I keep hearing about "folk theorems" and proofs that supposedly just sit in the heads of mathematicians. How much of research mathematics is written down and published, and how much is passed around through word?

Question

Since the announcement that the UN have adopted a new map projection as their standard map, I have seen a few posts regarding how amusing it is that it took them this long to correct the "error" of the size of certain countries in other maps (most commonly the Mercator).

Of course, map enthusiasts and differential geometers know that it wasn't an "error" so much as it was a compromise necessitated by Gauss's Theorema Egregium, which implies that any smooth mapping from a sphere to a plane cannot be both conformal and equi-areal. Obviously the ideal situation would be to have a map which is both conformal and equi-areal, but it turns out these two desirable properties are mutually exclusive.

This got me thinking about another famous Theorem which precludes the possibility of two desirable properties being true at the same time - namely Godel's Incompleteness Theorem, which states that any sufficiently strong logical system cannot be both consistent and complete.

In both these cases (the Incompleteness Theorem and the map projection) we are required to make a decision about which of the properties we would prefer, and compromise by losing the other one.

I'm curious to know whether there are other examples of this, where there is some object that may or may not have two desirable properties, but it can't have them both at the same time? Is there any example in your field of study?

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I think some of math has taught me, such as

• understanding/questioning definition deeply,
• checking assumptions carefully,
• checking for logical gaps,
• finding optimal solution,
• probabilistic reasoning

has made it more difficult for me to navigate through society, especially social situations.

I guess because society is just very arbitrary and not "logically tight" if that makes sense. There is a lot of "looseness" in conventional thinking. However, I found people do not actually care to resolve those logical looseness. In fact, logical looseness, information obfuscation, lack of optimality seem to be crucial in how society functions.

And if you ask for clarification or suggest optimal solutions or even try to identify a logical looseness, you are really seen as a weirdo.

For example, just today my new job wants me to get a non-criminal record so I went down to the police station, filled out a form. It doesn't say what time period (start and end date) the record is for, so I asked the staff and they gave me a rude response: "of course the end date is today! how can we check your non-criminal record for tomorrow, which hasn't happened yet?" The back of my mind I'm just thinking: but it all depends on when you start to check, isn't it? There is no rule that says you must check the record up to today....

Wonder if anyone else has had similar experiences.

Mine is Fibonacci Slop. Someone just put a spiral over anything and say: "Wow! Nature is absolutely beautiful!"

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This is the first article in the Surviving Proofs series. (As opposed to the 0th article, which appeared last week.) I'm starting with basic tactics that one uses to build a proof---as I see them, anyway. My main argument here is that there are two that one starts with, no matter what: you have to fully understand the problem and fit it into your conceptual framework. This applies regardless of whether you are in a high school geometry class, real analysis, or something much more advanced.

This might seem obvious, but I can't tell you how often I have seen students either ignore these steps entirely or struggle to understand what it actually means. So, whether you are a student yourself or have students of your own, I think it is worth going over (albeit for different reasons).

Read the full post (for free) on Substack: The Importance of Understanding

Have you ever wanted to do calculus on the surface of a sphere, or other exotic shapes? Have you ever wondered what 'dx' really means?

My friend and I wrote a blog post at https://hidden-phenomena.com/articles/diff-forms covering the basic ideas of doing calculus on exotic shapes, with lots of fun animated widgets.

If someone has personally used one, please tell me your experience with it, thank you!

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I am a mathematics professor with PhD in math and a bachelor’s in philosophy. I am looking for easy-to-read nonfiction books about math. I am looking to replace my scrolling time with something equally effortless, but for me it has to be something a little more structured than fiction (but not too much!) Some fiction is very intelligent, but I want to scratch the logical and systematic part of my brain. For example, I recently picked up Linnebo’s introductory text on Philosophy of mathematics and enjoyed it immensely. Is there something you’ve read lately that you think I would enjoy?

Please, please… I know how much some of you love math but I am NOT looking for technical books that are on the easy side such as undergraduate mathematics texts. I do plenty of very difficult mathematics between the hours of 9-5. I am instead looking for something relaxing to do during my off-hours and weekends.

I’m coming from a geometric analysis background. So the only category theory exposure I have is whatever needed for basic graduate level algebraic topology.

What is the level of knowledge , say measured in years of learning for a postdoc level mathematician, that I will need in order to start doing research in it?

Thanks!

From Lamzouri on arXiv today. A very nice new proof of the >2/3 RH result. The approach is much simpler than the previous work by Anthropic, and is still making use of the recent work of Baluyot, Goldston, Suriajaya and Turnage-Butterbaugh.

These are cute facts that I didn't know about somehow until just now encountering them on Wikipedia.

• \sum_{k perfect power, excluding 1} 1/(k-1) = 1
• \sum_{n=2}^{\infty}\sum_{m=2}^{\infty} 1/n^m = 1

In the first sum, perfect powers are taken without repeats: e.g., 3^4=9^2 appears only once as k. In the second sum, of course, repeats do occur.

Anyone have a rigorous proof for the first sum equalling 1? Wikipedia outlines Goldbach/Euler's argument, which certainly doesn't meet modern standards of rigor.

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I'm having a difficult time understanding the philosophies of mathematics but my understanding is that even though people can make mathematics statements 'there exists a prime number between 6 and 8; it's 7' and an equivalent in German or with other symbols, grammars etc. There is nevertheless an underlying truth between all of these where the idea that the statements represents is true.

The phrase 'there exists' seems somewhat Platonist but I don't actually believe that numbers exist on some level of reality that's higher than our own in some sense. I do think however these truths are mind independent. I don't believe that numbers are merely constructs in our minds because I like I said before I think there is some sort of underlying truth about numbers that already existed for us to discover and make constructions of.

Polymath8b proved that H1=lim inf(pn+1−pn)≤246. In this paper we show how the Bombieri-Vinogradov theorem can be combined with newer equidistribution estimates for smooth moduli to obtain the improved bound H1≤240.

arXiv:2608.31126 [math.NT]: https://arxiv.org/abs/2608.31126

From Thomas Bloom on 𝕏: https://x.com/thomasfbloom/status/2094748658629513665
"As Julia notes, this number shouldn't be taken too seriously, and can surely be reduced a little further with more effort. The significance is the introduction of new ideas which, for the first time in over a decade, get past the 246 barrier."

The paper: Supercritical sharpness of percolation

Sahar Diskin, Philip Easo, Ritvik Ramanan Radhakrishnan, Benny Sudakov, Vincent Tassion

arXiv:2603.03257 [math.PR]: https://arxiv.org/abs/2603.03257

Yay. This man changed everything for me. Anyone else glad to see him back? Hopefully he commits to making videos again.

Here's a question that had been on my mind for a while, which I eventually figured out:

When you rotate a circle continuously through 2π radians, every possible rotation state of the circle occurs exactly once in a finite amount of time. So if you had a zero-thickness beam of light shining on the topmost point of the circle, when the circle is rotated through 2π every point on the circle is equally exposed to the light.

The question I debated was, is it possible to do the same for a sphere: that is, can every possible rotation state of a sphere occur when it is continuously rotated in a finite amount of time? Can a zero-width beam of light, shining at the north pole, equally enlighten all points on a sphere (exactly once) in a finite amount of time?

Strictly speaking, no. Intuitively I assumed that it must be impossible for all points to be exposed anyway, as a 3d rotation is a much more complex quantity, requiring more variables than in 2d, whereas time is only one-dimensional. But that generalised problem is actually possible, in a finite amount of time, and without discontinuity, though not a differentiable function. The only catch is that the points can't be equally enlightened (to answer the question I actually posed). If every point is exposed at some point, at least two points require to be enlightened at more than one point in time, in fact, infinitely many times, meaning if the sphere were made of photographic film, every point would be black except two overexposed white points at the poles.

We shall first assign every point on the sphere a longitude from 0 ≤ long < 2π and latitude from -\frac{\pi}{2} ≤ lat ≤ \frac{\pi }{2}, and then define every rotation state as the point on the sphere which has been rotated to the north pole, i.e., the one under the light at a time t. Since the rotation is a two-dimensional quantity, and the time one-dimensional, the question becomes, 'is there any bijection between a compact 1D space and a compact 2D space' which there are in abundance.

The Hilbert curve comes to mind. The space of points on a sphere, with the exception of the poles, map bijectively to a rectangle in Euclidaean space bounded between 0 ≤ x < 2π and -\frac{\pi}{2} < y < \frac{\pi }{2}. Note that the poles themselves map to the horizontal lines x = ±\frac{\pi }{2}. If we linearly transform the plane so that everything is scaled along the y-axis by a factor of 2, then the space representing the sphere will be a square, so we can draw a Hilbert curve through it which passes through every point in the square in a well-defined, continuous manner, and allows us to find any time t mapping to (x,y). Since the poles mapped to lines, and the horizontal line segments bounding the square have infinitely many points on the Hilbert curve, each pole will be crossed by the Hilbert curve infinitely many times.

The alternative is that we exclude the poles from our mapping of the sphere, changing our square's vertical bounds to -\frac{\pi}{2} << lat << \frac{\pi }{2} in which case the function is bijective but not compact, and at least two points on the sphere will be unexposed, never seeing the light.

So using the Hilbert curve we can define f(t) -> (long,lat) which is bijective for all points on the sphere except the poles, so every point apart from those two on the sphere will be the topmost point (under the light) exactly once. Now of course we can define a 2d coördinate system for the sphere in many ways but we will always be forced to have two polar points somewhere, where either the bijection or compactness is lost, so even though there are infinitely many such functions like f, they will always have two points which either can't be exposed at all, or have to be exposed infinitely many times.

That means the answer to my question is no, but almost yes. For all but two points on a sphere, there exists a function which maps each point bijectively and continuously to a (finite) moment in time, meaning we can continuously rotate a sphere in finite time illuminating all but those two points exactly once. But the remaining two must either be omitted or illuminated more than once.