Do you think this is legit?

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 cognitively stimulating 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.

I took a look at my progress bar graph I keep where I track the scores of some mock tests. Over the course of 2 months, I have nothing. No improvement. At all. My score is still floating around the same range. 2 months of hard work, of dedicating my time and effort into this, and just nothing.

I had a mistake log, I practised every day, I tried my best to think about problems but often times would just end up staring and then seeing the solution. What more could I have possibly done? This feels like my dream of being a great problem solver and being able to learn the language of the universe is shattered. There must be something in me, something fundamentally broken that I haven't improved even a little bit even after giving my heart and soul for this.

And I break down every time I think about it. Honestly, I dont even know how many of these I can take. I dont know how long I can tolerate feeling this stupid, even when Im doing everything I can. I just never have the first step and end up staring at the problem, even when I change up the difficulties.

I don't want to give up, I really don't want to, but Im not sure if I'll even get somewhere if I keep going, maybe I'm just not meant to.

Have you all ever felt like this due to Math? Sounds stupid to ask, feeling so shattered because of a subject, but if you did and got through it, hearing it could really help right now

P.S: And for context I've been doing Olympiad styled Math:D

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.

In video games, they have built in levels, always at the start and they are called "tutorials."

In these levels you play a portion of the game while its also being explained to you. After finishing them, the point is that you know very little about the game BUT enough to "get" the basics.

The cheap is answer is high school 30 is the tutorial, but its really starting to feel that that wasn't even the complete tutorial honestly. It's kinda of embarrassing that our education doesn't cover such things as basic linear algebra and proofs. I'm still learning those.

Anyway to anyone who responds: thank you! I've been so happy latelty! Cyaaa.

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 co-ordinate 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.

This recurring thread will be for general discussion on whatever math-related topics you have been or will be working on this week. This can be anything, including:

* math-related arts and crafts,
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Just curious as to what people in the math community use for their research? Do you have your own systems just running in the background or do you utilize some of the web services that offer compute services?

If you have your own computer what is it?

I am launching a new series today, which I am calling Surviving Proofs. It's a little different than what I have done before---it's primarily intended for those who are stepping into a proof-heavy classroom for the first time, although I think it will have more general interest. It is not meant as a replacement for an Introduction to Proofs class---I trust the professor there to teach basic set theory and logical notation and so on. Rather, it is all about the underlying philosophy that one needs to read, write, and understand proofs and flourish in such an environment. We'll go through concrete examples, of course---we'll look at proof by induction, and so on---but we're after bigger lessons than just how to write a proof by contradiction.

Mathematicians on the whole are very good at teaching formalism and even specific applications. But, in my experience, this kind of big-picture philosophy is rarely discussed, and that is a great shame. This series is my attempt to correct this.

We begin with a simple question: why care about proofs? Very few of us are able to excel in something if we aren't convinced that it is interesting or useful, so it seems important to handle this first, before we do anything else. There is an obvious answer to this question, which is that proofs allow us to determine what is right. This is not... wrong, as such, but I think it misses what is primarily most important in proof-writing. (There is a particular Saturday Morning Breakfast Comic that is very relevant here---as usual, Zach Weinersmith is quite insightful. You'll see what I mean.)

Read the full post (for free) on Substack: Why Do We Care About Proofs?

There are many really good mathematics YouTubers nowadays like the popular 3Blue1Brown and Numberphile, but to me the one that shines above them all is Mathologer. Mathologer has been making mathematics accessible for almost a dozen years to a wide audience in a way that they can really understand and appreciate proofs that are may often be intimidating. A great example to this is the e and pi being transcendental video -- seriously who else can do anything like this?

Mathologer is really good at explaining concepts and carrying people through so undergraduate level students can understand and follow the work. Also, I love the mathematics history which I wish was not well represented in the mathematics textbooks of my generation. And it's very cool to see fun topics like Rubik's cube, a fine way to talk about the mathematics of permutations.

All the amazing work this guy has done, I just wanted to post a "shoutout" to him. Thank you Mathologer for all your amazing content.

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;
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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.

Pick's theorem allows calculating the area of any 2D polygon (including nonconvex polygons) whose vertices lie on an integer lattice from only the number of lattice points within it and on its boundary.

This feels like a minor miracle, and indeed there is no equivalent formula for the volume of polytopes in any higher dimension, even when restricted to convex polytopes.

What geometric/topological property of 2D space makes this magic possible that somehow fails in every other dimension?

It has always been somewhat strange to me how popular category theory and infinity-category theory are on the 'mathematical internet', despite how few working mathematicians actually need them.

However, over the past decade, infinity categories have grown in importance in more classical mathematics research -- especially in my own field of arithmetic geometry!

My friend and I wrote a blog post on infinity groupoids -- these are to infinity categories as sets are to ordinary categories. The goal of the blog post was to show, in as elementary a way as possible, what uses infinity groupoids have, to try and give readers a taste of why they've become so helpful in modern mathematics.

https://hidden-phenomena.com/articles/anima

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!

Does anyone know of works concerning number theory in fluid mechanics?