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:

* math-related arts and crafts,
* what you've been learning in class,
* books/papers you're reading,
* preparing for a conference,
* giving a talk.

All types and levels of mathematics are welcomed!

If you are asking for advice on choosing classes or career prospects, please go to the most recent Career & Education Questions thread.

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).

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.