You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.
John Onstead (Aug 06 2025 at 21:35):Hi! This is a general thread where we can discuss interesting news in math and mathematical physics. Feel free to share any interesting news you might have come across (does not have to directly relate to category theory). Every so often I'll post an interesting article along with (perhaps open ended) discussion questions laced in.
John Onstead (Aug 06 2025 at 21:35):I'll start it off with this article from one of my favorite science news sites, Quanta magazine (I'll probably be citing them a lot on this thread). In this article they cover ways of doing general relativity without calculus in what they call "non-smooth surfaces"- spacetimes where there can be edges and discontinuities like singularities. The goal is to try and extend results from GR (such as singularity theorems) to the non-smooth setting in the case that non-smooth geometry better describes the universe than smooth geometry (the article argues this to be the case, but I personally don't see why, since GR has achieved great success in its current smooth form). Though it is a little confusing that there'd be this research program when the major consensus among physicists is that singularities are mere mathematical artifacts and do not exist in any physical sense.
John Onstead (Aug 06 2025 at 21:35):From the mathematical perspective, the main object covered is "non-smooth spacetimes", which apparently are a generalization of smooth ones (so there's an embedding from the category of smooth Lorentzian manifolds into the category of these spaces). Usual differential geometry does not apply, but the article cites how some calculus-like results are still possible in these spaces. This makes me wonder how this effort might relate to synthetic differential geometry- might there be some variant of that which applies to this category, or maybe this will eventually inspire a new field of "synthetic non-smooth geometry" as a generalization of synthetic differential geometry.
One proposed definition for "non-smooth spacetime" is given here (on the official website, Quanta does not usually go into the mathematical details). These are called "Lorentzian pre-length spaces". It doesn't seem immediately clear what (if anything) these have to do with Minkowski space. Ideally, even a non-smooth spacetime should still, in some sense "locally look like" Minkowski space. So I'm not sure how that manifests here- something to think about!
John Baez (Aug 06 2025 at 22:06):Thanks for initiating this!
One thing worth noting is that there's no physical reason to expect solutions of the PDE in physics to be 'smooth' in the strict sense of infinitely differentiable. For example, for the vacuum Maxwell equations, the finite-energy solutions are typically not smooth: instead, the components of the electric and magnetic fields only need to be in at any given time. People have analyzed the hell out of the physically appropriate function spaces for solutions of the PDE of physics, using Sobolev spaces and such. It's particularly easy for the vacuum Maxwell's equations and other linear PDE, but people (like me, in my youth) have done a lot of work on nonlinear wave equations using this technology. Einstein's equations of general relativity are particularly challenging: e.g. Christodoulou and Klainerman won great acclaim for a 500-page book proving the stability of Minkowsksi spacetime, which necessarily uses Sobolev spaces rather than smooth solutions.
I haven't read the Quanta article yet, but I'm guessing that things like "Lorentzian pre-length spaces" go beyond the usual story.
Though it is a little confusing that there'd be this research program when the major consensus among physicists is that singularities are mere mathematical artifacts and do not exist in any physical sense.
They may or may not be "mere mathematical artifacts". There is not a unique definition of "singularity" in general relativity: there are many kinds of singular behavior, and it's far from obvious which are physically reasonable and which are not. Mathematical physicists approach theories of physics by studying the equations of physics, seeing what they predict. The question of which solutions are'"mere artifacts" slowly becomes easier to answer as we gain more knowledge. So there's been a huge research program of studying singularities in general relativity, starting with the Nobel-winning work of Penrose and parallel work of Hawking, and continuing to this day.
For example, an interesting question is whether singularities in general relativity are always hidden behind an event horizon. Various conjectures along these lines have been disproved.
John Onstead (Aug 06 2025 at 23:29):John Baez said:
Thanks for initiating this!
No problem!
John Baez said:
For example, for the vacuum Maxwell equations, the finite-energy solutions are typically not smooth: instead, the components of the electric and magnetic fields only need to be in at any given time.
That's quite interesting (yet unexpected). A while ago on here I tried to learn some basic bundle theory, but it was all in the context of smooth manifolds since those were the usual setting of differential geometry. This makes me wonder which category is the one where these Sobolev spaces become relevant- IE, where the spaces of sections of bundles form Sobolev spaces. That becomes tricky if the functions aren't even required to be continuous. Maybe you have to make use of local sections- for instance, the sign function is discontinuous over the reals but not over the reals minus the point at zero.
John Onstead (Aug 06 2025 at 23:50):John Baez said:
There is not a unique definition of "singularity" in general relativity: there are many kinds of singular behavior, and it's far from obvious which are physically reasonable and which are not.
From what I've learned, there's two kinds of singularity: coordinate and curvature singularities. A coordinate singularity is certainly a mathematical artifact even from within GR, since it isn't diffeomorphism invariant: you can change your coordinates to remove it (IE, the event horizon of a BH). A curvature singularity is the kind the article talks about, it's a diffeomorphism invariant point where GR predicts its own failure and "blows up". It's not technically an artifact in GR- it's a direct prediction of GR thanks to the singularity theorems. What I meant by "artifact" was that it seems (at least by my impression) to be a popular belief that the incorporation of quantum mechanics will resolve the singularity. For instance, in traditional QM, singularities are impossible because Heisenberg's uncertainty principle states it's impossible to infinitely localize something in space, hence it's not possible for there to be a point with infinite density and zero volume.
John Baez (Aug 07 2025 at 09:11):John Onstead said:
John Baez said:
For example, for the vacuum Maxwell equations, the finite-energy solutions are typically not smooth: instead, the components of the electric and magnetic fields only need to be in at any given time.
That's quite interesting (yet unexpected). A while ago on here I tried to learn some basic bundle theory, but it was all in the context of smooth manifolds since those were the usual setting of differential geometry.
Differential geometers like to assume everything is infinitely differentiable since that makes the math easy. Mathematical physicists start with differential geometry but then go further, adding analysis. There are a couple of reasons:
So, mathematical physicists, or even differential geometers who want to study solutions of PDE like the Yang-Mills equations or Einstein equations, can't work in the smooth context. The smooth context is just a first step before the real work starts.
John Baez (Aug 07 2025 at 09:24):John Onstead said:
From what I've learned, there's two kinds of singularity: coordinate and curvature singularities.
There are dozens of kinds of singularities in general relativity. Coordinate singularities are merely the result of a bad choice of coordinates, so we avoid them by avoiding bad choices. But saying that the curvature "blows up" is very vague, and when we try to make it precise we discover there are many different concepts of curvature singularity.
Furthermore, there are ways for solutions of GR to "suddenly end" even without the curvature blowing up. In fact the famous Hawking and Penrose singularity theorems only show "null geodesic incompletness", not that curvature blows up. Can we find physically realistic hypotheses that together with Einstein's equations guarantee that the curvature need to blow up in some sense? This is an area of active research.
What I meant by "artifact" was that it seems (at least by my impression) to be a popular belief that the incorporation of quantum mechanics will resolve the singularity.
Yes, I understood you meant that. This is indeed a popular belief. However, physicists working on general relativity don't say "quantum mechanics will change what happens, so we won't study the classical Einstein's equations too hard". Instead they study the hell out of Einstein's equations to see what they can learn - and just because it's fun!
John Baez (Aug 07 2025 at 09:33):By the way, if you check out Freed and Uhlenbeck's Instantons and Four-Manifolds you'll see that the classification of smooth 4-manifolds involves a lot of PDE and Sobolev spaces. Back then it was done using Yang-Mills theory; more recently the Seiberg-Witten equations have largely taken over.
(I mention this book not to be intimidating, but because it's rather pedagogical!)
David Michael Roberts (Aug 07 2025 at 11:06):Even the study of the S-W equations uses a lot of Sobolev space theory. We once had a seminar on them, and I had to be baffled by things like a lecture on the/a Sobolev embedding theorem, when I presumed we were supposed to be doing geometry!
John Baez (Aug 07 2025 at 11:28):Since I used to work on PDE, the Sobolev embedding theorem was my friend, back then. Seiberg-Witten is easier than Yang-Mills because it's 'less nonlinear', but even for linear elliptic PDE, Sobolev theory is how you prove the solutions are smooth.
John Onstead (Aug 07 2025 at 17:54):John Baez said:
So, mathematical physicists, or even differential geometers who want to study solutions of PDE like the Yang-Mills equations or Einstein equations, can't work in the smooth context. The smooth context is just a first step before the real work starts.
That's quite eye-opening! I'll have to look more into Sobolev spaces then, they sound quite fascinating!
John Baez said:
But saying that the curvature "blows up" is very vague, and when we try to make it precise we discover there are many different concepts of curvature singularity.
Furthermore, there are ways for solutions of GR to "suddenly end" even without the curvature blowing up. In fact the famous Hawking and Penrose singularity theorems only show "null geodesic incompletness", not that curvature blows up
Ah, I wasn't aware. I haven't studied much GR, but from what I've heard it's quite complicated and at times unintuitive (though arguably it's probably more intuitive than QFT, but that's not saying much). It's on my list to learn at some point in the future though!
John Baez (Aug 08 2025 at 12:21):There's something simple I forgot to say. If the only way for solutions of GR to "suddenly end" were for the curvature to blow up, this would be a strong hint that quantum gravity might save the day and eliminate this problem. After all, as curvature gets large we expect - by simple dimensional analysis - that quantum gravity effects should become large. But in fact, solutions can "suddenly end" without the curvature blowing up. This makes it more challenging to imagine how quantum gravity will save the day.
I was looking around for a good explanation of these issues, and I bumped into this quick but helpful passage:
In fact, there are different ways to define singularities. The most common ones include the curvature singularity and geodesic incompleteness. Curvature singularity usually means that the curvature invariants are divergent somewhere. Among all the invariants, seventeen curvature invariants, called Zakhary-Mcintosh (ZM) invariants, form a complete set. For spherically symmetric spacetimes, it has been shown that the Kretschmann scalar and the Ricci square are sufficient to determine whether all the seventeen ZM invariants are finite or infinite. Geodesic incompleteness refer to that the affine parameter along a timelike or null geodesic can not be extended to arbitrarily large values in either the future or past direction.
The two definitions are consistent in many cases, such as the central singularities in the Schwarzschild black hole and in the Kerr black hole. However, counterexamples have been found. For the well known C-metric, there exists a conic singularity while all polynomial curvature scalars are finite. On the other hand, [some] spacetimes with curvature singularities are found to be geodesically complete. This may seem incredible, since a metric usually cannot exist where a curvature singularity appears. In fact, a regular metric can generate a spacetime with curvature singularities and a series of solutions have been explicitly constructed. Unlike the usual curvature singularity where the metric cannot be defined, the metric is well defined where the curvature blows up. In such spacetimes, particles or light can reach or pass through the singularity.