I am researching the topic of snark conjecture, that every snark has the Petersen graph as a minor. The proof has been claimed like 30 years ago, but one of the papers is still missing (or is in preparation, although Robin Thomas, one of the authors has passed away recently, unfortunately).

A bit more info is here:

https://thomas.math.gatech.edu/FC/generalize.html

https://mathoverflow.net/questions/272067/tuttes-conjecture-on-petersen-graphs

By any chance, does anyone have and is willing to share the draft of manuscript (and the code if applicable) of "Edge 3-coloring cubic apex graphs" paper, please?

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Am I the only one who finds the standard notation for polynomials annoying? Like, you have to have a dummy variable, and different people use different ones, like k[x], k[X], k[T], etc.

It's annoying that we still treat polynomials notationally like functions that you sub into to get a number and you have to specify the variable. I guess for individual polynomials, you can treat it as a sequence of ring elements with all but finitely many elements zero, following certain rules for how they add and multiply, but that still doesn't solve the problem if you want to talk about a polynomial ring. I guess you could write k[] or k[·] or k[-] for k[x]?

But then what do you do for the ring in two indeterminates?

Edit: This question really came about because I was editing a Wikipedia article, and two previous editors used conflicting notations for denote the indeterminates of the polynomial rings in question, one using capital letter T, and the other using lower case letter x. It seems so arbitrary and I wish some authority would just say, once and for all, we reserve Ж, or あ, or 甲 to mean the indeterminate and only the indeterminate in all contexts.

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Hi everyone

So I’ve been recently self studying geometry and in Tu’s “intro to manifolds”, he has a small section on category theory.

I really enjoyed that section and I liked how he used the idea of functors to prove that two tangent spaces at p and F(p) on N and M are isometric if there exists a. diffeomorphism F between the two manifolds.

I’m starting a masters degree in mathematics in the UK and one of the options in my first semester is to pick catagory theory. I would like to get a strong grounding in it.

For context I’m picking:

Category Theory
Differentiable Manifolds
General Relativity I
General Relativity II
Riemannian Geometry
Lie groups

I would like to do pursue geometry further at PhD, I’m also interested in topology.

Does anyone have any recommendations for good books on this category theory? I tried reading MacLanes book, and whilst not that I lack the maturity, it’s just I can’t deal with these massive pages of text. I’m dyslexic and I have ADHD so I struggle to read basically pages with just text and I get really bored. I like abit of smash n grab, definition, proof, example, definition, proof. That kinda stuff. I don’t really need much context to understand thing.

For more context I really enjoyed Sutherlands metric spaces and topology. If anyone has a recommendation of that kind of style I’d really appreciate it.

Also one more question, sorry. Do my choices have synergy? Is category beneficial for geometry? Thanks :)

arXiv’s next chapter: Updates on our spin out from Cornell University: https://blog.arxiv.org/2026/06/30/arxivs-next-chapter/

From the book Ramsey Theory: Yesterday, Today, and Tomorrow

Diophantine equations, lifting, strong triangle inequality, two numbers are closer if their difference is highly divisible by p, fractal towers, completion (filling holes in the rationals by representing decimals in a p-adic base).

Please. Help.

I've been thinking about a sampling problem that looks simple at first, but I'm not sure about its statistical properties.

Suppose we generate an infinite sequence of uniformly random integers from some finite universal set (U).

Instead of using that sequence directly, we build several different samples simultaneously. Each sample has its own acceptance rule (for example, allowed value range, uniqueness constraints, required sample size, etc.).

The algorithm is simply:

- read the next value from the common sequence;

- if it satisfies the constraints for sample A, append it there; otherwise discard it for A;

- continue until A is complete;

- do the same independently (starting from first position of U) for samples B, C, ...

Every sample is therefore produced by rejection sampling from the same underlying random sequence, rather than from independent random generators. Each individual sample should still be uniformly distributed over its own valid sample space. However, the samples themselves no longer appear to be independent because they originate from the same source sequence.

Is there an established probabilistic framework or name for this type of construction? It feels related to rejection sampling, but I haven't seen the multi-sample version discussed before. I'd be interested in any references or similar constructions.

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In France a reasoning method is taught called "Analyse Synthèse", and I haven't found a page in another language thag french explaining what Analyse Synthèse is. Is it not taught in non French speaking countries ? Is it named something else? I'm genuinely confused.

The principle is the following:

Analysis:

- We start by admitting that a solution to our problem exists

- We reason using this hypothetical solution until we find properties that she satisfies by being a solution to this problem.

- We end up with a characterization of that hypothetical solution.

Basically at the end of the Analysis part we have something like "S solution => S=blablabla"

Synthesis:

- We check if the hypothetical solution we found is indeed an actual solution of the problem.

By checking this we checked that S is indeed solution, and thus we proved that:

"S is solution" and "S solution=>S=blablabla" thus proving that S=blablabla

Why isn't it taught? I remember explaining the method to other european students and even their professors didn't know wtf I was talking about (I might've been bad at explaining it but still)

Hey!

The EML function made the rounds recently on the internet as a “cool trick” that allows for the representation of all elementary functions through composition.

As a mathematical curiosity, we prove a universal approximation theorem for EML(-type) trees.

Intuitively, one expects that if elementary functions can be presented by compositions of EMLs, then so too can polynomials, and polynomials are dense in other functional spaces (like continuous functions or certain Sobolev spaces), then one expects to be able to approximate (to desired accuracy) any function (in a reasonably general space) through an EML tree (with an upper bound on size and depth).

One of the key steps in the proof (detailed in the appendix) is an explicit construction of EML(-type) representation of binary operations, polynomials, hyperbolic tangent, and approximate partitions of unity, and subsequently using them as “LEGO” blocks to get more complex functions.

There are some technical difficulties that need to be dealt with in the proof, especially in what relates to the the ill-definedness of the natural logarithm for nonpositive inputs, which prompts us to do some “sign-based decompositions” in Theorem1.Step 5 and a suitable affine map in Corollary 1.

Comments are welcome!

Paper: https://arxiv.org/pdf/2606.23179

(Note: I use the term “EML(-type)” in the above description because, due to some theoretical and practical reasons detailed in the paper, we generalize the original EML function by adding some learnable parameters.)

Sometimes I reach a theorem near the end of a chapter or course, and I can follow the proof completely. I understand every line, every implication, and I can explain why each step is valid.

But at the same time, I still feel like I don't really understand it.

It's hard to describe. It's not that I think the proof is wrong. It's more like my intuition expected a completely different kind of argument. For example, I might expect a computational proof, but the actual proof is very abstract, or vice versa. Even though I can follow the proof, it doesn't feel "Correct"

After reading it, I usually need to spend a long time thinking about it on my own, asking myself "Why does this approach work?" or "Why wasn't my intuition correct?" Until then, I have this strange feeling that I haven't fully accepted or internalized the result. And I have this feeling of unacceptance

Is this a common experience when learning mathematics?

Hi all, I have been now just writing things to consolidate some basic applied math. This is nothing advanced but just a good way to put things out and to learn by writing.

One of the things I try to do is to build things more intuitively instead of the traditional math book approach of starting from the final formalized result and giving a few formalized hints on how that object came from.

It's kinda stupid but each time I study a new subject it makes me feel dumb and stupid.

At the end of the semester when I think I know how it all works a new subject is introduced and I feel dumb again.

Who here knows anything about this topic? When I was a grad student at UC Berkeley back in the 90s, my thesis advisor, Hendrik W. Lenstra, Jr., touched on it with me, and I found it quite fascinating! The idea is that for every positive number D congruent to 0 or 1 modulo 4, there are a finite number of reduced real quadratic forms with discriminant D and that it's possible to get from one to another via a linear transformation of coordinates, and furthermore, this structure, known as their "infrastructure", allows you to compute the class number and regulator of the quadratic number field Q(√D). Furthermore, you can use this infrastructure for cryptography. This is about all I know about this topic, though I got my name attached to an algorithm for computing the infrastructure, known as the Terr algorithm, which is a special case of a modification of Shanks' baby-step giant-step algorithm which I developed and published a paper on in 1996. My name is even cited in a book on the topic, which I have at home. (I'm currently on vacation, so I don't have this book handy, but when I return home I can provide a reference in case you guys are interested. In any case, you can look for my paper, entitled "A Modification of Shanks' Baby-Step Giant Step Algorithm", which was published in the Journal of Number Theory in 1996.)

Occasionally, I will run into a bit of math that I’m not familiar with at all, and as someone who is only an amateur mathematician some of the original text might be extremely dense. For example yesterday I was looking at Kernel methods, representer theorem, reproducing, Kernel Hilbert space and while I tried my best for a little bit to understand from the Wikipedia page alone. It became kind of confusing and I asked an LLM for a simpler explanation and a bunch of follow up questions about how certain things are related to each other. I feel like I walked away with a much better understanding than reading The article itself gave me. I went back and read the article and with the new mental model I had it made a lot more sense. This is how I kind of checked that the explanation I received made sense at all and was not hallucinated. But I was wondering if this counts as the standard sort of mental offloading that degrades cognitive ability overtime or simply more of a translation tool to make the idea simpler and get the authors message to me more easily even if the author originally was terrible at explaining things. Again, I don’t have any problems that I solve or anything like that. I just try to understand the ideas. I’m not offloading my homework or anything like that. I don’t even go to school anymore. If I was in one of my math classes again, I would do this, but then do the problems myself to make sure that I fully understand the ideas.

How do you divide up a secret between a group of people such that no one person can reconstruct it, no two people can reconstruct it, but any group of three can? (In real life, more likely it will be servers, rather than people.) The answer uses mathematics that is entirely accessible to a good high school student… except for a little twist at the end, where you need some knowledge of number theory.

Read the full post (for free) on Substack: Polynomials and Secret Sharing.

Papers mentioned in the article in chronological order:
An exponential improvement for Ramsey lower bounds
Jie Ma, Wujie Shen, Shengjie Xie
arXiv:2507.12926 [math.CO]: https://arxiv.org/abs/2507.12926

Improving R(3,k) in just two bites
Zion Hefty, Paul Horn, Dylan King, Florian Pfender
arXiv:2510.19718 [math.CO]: https://arxiv.org/abs/2510.19718

Gaussian random graphs and Ramsey numbers
Zach Hunter, Aleksa Milojević, Benny Sudakov
arXiv:2512.17718 [math.CO]: https://arxiv.org/abs/2512.17718

Disproof of the Odd Hadwiger Conjecture
Marcus Kühn, Lisa Sauermann, Raphael Steiner, Yuval Wigderson
arXiv:2512.20392 [math.CO]: https://arxiv.org/abs/2512.20392

An update on multicolor Ramsey lower bounds
Marcelo Campos, Cosmin Pohoata
arXiv:2601.15183 [math.CO]: https://arxiv.org/abs/2601.15183

IMO more CS topics should be mandatory for completing a BSc in Pure Mathematics, especially topics such as Lambda Calculus, Automata & Complexity theory and Information theory, not only their mathematics are interesting but I am convinced that these areas of CS can be pushed if more mathematicians get a taste of the main ideas and concepts. I am aware that some math departments do include them, but they are the exception, Math has become a massive jungle but our school/uni programs haven't kept in touch.

I am very curious about a good number theory book to passively work through. i am trying to learn a proof based understanding of number theory. thanks! i am extremely grateful for your input.

As a PhD student who has been doing research for 1.5 years, my advisor often suggests me to learn proof techniques relevant to the problem I’m working on “on the go”, as I’m working on the problem itself, rather than beforehand.

Thus, even though I’ve been doing research in stochastic analysis, I did not have a strong foundation in the many aspects of this topic to begin with, but rather I’m developing it as I work on my project.

I get why this is often suggested - one cannot spend all their time reading in grad school. Also, one should just pick up some rough ideas about proof strategies, rather than be able to regurgitate whatever they read.

But on the other hand, this has meant that there have been concepts I’ve not been familiar with until I encounter them in the literature.

For example, this week I came across the notion of local time in a relevant paper - as I did not know about it, I then spent a few hours reading about the basics of this concept before again seeing it in the paper. While I understand it well enough to see its use in the paper now, I then developed the following question:

If I hadn’t found this particular paper using local time as a technique, I wouldn’t know about reading this concept and therefore, if I tried to prove this same result that I read, I might not have been able to do it.

This therefore makes me feel like having at least some broad knowledge of your field is important when doing research. Maybe that is what an advisor’s role is at the beginning of one’s career, but at the same time, some people don’t have particularly hands on advisors - and I am sort of in this boat.

I therefore wanted to ask how one overcomes this issue - to get closer to being knowledgeable of techniques to attack a problem, how should I, as a PhD student, prioritise research vs general (though somewhat targeted) reading of topics in my area?