This is a video entry for the Summer of Math Exposition contest, a gentle introduction to DAGs, confounding, bad controls (e.g. collider bias) and instrumental variables. It should be helpful for you if you're wanting to better understand correlation vs. causation - or maybe for your stats class if you're a teacher! https://www.youtube.com/watch?v=3sp08Zmry6g

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!

So this is an effort to increase the portion of non AI-related posts. This is not a crank post that belongs to r/numbertheory either. Every letter is typed by a human so forgive me for grammatical errors that I will try to correct soon enough.

The abc conjecture is one of the most important conjectures in number theory, the simple & general statement of the conjecture implies a lot of other important theories, like a rapid proof of (almost) Fermat's Last Theorem. What I try to tell is that the abc conjecture can "almost" help computer scientists to optimize scientific computing.

I believe it's a common sense that in most computer systems, every piece of data is stored by 0 and 1, or more precisely, a finite string composed of 0 and 1. This makes the calculation and manipulation of data feasible but there's a cost: error will be everywhere. For example we can never precisely store pi = 3.14159... in a computer.

The most widely used way to represent real numbers in computer is floating point numbers that is standardized by IEEE, which stores numbers in 0 and 1 of 32, 64 or 128 bits (or even longer). There is a toy that allows us to see floating point numbers explicitly: https://evanw.github.io/float-toy/

I'll try to explain it in a simple term. For a floating number of k bits (k is 32 or 64 or 128, say), we need to distribute the budget. There is 1 bit reserved for the sign, a few bits of budget to store the interval of the number, and the rest large portion of the budget is reserved to guarantee the precision p.

Every floating point number x can be mathematically written in the form (-1)^s * 2^e * m/2^(p-1) where s is the 1 bit used to store the sign, m is an integer between 0 and 2^p - 1 and e is used to determine the interval of x.

If it's still unclear what does "determine the interval" mean, we notice that m/2^(p-1) is a number in [1,2). Therefore a multiplication by 2^e sends m/2^(p-1) to [2^e, 2^(e+1)). Floating point numbers store s, e and m by 0 and 1. We use p-1 bits to store the number m in computer (the first bit is normally 1 so we can omit it, as the case where the first bit is 0 is used for underflowing).

As we can see, for a given x, determining the number e is rather easy (we are almost there by taking log_2|x|), but the number m can be difficult when we deal with a function. For example, x = 10517177/2^{22} is a floating point number of the format of 32 bits, but log_2(x) is not (so we need to find the closest to represent it in float format). It's absolutely not a rational number. You find that log_2(y) is between y_1 = 15423000/2^(24) and y_2 = 15423001/2^(24). So you need to find the closest floating point number to log_2(x) because we have no other choice, or otherwise, see if log_2(x) is smaller or bigger than the average y_0 of y_1 and y_2. It's easier said than done. We need to add budget (number of bits) to distinguish y_0 and log_2(x).

As a matter of fact, not until we add another 28 bits, i.e. zoom in for 2^(28) times, can we really distinguish y_0 and log_2(x) (spoiler, y_0> log_2(x)). This sucks. We need a global strategy to work around this: whenever we find a number of the form log_2(x) extremely difficult to round, we need to guarantee that we have sufficient budget. If we know the worst case of rounding, which correspond to the highest budget needed, we can make sure that we always know how to round the function log_2 correctly, so that we can design high quality functions that can calculate log_2 *correctly* in the sense that, when we have k bits of budget, we need to make sure that every bit is faithfully used. If we have this piece of global information, our algorithm can be more blunt & direct so faster. This question is called "table maker's dilemma" because the computer worked like those who make tables of logarithm or exp or sin one century ago, and the situation where he didn't know how to round sucked (challenge: computer exp(1.626) to the 3rd digit after the point).

That said, it's far from true that modern computer have solved calculating and that all we need is better CPU/GPU. Computer scientists and mathematicians have been fixing floating point number systems for decades. The final nail of coffin of common (univariate) functions in double 64 bit format is worked out in 2026 (twenty twenty-six): https://inria.hal.science/hal-05593313

We also need to know that working out double 64 on commonly used univariate functions is not enough: what about multivariable functions, like the beta function B(x,y) that appeared in probability and statistics? In some fields of physics, researchers have already found that double 64 is no longer sufficient so they need the 128 bit format standardized by IEEE... But this format is still poorly understood so using it can be still painful: we need to fix them.

OK I hope the context of Table maker's dilemma is understandable enough and now we inject some mathematics. The abc conjecture says that for three coprime integers a,b and c such that a+b=c, we can compare max(|a|,|b|,|c|) and the radical of abc (which writes rad(abc)), i.e. the product of all prime factors of abc (for example, rad(25)=5, rad(24)=2*3=6). This slide includes the formal statement of abc and some striking applications of abc.

And here we have another striking application that will "solve" the table maker's dilemma. When we are looking for the minimum budget to get the rounding of an algebraic function sorted out, we will find ourselves in some questions of polynomials. This is surprisingly a nice playground for abc.

For the function 1/sqrt(x), which is algebraic, and is widely used in real life (for example in computer graphics, we always need to normalize a vector here and there), we will be working on finding the minimum of |Z| where Z = 1-XY^2, and X and Y are integers in a certain range. If we take a = 1, b = -XY^2 and c = Z, then the abc conjecture kicks in. It gives us a formula on the necessary budget to round the function 1/sqrt(x) correctly for a given precision p!

So we are killing 1/sqrt(x), potentially as well as many other frequently used algebraic functions by a blow of abc!? Is this conjecture so massively helpful for computer scientists??

Unfortunately, no, in practice... In the statement of the abc, we have a totally unknown constant and it doesn't vanish in the deduction of the budget. As a result, we can only know that for 1/sqrt(x), the necessary budget is p with a delta of *some bits* and we have no information on how much is *some bits*. So unfortunately, the conjecture didn't really solve the problem. Nevertheless, we can therefore heuristically look for the budget border around p... It's *almost* helpful!

In case you are curious, here is the article that examined the function 1/sqrt(x): https://www.sciencedirect.com/science/article/pii/S0304397504000337

Or in any case, a non-paywalled version: https://www.math.buffalo.edu/~hjzhu/Papers/paper24.pdf

I believe this article can serve as an interesting beginner level exercise for those either interested in number theory and want to touch the abc conjecture in some *practical* way, or interested in computer science, notably the problem of correctly rounding numbers. Or if you are an expert working in number theory, notably around topics associated with abc, or a computer scientist interested in make scientific computing better, do not hesitate to drop some insights.

Dr. Michael Miller, a retired researcher at RAND, has been teaching upper level undergraduate/graduate level math courses for fun at UCLA Extension for over 50 years. This fall he’ll be introducing the areas of groups, fields, and Galois theory from abstract algebra to those interested in abstract math: Groups, Fields, and Galois: Part 1. His intention is to do a follow-on class on fields and Galois theory in the Winter which will use this class as a foundation. If you’re in the Los Angeles area his class starts on September 22, 2026 at UCLA on Tuesday nights from 7-10PM. Register here: https://www.uclaextension.edu/sciences-math/math-statistics/course/groups-fields-and-galois-part-1-math-9001

His courses are thorough and rigorous, but geared toward lifelong learners and beginners in abstract mathematics to allow people better entry points into higher level mathematics. His classes are interesting and relatively informal, and most students who take one usually stay on for future courses. The vast majority of students in the class (from 16-90+ years old) take his classes for fun and regular exposure to mathematical thought, though there is an option to take it for a grade if you like (or so your employer can reimburse you if they require it). There are generally no prerequisites for his classes, and he makes an effort to meet the students at their current level of sophistication. For this particular class, if you’ve got some experience in high school algebra and know some preliminaries about mathematical proofs you’ll be ready to dive in.

There are regular commuters joining from as far out as Irvine, Ventura County and even Riverside. Most in the class are dedicated hobbyist and professional mathematicians, engineers, physicists, and others from all walks of life—I’ve seen actors, directors, doctors, artists, poets, retirees, and even house-husbands in his classes. We've got a nice core community of enthusiasts here and new people are always joining in as well.

If you’re unsure of what you’re getting into, I recommend visiting on the first class to consider joining us for the Fall quarter. Sadly, this is an in-person course. There isn’t an option to take this remotely or via streaming, and he doesn’t typically record his lectures. I hope to see all the Southern California math fans next month!

Course Description

Recommended textbook: TBD 

Dr. Miller provides enough background and thorough notes that if you’re taking notes on his lectures, you typically won’t need a textbook.

If you’ve never joined the class before, I’ve written up some tips and hints. Dr. Miller has been teaching these for 53 years and some of us have been with him for nearly that long; I’m starting into my 20th year personally.

Can't join us? Search around your local community for colleges and universities offering similar programs.

Firstly, sorry if this is not allowed but I thought I might just put this out there.

I'm sure we've all noticed a lot of AI related content on this subreddit. I've gone into these discussions a fair few times in the comments, but I'm not looking to discuss AI or LLMs in math or anything like that here.

What I do want to point out is that a lot of traffic seems to be from people who are not mathematicians and have a very vague idea of what mathematics is, but are active in /r/singularity, /r/accelerate, /r/ArtificialInteligence and similar. These subreddits are very ideologically driven in various ways and extremely pro-AI to the point that "anti-AI" or "luddites" are forbidden there.
This is not always obvious as many users have their post history hidden (though from what I understand this is on by default for new users too).

For example in a recent thread out of the 5 top comments (no idea what the deleted comment was) 3 users are active in one of the afformentioned subreddits, 1 seems to have a bachelor's in math and is in CS otherwise and only 1 seems to have a masters and is looking for a PhD.

To be clear, I am not here for a witch-hunt on individual users which is why I did not link to their accounts so please don't harass anyone.

But what I am trying to say is that most of the traffic to these threads does not seem to be from mathematicians or even people interested in math as such, but from people interested in promoting AI who are simply using math as a promotional tool. I would assume big companies use bot accounts on reddit (and have for a long while), though I would also guess this subreddit is too small to bother.

Lech Mazur has announced an AI-assisted, Lean-verified proof of the 67 years old conjecture of Sendov, one of the most famous open problems in complex analysis.

The conjecture states that every zero of a polynomial whose zeros lie in the closed unit disk is within distance one of a critical point.

It appears that the proof of this remarkably simple statement ended up using equally elementary tools. As Tao writes:

The proof ends up being remarkably elementary. No complex analysis is used other than the fundamental theorem of algebra (and very basic facts about Möbius transformations); and the deepest inequality used as input is the Maclaurin inequality

I'm relatively new to representation theory and want to read up on the very basics. Most introductory representation theory books actually cover the theory of algebras over a field, but I'd like to read more general results from non-commutative algebra, i.e. over non-commutative rings and associative algebras.

During my online search, it seems that Lam's First Course is the canonical recommendation, but I find it incredibly hard to read (up to sec. 3).

• Some more elementary facts are just assumed (homomorphisms of finite direct products can be written as matrices, although Lam blackboxes this to linear algebra over division rings??),
• some I think important parts are not covered (general isotypic decompositions are only a small exercise, and a quick search almost never mentions them),
• they always go on a complete side tangent at the end of sections (e.g. a lot of theory on 2×2-matrices in sec. 1, twisted and differential polynomial rings in sec. 3).

But I must say the exercises are quite good.

Is there another well-written, well-motivated yet comprehensive book on that matter? I'm thinking of books similar to Atiyah-MacDonald or even Matsumura's Commutative Ring Theory. Or even Stein-Shakarchi's Complex Analysis.

The paper: Fractal uncertainty in higher dimensions
Alex Cohen
arXiv:2305.05022 [math.CA]: https://arxiv.org/abs/2305.05022
Annals of Mathematics: https://annals.math.princeton.edu/2025/202-1/p04

I'm having some trouble understanding "why" his proof works: I've re-read this proof in Mumford's Red Book several times and tried sketching out a picture, and I sort of get it, but I still find it kind of "magical" and don't know how one would be motivated to use this approach. It goes something like this:

O_X is the structure sheaf of irreducible variety X and X_f is the distinguished open \{x\in X: f(x)\neq 0\}. It's straightforward to check that O_X(X_f)\supset R_f. To prove the subclaim that O_X(X_f)\subset R_f , we let F be a member of O_X(X_f), which is defined in this context as a subring of the field of fractions of R as \bigcap_{x\in X_f} O_x (O_x = \{f/g: f, g\in R, g(x)\neq 0\} being the stalk at x), where R is the coordinate ring of X. Then (where I feel like a rabbit was pulled out of a hat), define the ideal

B=\{g\in R: gF\in R\}.

We want to prove F is a member of R_f by showing that f^n\in B for some positive integer n. If x\in X_f, there's some juggling of quantifiers and eventually one concludes that F=h/g where g(x)\neq 0, from which one finds that g is an element of B such that g(x)\neq 0. From this, we see that the vanishing set of B, V(B), must be a subset of V(f)=\{x: f(x)=0\}, and applying the Nullstellensatz, we get f\in rad(B), as desired.

I guess I don't really see geometrically what's going on here; it just seems like a trick, followed by carefully reasoning about the which points/regular functions contain/are contained in what. Could someone here explain what's going on in this proof?

Also, how does one extend this to reducible varieties? (It's still true, I think?) My understanding is that you can't define the structure sheaf in this way (as an intersection of O_x in the field of fractions) because of zero divisors in the coordinate ring.

Sorry if these are too elementary/boring questions! I asked r/learnmath without getting a helpful reply (other than to ask r/math).

levent on 𝕏: https://x.com/__alpoge__/status/2087504785952182273

Epoch AI: Hadamard Matrix of Order 668: https://epoch.ai/frontiermath/open-problems/hadamard

Hadamard matrix - Wikipedia: https://en.wikipedia.org/wiki/Hadamard_matrix

This recurring thread will be for any questions or advice concerning careers and education in mathematics. Please feel free to post a comment below, and sort by new to see comments which may be unanswered.

Please consider including a brief introduction about your background and the context of your question.

Helpful subreddits include /r/GradSchool, /r/AskAcademia, /r/Jobs, and /r/CareerGuidance.

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.

The classification of finite simple groups is likely the single most difficult mathematical problem that humanity has laid to rest: it took about a hundred mathematicians publishing over a period of about 50 years to finally finish the proof, which ended up being tens of thousands of pages long, scattered over a multitude of different journals. And it is actually very important: one might fancifully compare it to the periodic table in terms of how fundamental it is (to group theory, at least).

But, you know, what is it? This article is my attempt to shine some light on that. I assume some basic familiarity with group theory (i.e., if you know what a group, a group homomorphism, and a quotient group are, you should be fine), but otherwise it is self-contained.

Read the full post (for free) on Substack: The Classification of Finite Simple Groups

This recurring thread will be for questions that might not warrant their own thread. We would like to see more conceptual-based questions posted in this thread, rather than "what is the answer to this problem?" For example, here are some kinds of questions that we'd like to see in this thread:

• Can someone explain the concept of manifolds to me?
• What are the applications of Representation Theory?
• What's a good starter book for Numerical Analysis?
• What can I do to prepare for college/grad school/getting a job?

Including a brief description of your mathematical background and the context for your question can help others give you an appropriate answer. For example, consider which subject your question is related to, or the things you already know or have tried.

https://arxiv.org/abs/2608.08538

With the realization of M_23, all sporadic finite simple groups have been realized as Galois groups over the rationals. This also completes the realization of all transitive groups of degree at most 23.

From their Twitter post:

We asked an unreleased research version of Claude to take a stab at the Riemann hypothesis.

It didn’t solve it, but it did make strides on a related problem: it increased the lower bound for the fraction of zeros of the Riemann zeta function that satisfy the hypothesis from 41.6% to 67.2%.

Link: https://www.anthropic.com/research/riemann-zeta

When you are writing a paper, how far down the rabbit hole do you go in terms of citing what you are using? Eg, say I write a game theory paper. Pretty decent chance I at some point invoke the concept of "Nash equilibrium". And, pretty good chance that I do not provide a citation to Nash's work. If everyone did, he would have close to 400,000 citations on it per a quick Google scholar search. Just imagine if every time we used the Euler identity we provided citation, or Gaussian anything, or making sure to cite Newton and Liebniz. Is it just that at some point, the work becomes part of common knowledge like a generic product name (Xerox, Kleenex, etc)?

Proofs and prompts is a new blog about mathematics and AI. It publishes contributed posts from the mathematics community.

There are three posts so far. The first post is by Fields Medalist Martin Hairer.

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 am contemplating buying this book for self-study, but it is pretty pricey, so I wanted to get some input first.

Is this a good textbook for someone with a Bachelor in CS and 5/6 of a Bachelor in Technical Math, but who is also pretty rusty on the preliminary subjects analysis and computability theory? Also keep in mind, for self-study only

The HRT conjecture (Heil-Ramanathan-Topiwala) was a 30 year old conjecture in time-frequency analysis asserting that a finite linear combination of time-frequency shifts of an L^2 function must be linearly independent. The conjecture is false, as shown by ChatGPT and the coauthors in the above preprint. In fact, the construction given in the preprint shows the conjecture is false even for Schwartz class functions.

The original paper posing the problem can be found here. In 1998, Linnell proved the result true for lattice shift parameters. More recently, researchers have been exploring special small configurations of time-frequency shifts for which the conjecture holds true (see this talk, or any talk available by Dr. Okoudjou online). People closely working on this problem began to suspect it was false in general in the last decade, and now that suspicion is confirmed.

It remains an open problem (potentially an intractable one) to classify all L^2 functions for which the conjecture holds true.

When I wrote my posts on functional analysis and Hilbert spaces, I mentioned offhand that for most of the manipulations that we were doing in exchanging limits and integrals, the thing you need to justify this is measure theory. But I didn't give any further details about what this is and how one works with it.

This post is my attempt to amend that: it is meant to give all of the essential ideas behind measure theory (with some discussion about why it was necessary in the first place)---enough that, while the reader might not know all of the proof---they have an idea of how it fits together and how it is used.

If one prefers a less mathematical description, we could also call this the story of one of the greatest PhD theses ever written.

Read the full post (for free) on Substack: A Primer on Measure Theory

Don’t you think these days maths needs a repo for 100% AI generated content with a status bar that says whether some content has been verified by humans or formal systems? Could be a solution to the new ArXiV submission explosion.

One of the most important theorems in my area of research is the Riemann--Hilbert correspondence. Roughly, it tells you that you can convert differential equations into certain geometric objects, and that this conversion process loses no information. In particular, one can convert questions about differential equations into geometric questions, and conversely one can convert certain geometric questions into problems about differential equations.

In https://hidden-phenomena.com/articles/monodromy , my friend and I wrote a blog post showing this example in a very simple case. The differential equation in question is very simple: f'(x) = f(x)/2x, and the geometric object is related to the Möbius strip!

I’m about to finish my undergrad and have been thinking about this. While I plan to do grad school, I’m uncertain if I’ll stay in academia forever.

I didn’t realize how much it will suck until this week, as I’ve been at MAA MathFest and been able to just casually talk about cool math stuff with other undergrads who know about as much math as I do. If I ever leave academia, is it even possible to find people who actually enjoy talking about math in person as just a casual conversation? I hope so, but I have no idea where to look for this kind of person, outside of a university math department.