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, … “
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.