. The 1-D case that the Langlands program is trying to generalize is https://en.wikipedia.org/wiki/Tate%27s_thesis, also called "Fourier analysis on number fields," one of the most important ideas in number theory in the 20th century.

3. One of the citations in the Guth Maynard paper is the following 1994 book: H. Montgomery, Ten Lectures On The Interface Between Analytic Number Theory And Harmonic Analysis, No. 84. American Mathematical Soc., 1994. There was already enough interface in 1994 for ten lectures, and judging by the number of citations of that book (I've cited it myself in over half of my papers), much more interface than just that!

What's surprising isn't that they used harmonic analysis at all, but where in particular they applied harmonic analysis and how (which are genuinely impossible to communicate to a popular audience, so I don't fault the author at all).

To me your comment sounds a bit like saying "why is it surprising to make a connection." Well, breakthroughs are often the result of novel connections, and breakthroughs do happen every now and then, but that doesn't make the novel connections not surprising!

"It has been proved all the zeroes are within a narrow strip centred on the line and you can make the strip as arbitrarily narrow as you like."

Nothing close to this is known.

The nontrivial zeros of zeta lie within the critical strip, i.e., 0 <= Re(s) <= 1 (in analytic number theory, the convention, going back to Riemann's paper is to write a complex variable as s = sigma + it)*. The Riemann Hypothesis states that all zeros of zeta are on the line Re(s) = 1/2. The functional equation implies that the zeros of zeta are symmetric about the line Re(s) = 1/2. Consequently, RH is equivalent to the assertion that zeta has no zeros for Re(s) > 1/2. A "zero-free region" is a region in the critical strip that is known to have no zeros of the Riemann zeta function. RH is equivalent to the assertion that Re(s) > 1/2 is a "zero-free region." The main reason that we care about RH is that RH would give essentially the best possible error term in the prime number theorem (PNT) https://en.wikipedia.org/wiki/Prime_number_theorem. A weaker zero-free region gives a weaker error term in the PNT. The PNT in its weakest, ineffective form is equivalent to the assertion that Re(s) >= 1 is a zero free region (i.e., that there are no zeros on the line Re(s) = 1).

The best-known zero-free region for zeta is the Vinogradov--Korobov zero-free region. This is the best explicit form of Vinogradov--Korobov known today https://arxiv.org/abs/2212.06867 (a slight improvement of https://arxiv.org/abs/1910.08205).

I think your confusion stems from the fact that approximately the reverse of what you said above is true. That is, the best zero-free regions that we know get arbitrarily close to the Re(s) = 1 line (i.e., get increasingly "useless") as the imaginary part tends to infinity. Your statement seems to suggest that the the area we know contains the zeros gets arbitrarily close to the 1/2 line (which would be amazing). In other words, rather than our knowledge being about as close to RH as possible (as you suggested), our knowledge is about as weak as it could be. (See this image: https://commons.wikimedia.org/wiki/File:Zero-free_region_for.... The blue area is the zero-free region.)

* I don't like this convention; why is it s = sigma + it instead of sigma = s + it? Blame Riemann.

You don't think that a model is something abstract? Abstract doesn't have to imply nonphysical in the sense that people think souls or God are nonphysical. I mean abstract in the sense that language, mathematics, or a sketch are abstract.

To expand on this: I think models are representations, and whether or not something is a model depends in some way on human minds. (In particular, it depends on whether a something would be understood by a human mind to be a representation.)

I don't think that any correlation between physical systems qualifies one as a model for the other. Your definition as written would include any two things that are connected causally, or have a common cause, as models for one another. One problem (though not the only one) that I have is that your definition removes any mention of human minds.

In particular, I think "representation" is, broadly speaking, some kind of correspondence relationship between linguistic or pictorial things (where I include mathematics as "linguistic") and physical reality, and "a representation" is some linguistic or pictorial thing that corresponds to reality. I think that a model is a kind of representation.

A model is a kind of representation where for convenience and tractability, certain aspects of reality are left out or "abstracted away" (deliberately), with the goal of understanding the real world by understanding the simpler representation of the real world.

Well, it is an empirical question whether or not matter is continuous and infinitely divisible, or discrete. Our best theories tentatively suggest that matter is discrete (though it is hard to say how much confidence to put in that, or how we could really know either way).

I agree that this is a problem with the above definition of model, analogous to the (IMO fatal) problems with https://en.wikipedia.org/wiki/Deductive-nomological_model#We....

No, that's not exactly a sci-fi concoction. In special and general relativity, there are three dimensions for space and one dimension for time, and this is not something that is of "incidental" importance to special / general relativity, it's a pretty essential shift in perspective to these theories to think of the universe as (curved) four-dimensional spacetime.

But "dimension" is something mathematical. I would say it doesn't quite make sense to say "is the fourth dimension time" in the same way as it wouldn't make sense to say "is the fifth an apple?" The same way that numbers can refer to different things in different contexts (including in the context of different scientific theories), dimensions can correspond to different things in different contexts. For example, statistics and machine learning heavily use "high dimensional" mathematics, but there the "dimensions" would correspond to different variables you are trying to predict or explain. E.g. if you were trying to predict chance of heart attack from 1000 different factors, then you would have 1000+1 total "dimensions," and in that case the "fourth dimension" might be "cigarettes smoked per week" (rather than time).