Inspired by John Baez’s long running series, I thought I would start writing about mathematical things I stumble upon. I’m not sure how frequently these posts will be (certainly not every week), but it might be a way to share things I find interesting.
A series on the Riemann Hypothesis
Recently, Youtube suggested to me an ongoing series of videos by PeakMath on the Riemann hypothesis and F1. PeakMath seems to be a paid subscription service which will run courses on L-functions, but also have a Youtube channel which posts their videos after a time delay. I really enjoyed the presentation; Andreas Holmström is an engaging and entertaining lecturer and it is great to see advanced math presented in such a polished and clear way. The first video has an excellent explanation of the relationship between the zeros of the zeta function and the distribution of primes, which I think is very accessible to a general audience. Thus far, there are three videos out in the series and I look forward to seeing more.
The field of one element
The field of one element (denoted F1 or Fun) is a hypothetical and seemingly impossible object which has generated a huge amount of interest in the past 70 years. In abstract algebra, a finite field is an algebraic structure which has addition, subtraction, multiplication, and division (except by 0) in all the ways that we are familiar with. However, the standard definition of a field requires at least two elements (the additive and multiplicative identities), so there is no such thing as a field with one element in the usual sense. In the 1950s Jacques Tits developed a complicated construction in algebraic geometry for finite fields which are now known as buildings. Furthermore, there was a degenerate case of this geometry that seemed to fit into this picture, but would correspond to a field with only a single element. Using this idea, he was able to establish some properties for this mysterious object, but had no way to construct it.
In the late 1980s/early 1990s, Alexander Smirnov proposed that a fully developed theory of F1 could yield a proof of the Riemann hypothesis. I’m a total novice on the subject, but I believe an over-simplified synopsis is that if one interprets the rational numbers as the function field of the spectrum of the integers (whose closed points correspond to prime numbers), it might be possible to understood the rational numbers as a function field extension over the field of one element. The previous sentence is likely incomprehensible unless you’ve studied abstract algebra, but there is a famous result by André Weil which proves a version of the Riemann Hypothesis for function fields (see this survey by James Milne). So the hope is that it might be possible to translate the result for function fields to the rational numbers, which would establish the Riemann Hypothesis in the usual sense.
Since then, there has been a substantial amount of interest in F1 since then and it is a flourishing area of research. In particular, Alain Connes, Caterina Consani and others have studied the relationship between F1 and non-commutative geometry and others have made connections to tropical geometry, computational complexity, etc. I found this survey to be nice introduction to the field, but is definitely written for mathematicians (and not necessary “everyone” in the broader sense).
My uninformed opinion
For now, I’ll confess to some skepticism about solving the Riemann hypothesis by way of F1, at least for the foreseeable future. I am a complete non-expert so it’s very possible that such a proof of the Riemann hypothesis could come out tomorrow and make me look like a fool. But in that case there would be a proof of the Riemann hypothesis, so I would be thrilled to be wrong here.
At least to my outsider perspective, the sharpest results on zeta functions come from more traditional analytic/algebraic number theoretic methods, such as Brian Conrey’s density result for zeros on the critical line or Pierre Deligne‘s proof of the RH for finite fields (the survey by James Milne covers this as well). For instance, a substantially weaker result than the Riemann hypothesis is that all the non-trivial zeros have real part less then 1. It turns out that this fact is essentially equivalent to the prime number theorem. So using classical analytic number theory we can bound the real parts of the non-trivial zeroes a distance 1/2 from the critical line. Improving on that bound is brutally difficult and finding zero-free regions of the zeta function is an active area of research. However, results like these give fairly convincing evidence that analytic number theory plays an important role for studying the zeta function (not that this was ever in doubt).
Now, F1 is an active area of research which has made a number of breakthroughs, although my impression is that thus far the results are mostly of interest to algebraic/non-commutative geometers. To the best of my knowledge, this progress has not yet yielded concrete insight into the behavior of the zeta function or other L-functions. For instance, while writing this post I could not find a proof of the prime number theorem in terms of F1 geometry (please send me one if it exists). I should add that there’s nothing wrong with that; math often develops in this way and there are plenty of reasons to study F1 apart from the RH.
However, it is very unusual for a major open problem to be solved without intermediate/weaker results being found along the way. Proofs of major theorems tend to incorporate many new ideas and those invariably have other (often simpler) applications. This has been the case for nearly all the major breakthroughs in the last few decades (e.g., Ricci flow/Poincaré1, Modularity/FLT, etc.) In fact, one of the clearest red flags in the IUTT debacle of the past decade was the claim that the theory did not have any simpler consequences short of the abc conjecture.
In defense of the field with one element, there are a number of visionary conjectures (such as the Langlands program or Geometrization conjecture) which were instrumental in guiding a large amount of research, even though there was no way to prove them when they were first proposed. And with hindsight, we can see how these conjectures illuminated the underlying mathematical landscape and lead to developments which solve major open problems. So it is entirely possible that an eventual proof of the RH will use ideas from F1 geometry and we will view the theory in the same way as those other grand conjectures. Nonetheless, until there are more definite results where F1 gives some new insight into zeta or L-functions, I don’t think a breakthrough on the Riemann hypothesis is imminent. But again, math has a way of surprising everyone, and I hope that this post ages poorly.
Other finds
- Hiranya Dey and Soumyajit Saha (who is currently a postdoc at ISU working with me) recently wrote a preprint on the number of nodal domains for low frequency eigenfunctions of the Laplacian on graphs.
- Benjamin Walker, Adam Townsend and Andrew Krause made a website where you can visualize and interact with solutions to various PDEs. It’s really cool and definitely worth checking out. visualpde.com
1Richard Hamilton’s first paper on the Ricci flow proves that any simply connected three-manifold with positive Ricci curvature is diffeomorphic to a three-sphere. On its own, this is not enough to prove the Poincaré conjecture (that all simply connected three-manifolds are diffeomorphic to spheres), but gave an immediate proof of concept that Ricci flow might be useful to this problem. Even forty years later, there is no other known method to prove that simply connected three-manifolds with positive Ricci curvature are spheres. And in the 20 years between Hamilton’s first Ricci flow paper and Perelman’s work, Hamilton and others proved a number of major results using this approach. For instance, they even found a proof of the Uniformization theorem by way of Ricci flow. Furthermore, the first few pages of Grisha Perelman’s work presented a new breakthrough on the flow which resolved a number of outstanding questions.
2Mathematica can make really nice plots of complex functions. The code for this was ComplexPlot3D[E^(-1.9 I) Zeta[z], {z, -5- 4.5 I,12 + 30 I}]. I then removed the axes and did some color correction in a separate program. The E^(-1.9 I) term simply serves to make the right half of the picture purple as opposed to the default of red.
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