A few quick math/physics items (OK, mostly math…):
- Contributions to next year’s ICM have already been written up by many speakers, and posted on the arXiv. Try this link to find them.
- There’s a wonderful new result from Kevin Costello that he talks about here. A central part of our understanding of the Standard Model is the computation of the beta-function of QCD. The beta-function determines the running of the effective strong coupling with energy, and this has been convincingly tested in many processes over a wide range of energy scales.
The usual way of calculating this is a Feynman diagram calculation that can be found in any QFT textbook that shows how to do calculations in gauge theory. Costello explains how to get the result in a very different way, using the self-dual theory, twistor space, and the Grothendieck-Riemann-Roch theorem.
- There’s a new volume of articles in honor of Gerard Laumon (who passed away on October 4) about algebraic geometry and the Langlands program, available at this website.
- On the Peter Scholze front, in this interview he explains in general terms some of the fundamental ideas he has been pursuing in his recent research, including the motivation of finding new ideas about geometry to describe Spec Z.
This semester at Bonn, he’s pursuing a project of generalizing geometry (lecture notes in progress here) by defining and studying “Gestalten”, which are supposed to be a new sort of geometric object, for which there is “a perfect duality between geometry and algebra!”
For a nice write-up about Scholze’s work on a geometrization of real local Langlands, see here.
At the late March 2026 Seminaire Bourbaki, Scholze will be lecturing on “Geometric Langlands, after Gaitsgory, Raskin, … “
Update: Both the Clay Mathematics Institute in Oxford and the CMSA at Harvard have organized talks about the Millenium problems to celebrate their 25th anniversary. The Clay Math talks are here. CMSA so far has had talks on Poincare (Mike Freedman) and the Yang-Mills mass gap (Sourav Chatterjee). Next up: Pierre Deligne on the Hodge Conjecture.
I’ll eventually watch the video, but I’ll ask now: does Costello give a new way to compute just the one-loop beta function, or the n-loop beta function for arbitrary n? Or something else? The “exact” beta function is probably too complicated to compute by hand.
John Baez,
It’s the one loop coefficient of the beta function.
To be clear, what I think is important here is not the result of the calculation, but the calculational method, which gives this a new interpretation. As a student I remember doing this calculation, wondering if this important number had any better explanation than what drops out of a slightly obscure calculation. Costello gives such an explanation, and, even better, it’s in terms of reformulating Yang-Mills on twistor space, something that fits in with the things I’ve been trying to do as “twistor unification”.
… but there is a very intuitive simple way to understand the beta function. This was worked out by N.K. Nielsen and Richard Hughes way back in the 70’s. In the presence of an external background magnetic field, Landau diamagnetism gives a positive contribution. But the spin of the gluon gives a contribution from Pauli paramagnetism, of the opposite sign, which is also larger by a factor of 12, resulting in an overall coefficient of -11.
Not to say Costello’s method has no value, but I think it requires more sophistication than the old explanation above.
Peter Orland,
Yes, thanks, that is a good physical way of understanding the result of the calculation.
Costello is providing not a new physical interpretation like this, but a new mathematical interpretation. What’s very unusual here is that we’re seeing a new and very non-trivial mathematical interpretation not of some random aspect of some random quantum field theory, but of a central aspect of a central part of the Standard Model. Such a thing doesn’t come along very often…
Hi Peter,
Thanks for the links
If anyone can explain how the “Spectral Algebraic Geometry” of Lurie and the “Gestalten” of Scholze relate, please do so.
I just watched Costello’s talk. Although I am far from an expert on the necessary background, I suspect that his instanton scaling somehow has a (very obscure to me) regularization involved (this would make a connection with 1-loop Feynman diagrams). He is studying the dilatation anomaly, and I can’t see how he can distinguish the classical and quantum theories without something like this going on. Maybe someone who understands the families index theorem or topological string theory better than me can explain this.
Hi Peter
Thanks for highlighting my talk! This is all joint work with Roland Bittleston (I’m not sure I did a good job crediting him in my talk).
Regarding Peter Orland’s comments, there are many one-loop quantities in QCD that one can compute using the index theorem, for example the ABJ anomaly. The one-loop beta-function is also a kind of an anomaly, so it should not be so shocking that it can be computed using an index theorem.
Peter O. is of course correct that there is a good understanding of the one-loop beta function using the background field method, so physicists might not be so excited by yet another way to compute this.
To answer John’s question, beyond one loop things are quite subtle. We do also have a two-loop result. However, since we work in perturbation theory around the self-dual sector, this is different from the known result that is obtained when you perturb around a free theory. (This kind of phenomenon is known in the literature — there is an old paper of Nima on this point). Beyond two loops things are scheme dependent.
Kevin
Terry Tao contemplates leaving the US.
https://youtu.be/yLvO070E_dI?si=nHtEynj11CWRk6ic
I watched it. Tao says he would have to consider moving if he lost funding but that so far he’d rejected (presumably lucrative) offers from around the world.
It’s mind-boggling that they make Tao waste his time worrying about and seeking funding, and planning for funding cuts. Imagine you’re the boss, and that the smartest or one of the smartest humans in the world works for you & that’s what you make them do. Why can’t academia guarantee him what he needs & let him get on with it? His work is surely cheap compared to other kinds of research.
Maybe it’s always been this way. Have all the truly great mathematicians in academia had to waste brainpower on university business?
Andrew,
There are some things about what is going on that should be clarified:
1. The few mathematicians in the US at Terry Tao’s level can easily find very well-paid positions where they can do whatever they want and pursue their own research with minimal distraction. Just as one example: I’m sure the IAS in Princeton would happily offer him a professorship if he were interested.
2. The need for research funding and the associated involvement with satisfying whoever is providing it comes when one wants to fund others to create a larger environment to further one’s research: grad students, postdocs, visitors, seminars, etc. etc. At UCLA, a lot of this is hosted at IPAM, which Tao has been heavily involved with.
3. There’s a larger effort by the Trump administration to defund scientific research agencies as part of their war against universities, which they consider their political enemies. While they have proposed huge funding cuts to these agencies, my impression is that Congress will not go along with this. If the US ever again has a functioning Congress and a budget, it quite possibly will not include those funding cuts.
4. The immediate problem for Tao has been the Trump administration specific attack on UCLA, defunding its research because of bogus accusations of “antisemitism”. This dishonest and illegal attack has gotten traction because of the awful behavior of the Columbia University trustees, who caved in to it when it first happened here (and was supported by Scott Aaronson and many other obsessed with stopping any criticism of the Israeli slaughter of civilians in Gaza).
5. My understanding of the current situation is that UCLA, like Harvard, has decided to fight the illegality and is doing well in court, with funding at least temporarily restored.
I don’t know if this has been mentioned before but John Baez has a new popular lecture series on the Standard Model and it’s really excellent: https://www.youtube.com/watch?v=0yjxqMoX-y8. The level of detail he goes into just seems to be at the sweet spot for those familiar with the basic concepts but do not have a physics degree.
Thanks for the info Peter. Have you been paying attention to recent discussions on strong CP problem? E.g., https://arxiv.org/pdf/2511.04216 from today’s arXiv.
They are debating QCD predictions and correct mathematical treatment of infinite volume limits in QCD. It might be interesting to you as an example of the lack of rigorous mathematical understanding of QCD and QFT. I also wonder if the sophisticated techniques from maths shed light on this issue.
Andrew,
Thanks, but I haven’t been following this sort of argument, which has been going on for a long time. Last time I paid any attention was 30 years ago, talking to Hidenaga Yamagishi, who had been a fellow Princeton student and was writing this:
https://arxiv.org/abs/hep-ph/9507296
The first CMSA lecture struck me as odd, since it did not go into detail about how Perelman solved the Poincare problem, and how that solution relates to further activity. The second lecture did not indicate any preferred direction into solving the problem; maybe that IS the point.
Deligne’s lecture has just shown up at: https://www.youtube.com/watch?v=d23FICTAC1Y It is delightful how easy it is to follow so far, waiting to be suddenly left in the dust …