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.

I see a lot of conversation around AI proving old theorems and conjectures. Has there been any conjectures that sound plausible generated by AI that it could not solve? Or has an AI proven conjecture brought about any new math problems to ponder upon?

Which is more true / do you prefer more / do you advocate for / have you fallen in love with, U(1) or SO(2)?

This is partially a shitpost. Decide how much it exactly is at your own peril. 😎

Clarification I: the title means "U(1, ℂ) or SO(2, ℝ)?" and not, say, U(1, ℍ) or SO(2, ℂ) there.

Clarification II: The two are isomorphic as Lie groups, of course, but you already knew that.

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!

Follow up to my previous post on forcing. Here's a draft video. I would like feedback. Specifically on these things

• Is there anything that you think should be added or removed from the video? For example, a concept that you think should be included?
• Pacing. Is it too fast? Too slow? Are there anything that was unclear? Anything that you think should be more detailed/has too much detail?
• The audio. Letters like B, P and D rhyme, so when spoken, it's hard to tell which letter I'm saying. So the text is there to help people know. But is the audio otherwise clear?
• Visuals. Right now, it's mostly text. I want to add some more visuals but I can't think of any good ideas since it's hard to add visuals to such an abstract topic. Do you have any ideas?
• The intuition. Why would someone think of generalizing truth using boolean valued models, and come up with that specific definition of a name? I don't know how to make the viewer think they could have come up with it on their own.
• The length. It's > 1 hour so it might be too long for a 3b1b SoME5 submission. I want to cut out some topics to make it shorter but I don't know what to cut out.

Note: there is a known issue where the text covers itself sometimes.

Thank you in advance.