He is famous for his undergraduate Topology book but he also wrote a book on linear algebra, one called Analysis on Manifolds, which develops multivariable calculus in n dimensions, one on differential topology, and one on algebraic topology. I like his topology book but I am also a big fan of his lesser-known Analysis on Manifolds book. He clearly put a lot of effort into his exposition.

When working (not teaching) things on a whiteboard, whether standing at a large board or sitting at the desk (small whiteboard), do you find it useful to use markers of different colors? The pedagogical value of colored markers is clear. I'm asking about working exercises for myself.

Typically I've used pencil and paper and am newly switching to a handheld whiteboard. So I'm used to monochrome. I dont want to look like a colorfest either. But if people find it useful and not too cumbersome to use 2-3 colors, I'm happy to order the markers in a couple of different colors.

I'll mostly be doing real and eventually complex analysis, linear algebra, probability, abstract algebra, and a bit of 3d.

Would you enjoy it?

The mathematician Emmy Noether made a fundamental, and very beautiful, discovery: continuous symmetries in the laws of physics give rise to conservation laws in physics! In this way, conservation of energy, conservation of momentum, and conservation of angular momentum all come from symmetries of the laws of physics: energy is conserved because the laws of physics are independent of time; momentum is conserved because the laws of physics are translation invariant; and angular momentum is conserved because the laws of physics are rotation invariant.

At the end of the article, we also say a little about Lie groups and Lie algebras, because secretly they are the mechanism by which mathematicians formalize continuous symmetries; for conservation of energy and conservation of momentum, it's easy to get by without them, but to really understand conservation of angular momentum, it is very helpful to think about the Lie algebra of the group SO(3).

The linked paper read as complete slop to me as a geometer, but am I just out of my depth here? I am just bewildered by such a massive pdf making it through the arxiv checks. I assumed anything that big would get flagged for manual review.

I figured it was high time that I did a follow-up on my original post on category theory---this time, to discuss universal properties and natural transformations.

Why is this of any interest? Simple: those two notions give a framework for how to think about coordinate-free definitions. If you are unfamiliar with the concept, I can give a very concrete example. In linear algebra, one defines the trace of a square matrix as the sum of its (main) diagonal entries. A priori, this seems entirely random and it is perhaps a great surprise that this turns out to be coordinate-independent---you will get exactly the same result if you choose a different basis in which to express your matrix. This can be gainfully exploited (to aid with calculating eigenvalues, for example), but one is still left with the uneasy question of why exactly this just happens to work out.

Alternatively, it is possible to give a coordinate-free definition of the trace (and I do so in this post), which doesn't make use of any particular basis. It is then immediately obvious why the trace doesn't depend on a choice of coordinates, but there are other benefits as well: one of them is that it offers some insight into what you need to extend this definition to work beyond simply finite-dimensional spaces. (The key property turns out to be that you need the vector space to be naturally isomorphic to its dual. This occurs, for instance, for Hilbert spaces.)

Read the full post (for free) on Substack: How to be Universal and Natural

This recurring thread will be for discussion of AI in mathematics. This includes, but is not limited to, the following:

• informal announcements of AI-assisted discoveries, such as those not yet published in a peer-reviewed journal, or not uploaded as a paper to arXiv;
• informal announcements of discoveries related to AI architecture (if relevant to mathematics);
• discussion of such announcements, such as proof breakdowns or other opinion pieces;
• discussion of the impact of AI in mathematics in general.

AI-assisted mathematical papers published in peer-reviewed journals or as arXiv preprints may be submitted as their own posts.

Please keep in mind rules 1 and 6 of our subreddit.

Having concluded my bachelor's in physics I'm transitioning towards mathematical physics for my master's and one of the first questions that made me realize I'm actually quite interested in the field of mathematical physics is this one (besides of course that I'm fascinated by rigorous and unambiguous approaches to a field that sits so close to my heart).

I know Lorentz invariance in itself causes a whole lot of issues, the big one that immediately comes to my mind is covariant quantization of gauge theories: fix a gauge that is not lorentz invariant and the naive approach to canonical quantization works just fine, but add this constraint back and suddendly you get a physicist to ramble about negative norms in a Hilbert space (the slander comes from a place of love, I like teasing my physics dept friends).

But these kinds of problems seem, to some extent, secondary. In fact, with some clever "work arounds" one can formally solve these issues, whereas the more fundamental task of simply _defining_ an interacting theory in 3+1 dimensions seems to be still out of reach. Why? I still know very little about constructive QFT but as far as I understand it interacting QFTs in lower dimensions have successfully been defined, while in 3+1 dimensions we only have rigorous constructions of free theories, so I guess the problem isn't the presence of the interactions per se but it's specifically the number of dimensions?

The paper: Decoupling via Affine Spectral-Independence: Beck-Fiala and Komlós Bounds Beyond Banaszczyk
Nikhil Bansal, Haotian Jiang
arXiv:2508.03961 [math.CO]: https://arxiv.org/abs/2508.03961

I'm wondering what their strategy is for being this productive.

I'm not an expert in complex geometry, but as far as I know this is a pretty big conjecture. It was settled for Fano manifolds by Chen-Donaldson-Sun some time ago, and there are other similar conjectures/theorems like the Donaldson-Uhlenbeck-Yau theorem, which says under algebraic conditions conditions that one can find a Hermite-Einstein metric on a holomorphic vector bundle. Unfortunately, it seems to be completely proven by AI. The length of the paper is also a lot longer than previous AI papers.

Edit: Apparently this was expected to be false in this generality anyway, and multiple people had claimed they could write down counterexamples but hadn't done so.

The counter example was given by Danielewski in 1989 but I don't have the copy. It's not even on the internet and my college doesn't have access. I need it for my thesis. Any help is appreciated.

Is there any area of study or conjecture in research mathematics where it is either proven or highly suspected that a unique proof is required for an infinite (or extremely large) number of cases? For example, something where “proven for all dimensions” is known to not be possible.

By hanging fruit I don’t mean the proof has to be easy, in fact infinitely many difficult proofs is more of what I’m curious about.

This recurring thread is meant for users to share cool recently discovered facts, observations, proofs or concepts which that might not warrant their own threads. Please be encouraging and share as many details as possible as we would like this to be a good place for people to learn!

While doing exercises or examples, do you prefer to be sitting or standing?

Feel free to elaborate on your problem solving setup.

This is inspired by a recent comment about some people understanding entire fields. So I'm wondering what it could even mean to understand an entire field. Certainly the average mathematician doesn't satisfy that. I suspect most mathematicians do not have an entire field mastered. Again, what does that even mean?

I propose one possible interpretation. Let's just take something that is arguably a "field of math". I'll use probability theory as an example since it's what I'm familiar with. It's a big field. So the best pretty theory experts really understand the entire field though? It's a big field. Probably even then best experts still have a long list of results they have never heard of. They'll likely literally know everything in their little sub discipline, but the *entire* field of probability theory?!?

I could be wrong though.

Also, this is likely simply asking too much. Rather than literally knowing every result and every proof, maybe we should set the bar at something like: they can read an arbitrary new-to-them result in that field and understand it nearly instantly and to be able to breeze through the proof and then explain it without much study. That is probably feasible for a really smart expert, but I'm not really sure. This is less than asking them to produce a fully rigorous proof, more like a satisfactory sketch.

I'm not at all an example here. I know very little compared to such folks. I suspect my level of knowledge is not that unusual though, even if somewhat on the low end. But one can know orders of magnitude more than me and still not approach the entire field of probability theory.

I home at least some find this question interesting.

Recently, I asked about the usual proof that the ring of regular functions on distinguished open set D(f) on variety X \subset \mathbb{A}^n is A(X)_f where A(X) is the coordinate ring of X. In several places, including Hartshorne, there's a statement that looks highly related but is different: that D(f) is isomorphic to an affine variety X' \subset \mathbb{A}^{n+1}, defined by \{(x,t): (x,t)\in A^{n+1}, x \in X, tf(x)-1 = 0\}, and the coordinate ring of X' is A(X)_f.

I have two questions:

1. Does this constitute another proof that the ring of regular functions on D(f) is A(X)_f? It seems like it "obviously" should be, although I'm not sure what you need to do to formally show it.

2. This feels geometrically very unintuitive to me, though the algebra seems reasonably well motivated. How does one "see" that a distinguished open set isomorphic to an affine variety embedded in a space of one higher dimension?

What the title says: is there a clear, unambiguous, systematic way of writing diagram chasing arguments? What I generally do is use different colours for arrows on different paths but I’m not entirely satisfied with this. There’s also the issue of denoting when an element is being mapped to from another vs when it is being lifted from another etc

This recurring thread will be for any questions or advice concerning careers and education in mathematics. Please feel free to post a comment below, and sort by new to see comments which may be unanswered.

Please consider including a brief introduction about your background and the context of your question.

Helpful subreddits include /r/GradSchool, /r/AskAcademia, /r/Jobs, and /r/CareerGuidance.

If you wish to discuss the math you've been thinking about, you should post in the most recent What Are You Working On? thread.

My YT algorithm has recently recommended to me the Youtube channel Zeta Explained. Contrary to most amateur content on the Zeta function and other Riemann hypothesis attempts, these videos seem highly professional and (for me as a grad student) mathematically sound (history of results, examples, proof sketches, etc.).

What intrigues me is that I don't know who is doing these 100+ videos on the Riemann Zeta function, quite impressive. The videos are definitely not AI generated. Do we in this sub know who this is, or at least at which institution they're affiliated with?

And for my personal curiosity: is this channel "legit"? I myself am not an analytic number theorist. So, is their content meaningful, or rather, what's the intended audience?

Appeared on the arxiv today. It still uses the CW-tensor/laser method approach.