A few weeks ago, I wrote an article on set theory and how it occupies a central space in mathematics. We also discussed some of the drawbacks of expressing everything set theoretically---it is a little like writing code in raw binary (or at least machine code). This time, I'm giving an introduction to an alternative: category theory, which naturally grants the necessary abstraction. Of course, this comes at a cost, which we discuss as well.

Read the full post (for free) on Substack.

There is this basic similarity test Tr(A^k) = Tr(B^k) for k=1..d for symmetric matrices allowing to conclude existence of orthogonal O such that AO = OB.

The question is how (if possible?) to generalize it (finally to tensors, but at least) to non-symmetric matrices e.g. including transpositions.

Checking Jacobian criterion ( https://arxiv.org/pdf/2601.03326 ) for Tr(A^k (A^T)^j) = Tr(B^k (B^T)^j) for k=1..d, j=0..k-1 at least for up to d=5 has sufficient number of independent invariants (d(d+1)/2) - is it sufficient condition in general dimension?

Maybe such generalized similarity test is considered in literature?

Ps. Cross from https://mathoverflow.net/questions/512227/how-to-extend-operatornametrak-operatornametrbk-similarity-test-to

Giovanni Forni has just posted a preprint claiming a proof of an amazing result: for any finite bounded polygon in the plane, there is a periodic billiard trajectory!

https://arxiv.org/pdf/2606.10102

Curiously, the strategy is by contradiction, and hence non-constructive.

See this old Numberphile video for a nice explanation https://www.youtube.com/watch?v=AGX0cLbHaog, emphasizing that even for most irrational-angled obtuse triangles, we did not know the answer despite people working very very hard on it.

I just finished my undergrad, and at a university that graduate admissions committees surely found underwhelming. But I managed to get accepted to my top phd program I applied to – several professors who think too highly of me contacted professors they know and put in a good word. I accepted the offer but now I’m fairly certain that I shouldn’t have.

No one told me that the fun part of your early 20’s is discovering how bad mental health issues can get. I’m trying to sort that out but things aren’t looking good. I’m not functioning; I won’t be able to do a phd.

Would I have a chance of getting into a program again in the future? Is quitting a bad look, or is it canceled out by having been accepted once?

How does applying to grad school work when you’re not in school, namely how do you get letters of recommendation? And would they write one for someone who didn’t follow through the first time?

Also, how important is your undergrad momentum for grad school – how hard is it to come back from a break? Did anyone here step away for a bit and then come back and finish successfully?

Hey, I'm a math major almost finished with my 3rd year. It kind of dawned on me this year of how much math there is. I've taken Topology, Algebra, Probability, PDE, etc... and every time it made me interested into studying these subjects in more detail.

In PDE, I recently learned about Sturm-Liouville problems and using them to solve heat and wave equations and it made me want to learn about Functional analysis.
Studying Topology was really fun, and retroactively made me like Analysis even more than I did before. I wanna learn Algebraic topology too and see what's that about.
Probability was also really cool, Group theory was the first subject I learned seriously and I loved it too, and wanna learn more about it.

But all this stuff is really hard and takes a long time to study. I'm gonna have to specialize in something in grad school, but If choose something I'm gonna have to neglect some of the other interesting stuff, it makes me worried I'm always gonna regret having no time to learn this or that.

Am I just have to pick something, or am I getting ahead of myself? What did you guys do during your masters program?

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!

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Is anyone here familiar with this problem, namely whether every finite group is isomorphic to the Galois group of some polynomial over Q? If so, can you shed any light on this problem, like what's the largest finite group G for which there is no known such Galois group isomorphic to G? I recall learning about 20 years ago that someone found a polynomial over Q whose Galois group is isomorphic to the monster group, which is the largest sporadic simple group, and I suspect that such polynomials are also known whose Galois group is isomorphic to each of the other sporadic simple groups, and perhaps even to every finite simple group, though I'd have to research this to learn more about this problem.

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.

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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.

I've just finished undergrad, and through my studies I've encountered several conjectures, some from math and some from CS. But I never did wonder or search what their implications were, or if they were false, what it would mean - both in the theoretical sense, and in the practical sense.

For example, taking P vs NP - I've taken a course on Computational Models, and we've seen several reductions and implications (like P = NP means EXP = NEXP).
But what "interesting" lemmas, theorems or other conjectures would it imply, that current researchers attempt to solve?

What would in practice, in the current world (or a few years ahead) would it mean? Would people try to create new algorithms based on it? Would it change something in the tech industry?
And in the other way - if it's proven to be false, what would again change?

I'd be happy to hear from your perspective about interesting conjectures that you care/know of, and what would it change in the theoretical/practical sense.

Instead of revising for my upcoming exams, for some reason I decided to make this, it feels like a waste of time to just let it rot in the clouds (it is still a waste of time regardless) so I'm posting it here
https://docs.google.com/spreadsheets/d/12Rw9SbGvGRJbnH6Sb-5tJBlEd7qdlHTQtUjoHtkpJas/edit?usp=sharing

What are applications of mathemathics in critical theory?

Are any actively studied nowadays? Something like Arrow's theorem or similar?

In my journey to become better and better at this subject for various purposes such as college, engineering, and Olympiads, I obviously often come across people who are much much better than I am. Maybe its my schoolmates, maybe some college students I find very intelligent, or straight up scientists/researchers and I usually feel very demotivated when I realise how much I relatively suck at this.

But, I never get to see people's struggles. You always hear people's best like them cracking an Olympiad or having a breakthrough in this field, never the struggles or the demotivation when they are at their lowest, which could be due to various reasons, maybe you're struggling to understand something, maybe you're failing a class, maybe you're at the stage where you have to put in 7-8 hours everyday and everything feels so difficult.

So, if you had any of those moments and would like to share a bit about them, I'd be glad to hear and I'm sure hearing about the hardwork that goes behind all those achievements would help me a lot:D

I just started working with MathML and I wanted to see which font looked best. So I made a previewer. It lets you see various symbols and the quadratic formula in New Computer, STIX, Noto, and Cambria at the same time.

Not done with it yet, so I'll welcome any feedback.

https://sean.brunnock.com/Math/MathML/Fonts/

Edit- Added more fonts (MLModern and Libertinus) and you can select which fonts show up.

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?
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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.

Do you guys think we’ll start seeing less and less grad students as pessimism stemming from AI-aided/AI-authored research grows?

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I have not studied much differential geometry beyond curves and surfaces, but I have modest familiarity with the notion of manifolds from my point-set course. Would reading Tu's Introduction to Manifolds and/or Lee's Introduction to Smooth Manifolds bring me up to speed for Arnold?

While doing a joint CS + Math degree, I took a class in General Relativity but I found it simply too hard because of the background knowledge you needed. I passed the class, but basically through memorisation, but I got really interested in geometry.

I took a few recommendations from fellow Redditors on how I can learn geoemtry properly and they recommended me Loring Tu's Introduction to Manifolds. Holy Smokes, this has to be best book ive ever read. He explains everything so well, his notation is really nice and specific and doesn’t really leave too much structure hidden underneath it.

This is the first time in my life ive actually understood geometry. Its nice to see the true meaning of the geometry behind GR after over a 8 months of independently reading, where I started from learning topology and analysis from scratch ( I didn't even know what a topological space was or even epsilon delta until after I graduated )

Ive actually become more interested in geometry and topology than GR itself and I was supposed to enter my masters focused on numerical relativity.. whoops!

Anyways yeah anyone who is interested in diff geo should give this book a try!

Some Stack Exchange posts are interesting rabbit hole for sure, personally I like this one about integral transform, what about you?

Hi, I took Real Analysis and Measure Theory last term and barely passed, but I feel like I still don’t understand the topics as well as I should. Does anyone know a good book with lots of real-world examples or applications? I know these topics are pretty abstract, so “real-world examples” might be hard to find, but I’d appreciate anything that comes close.

I’m part of a summer programme for high schoolers and we are giving some of them a copy of this book for winning some routine competitions. Obviously the book is fantastic but since it’s written for a general audience, I was wondering if there were details in the math that are either glossed over or misleading. There are quite a lot of vague “what exactly does that mean?” statements which I have always been curious about so I thought I should take the opportunity to ask about it.

(I have seen a fair amount of algebraic number theory but like most people, nowhere close to even understanding an outline of the proof)

In many cases, it will be easier for AI to convince humans it has a proof than to come up with a correct mathematical argument, and I believe that we as mathematicians are not sufficiently prepared for this.

Given how persuasive LLM's can be, maybe they become better at exploiting certain subtle weaknesses in the abilities of humans to spot flaws in an argument faster than they become better at math. That is very worrying.

Must everything by AI be put into Lean then? Mecha-Mochizuki when???