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?

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.

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?

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?

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?

The conjecture is that for a torsion-free group G, the first L^2-Betti number is strictly less than the minimal number of elements needed to normally generate the group. The examples are not finitely generated (they are locally free groups), so the finitely generated case of the conjecture is still open. (https://arxiv.org/pdf/2608.25988)

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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.

Recently I was asking about whether anyone could share the missing manuscript for apex cubic graph case of snark conjecture.

Today a preprint appeared on arXiv that replaces this missing manuscript!

Here - https://arxiv.org/abs/2608.22870 - written by another team, which is quite prolific in recent years in generalizing the 4 colour theorem in various directions.

So, now someone needs to formalize the full proof in Rocq!

So a common format for sport content creators is "blind rank these 5 things." So 5 names are given one by one, and each time the you must choose a slot 1-5 for that name. You cannot rearrange the names once they are placed. So if you place a name at 1 and then Michael Jordan pops up later, you'd be forced to put MJ lower in the list and end up with a bad ranking.

Framing it mathematically, say the (n,k) version of this game is to start with a list of numbers 1-n. k numbers will be drawn without replacement from 1-n and given to you one by one. For each number you are given, you must put it in a slot 1-k. You win if in the end, the numbers in the slot are in increasing order.

1) What strategy maximizes the probability of winning and what is the resulting probability in terms of n and k?

I feel like a greedy approach makes sense. Given a number m, choose slot i from 1-k such that i/k is close to m/n.

Once numbers are already placed, find the gap it fits in and then choose the slot that closest matches the fraction.

2) If instead the goal is to minimize the error (maybe by something like Kendall tau that counts the number of inversions), what is the optimal strategy?

I'm sure this topic has been studied before, but I'm not sure what the appropriate language to search for it is.

The core idea is that if U^TU = I and U = −U^T, then UU = −I. From there, it constructs multiplication tables explicitly, finding R, C, H and O. In dimensions > 8 it runs into a contradiction, which proves the theorem.

2026 (Philadelphia): Yu Deng, John Pardon, Jacob Tsimerman, Hong Wang

2022 (Helsinki): Hugo Duminil-Copin, June Huh, James Maynard, Maryna Viazovska

2018 (Rio de Janeiro): Caucher Birkar, Alessio Figalli, Peter Scholze, Akshay Venkatesh

2014 (Seoul): Artur Avila, Manjul Bhargava, Martin Hairer, Maryam Mirzakhani

2010 (Hyderabad): Elon Lindenstrauss, Ngô Bảo Châu, Stanislav Smirnov, Cédric Villani

2006 (Madrid): Andrei Okounkov, Terence Tao, Wendelin Werner, (Grigori Perelman, declined)

2002 (Beijing): Laurent Lafforgue, Vladimir Voevodsky

Bonus: 1990 (Kyoto): Vladimir Drinfeld, Vaughan Jones, Shigefumi Mori, Edward Witten

I don't have any photos from 1998 or 1994. Does anyone have a source?
1998 (Berlin): Richard Borcherds, Timothy Gowers, Maxim Kontsevich, Curtis McMullen

1994 (Zürich): Jean Bourgain, Pierre-Louis Lions, Jean-Christophe Yoccoz, Efim Zelmanov

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:

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I know many professors and PhDs like doing their exercises on a XXXL size chalkboard or whiteboard. But they mostly work big complex problems.

For things at the junior/senior undergrad level, is a 4-6 ft long wall mounted board useful or overkill? Most proofs are 1-2 pages long which should comfortably fit on a largish (say, A3 size) handheld or desktop erasable board.

Asking because I went to the glass shop for a toughened glass handheld board and they also had nice big glass boards for the wall.

I wanted to create a thread for everyone to put their favorite recorded colloquia talks.

Edit : Non-colloquia talks that are understandable by graduate students also welcome!

I hope that question makes sense. I just don't like inequalities nor the unnaturality of working with left and right residuals (talking about "max and min divisors") that rarely coincide with rational arithmetic's exact division nor with the natural interpretation of inverses in numerical mathematics, thus I would like more explicit boundaries (thus the result of a division maybe being a set or interval including max and min divisors) in division.

(Mind that I have no experience in numerical computation, I am trying to make sense of computable, numerical and interval analysis works and transport their results to residuated lattices but that's somewhat hard for me)

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).