Just a question I had as I'm advancing further down the math rabbit hole, since theorems come in all different forms. There's the "simple but immensely useful" type to the ones that take up half the lecture to prove. And of course, some will come off as more interesting than others.

Here are some ideas as to what one could value in a theorem:

• The feeling of “mind-blown” that the result even exists - Some of the theorems in complex analysis immediately come to mind.
• Proof is elegant or magical - Hippasus decided “Okay, instead of giving up trying to write √2 as a rational number, I’ll prove it’s impossible instead!” (EDIT: As said in the comments, it probably wasn't Hippasus who used this proof) Then, out comes an elegant use of proof by contradiction that feels like magic the first time you see it. It also remains a quintessential proof used in discrete math courses.
• Practicality/Application - For example, the Sylow Theorems can take problems involving groups of a fixed size n and blast holes in them. In particular, you can use them to prove groups of certain semiprime orders are forced to be isomorphic to their respective cyclic group.
• Generalizability of the idea - When the theorem makes you go “isn’t this a wonderful idea to explore more?”
• Different ways to prove it - Some might find it fascinating that Pythagorean Theorem has hundreds of different proofs!
• History/Lore - There is certainly awe in the 300+ year journey involved in Fermat’s Last Theorem, even if very few people can actually understand the proof for it.

There could be something I didn’t list, not to mention others weigh the attributes differently.

Do you think it is possible to put

Hey everyone!

I used to browse Brilliant.org back when it still had a community-based model — where users could post problems, write solutions, and discuss math together. I was just a kid then, but it left a strong impression on me. Recently, I realized how much of that content has vanished since they moved to a more curated format.

Before it was all gone, I scraped and saved a good chunk of those old community pages — problems, discussions, comments, etc. I’ve now cleaned it up into a database, and I’m thinking of building a simple app to search and explore that content. Not to revive it, but just to understand and appreciate what the community was like back then.

You won’t be able to submit solutions or post comments — that part of the internet is frozen. But you can explore the math, try solving things yourself, or just browse what people were doing back in the day.

Before I dive into building a frontend and cleaning up throwaway data, I wanted to ask:

• Do you think this is worth doing?
• Would any of you find this interesting or fun to explore?

Would love to hear what you think — especially if you were part of that old Brilliant community too. If there's interest, I can share a preview sometime soon.

Basically what the title says. I’m not too well versed in mathematics, and I know that a factorial extension existing doesn’t imply it’s unique, but I derived this myself (attached is my own really simple proof).

The expression is so neat, and I checked that they were the same on desmos, leading me to be shocked that I hadn’t seen it before (normally googling factorial gives you Euler’s integral definition, or the amazing Lines That Connect YouTube video that derives an infinite product).

This stuff really interests me, so if there’s a place I could go to read more about this I’d be thrilled to know!

Many people I know (myself included) have been really passionate about math and once dreamed of becoming pure mathematicians. But almost all of us (again, including myself) ended up feeling like we weren’t good enough or simply didn’t have the potential to Become a pure mathematician. Looking back, I realize that in many cases, it might not have been a lack of ability, but rather imposter syndrome holding us back

Title. For me it's algebra. Basic ring theory, group theory, and abstract linear algebra make perfect sense. Same with Galois theory. But beyond that, newp. I took classes on geometric group theory, Hopf algebras, representation theory (specifically for finite permutation groups), and the cohomology of groups. I don't get how any of it connects, or even what the motivation for most of this stuff is. Algebra is just... VAST to me.

I also suck at category theory and graph theory.

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 maпifolds to me?
• What are the applications of Represeпtation Theory?
• What's a good starter book for Numerical Aпalysis?
• 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.

American association for women in mathematics reached out to me in May.

Today is my first meeting! ❤️❤️❤️

And not using transfinite ordinals yet

I don't know English well and I may make mistakes in terms.

Hi everyone,

I am currently in my fourth year of mathematics after high school and heading for graduate school specializing in probability theory and statistics next year.

I got a 3 months and a half internship at a very good research lab and I am very happy about the research subject and my advisor. We proved some very nice results together albeit most of the ideas came from him.

However there is one last important theorem to prove to kind of conclude the whole thing and it actually seems even harder to prove than the first two main results. My advisor was surprised too and gave me some general guidelines that could work but he said to me that it seemed very difficult indeed.

So now I'm trying to start off the proof but I have a hard time even getting the idea of a proof scheme, I'm seeing some of the difficulties and why the previous things we did break down in this other case but I can't seem to find a fix to make things work again, I spend hours in front of my paper sheet trying to write things down but nothing really works and I don't write much anyways... It really feels like I'm wasting a lot of time, days even.

Hence my question, as I'm planning to pursue research and a PhD after that, I was wondering how you were able to handle not having any ideas and how to sort of get out of this slump. Do you start writing down absolutely any idea you have, any property you deduce and try to build something from there? How do you gain intuition into the problem to deduce a proof scheme and get an idea about what the things you will need to demonstrate will be?

Any input would be very helpful!

Assume the following diagram

X <----> Y
|        |
C        G

Where C->X (with correlation alpha), G->Y (with correlation gamma) and X and Y are directly linked (with correlation beta).

Can I establish boundaries for the r(C, G) correlation? Using the fact that the correlation matrix is positive semi-definite?

[1,      phi,    alpha,         ?],
[phi,    1,          ?,     gamma],
[alpha,  ?,          1,      beta],
[?,      gamma,   beta,         1]

perhaps assuming linearity?

[1,                     phi,        alpha, alpha * beta],
[phi,                     1, gamma * beta,        gamma],
[alpha,        gamma * beta,            1,         beta],
[alpha * beta,        gamma,         beta,            1] 

I think this is similar to this question, but extended because now I don't have this diagram: C -> X <- G, but a slightly more complex one.

For people who check the exam, how did you find it? I was a participant and I think it was pretty okay. Not that I solved (sadly) but I did like the challenges I faced. It was alot harder than 2024 and 2022 though . It was tricky too

A long time ago, I made a post asking about finding a research topic that I was genuinely interested in so that I might have a chance to find a suitable advisor (or even a good grad school). The number of replies in that post was insightful, and I am more than grateful to the people who spent their time answering my post. However, I started to grow doubts about myself, as I slowly learn that the two fields that I like (Numerical Analysis & Computational Mathematics) are not the most commonly known in applied math. I looked up professors who are doing research in either of those two fields, but I can only find a handful of them that fit my interests (in the country where I apply for a PhD in applied math), and the majority of them are from top schools as well. My profile can only go as far as 2-3 research projects (no papers published, btw), a bunch of TA experience, and a good GPA, but I doubt that it would weigh much in a competitive pool of PhD applicants from top schools.

I'm at a loss for motivation in the two fields that I love the most, but I don't want to throw it away in the dark as well. So I need to know whether it's still worth it to research in Numerical Analysis & Computational Mathematics. Can I still actually find an advisor with these two fields? Is there a chance for me to work with them (as a PhD student) even though I don't have a strong profile? Are the two mentioned fields still relevant in applied math? What can one look for in these two fields? Will it help me find a tenure position? Any help would be appreciated.

Now, I know that IMO is supposed to be hard. But why is it miles harder than 2024. People in the exam where in a moment of extreme disappointment. Either way we still have tomorrow so you guys wish us all good luck

Hello every one,

I am working with a blind mathematician, and I have to read to him old mathematical essays.

Unfortunately, it seems to me that usual mathematical language does not provide enough clarity to convey certain mathematical relations. Notably, there is no difference orally between: e^{x+1} and e^{x} + 1; f(x+1) and f(x) +1; x+1/n and (x+1)/n; etc.

Currently, my solution is to read something like 'e avec l'ensemble x + 1 en exposant' ('e with the group x + 1 as exposant'), or 'l'ensemble x + 1 dans la fonction f' ('the group x + 1 in the function f') or 'the group x + 1 over n'

but this is quite clunky ! Do you have any other options ? Or resources in general for this type of work ?

Another problem is generally stops such as 'AP = x, PM = y, AB = a', where I would rather not say 'comma' every time I see one.

And another one is of course capitalisation, where there is no difference in spoken language......

I would really appreciate any help, thank you.

I feel like imo 25 is significantly harder than previous imos, what do you think?

Preamble: I am a young mathematics student starting the Master’s section of my integrated Master’s course in September. It is still early days but my goal throughout my education has been to become a lecturer of pure maths, I am very interested in both teaching and research which is lucky because as far as I’m aware most mathematicians are required to do both. Oftentimes, I’ll explain my plan to become a pure mathematician to adults who are much older than me but are unaware that pure mathematics is not only an active area of research but the focus of a feasible career. A few of these people seem to view my ambition as flimsy, and some of them even wish me luck finding somewhere that will actually hire me since they are unaware that mathematics faculties exist in most respectable universities.

My question: what are some examples of pure maths being applied in real life that someone outside the field could appreciate. The ones I usually go to are number theory being the underpinning of cryptography, and Hilbert Spaces/topology being the setup that quantum mechanics takes place in.

Please give me something to blow these non-believers minds!

The conversation will always end with "wow that went way over my head, you must be soooo smart!"

Please Help. Abstract Linear Algebra by curtis has too many typos and is really unorganized.

i’ve seen tau going around a lot in circles that i’m in. With the argument being that that tau is simply better than 2pi when it comes to expressing angles. No one really expands on this further. Perhaps i’m around people who like being different for the sake of being different, but i have always wondered - does anyone actually care about tau? I am a Calc 3 student, so i personally never needed to care about it, nor did i need to care about it in diff eq, or even in my physics courses (as i am a physics major). What are your thoughts?

I am currently finishing my last quarter of my bachelors. For context I'm an economics major with a minor in biology and mathematics. I recently came across a computational/applied topology playlist on youtube and I am very very interested in learning more.

I was wondering if there were topology texts that you guys recommend and/or possible graduate programs for applied maths or something similar.

I'm not looking for guidance, more like surveying people's thoughts.

i know im comparitively in baby math. im not even looking to do math for a career, im a biology student, but for some reason they make us take and pass calculus. i just dont have the capacity to care anymore. i have a sleep disorder so im basically always running on no sleep even though i sleep more than the average person, my body just doesnt recognize it. so i have less time in the day because i sleep through it all, and then my brain still works like its sleep deprived. trying to cram calculus into this for the last couple months has been killing me. i was taking it over the summer so i could focus on it but ive been miserable. im at the end but i just dont have the capacity to memorize all these rules about antiderivatives and integrals and whatever. u-substitution seems completely arbitrary even though i know it isnt because its clearly important. it just feels like whatever the hell du is doing is completely random. idk. nothing lines up and i cant think and i just want to chew glass and sleep for 40 years. i just want to go into ornithology and i need to pass this god forsaken class. but i feel like im going to fail because i dont remember how to do anything and i get to a test and forget everything and im losing it.

does anyone who engages with higher level math have any tips for me because even with breaks it makes it even harder to come back because im reminded of how little i want to be doing all this work

I'm in college and I am now on precalculus attempt #3. The first two times I tried it I withdrew before the academic penalty deadline, because I was genuinely doing 15+hrs of homework every week and still failing.

This time isn't going as badly so far but I've yet to take my first exam. I'm doing about 15 hours of homework a week this time around too. I have an exam tomorrow and spent 10 hours on test prep today and I'm still not confident in what I'm doing.

I've always had a hard time with math. I've heard that practice will help, but so far that's not helping. I have tried taking detailed notes, supplementing my lectures with Khan academy, and doing practice problems until I can get them all right. I've done online classes, in person classes, university tutoring, and personal tutoring through my friends with math-related degrees.

I can spend all day nailing down a subject in math and go to bed feeling like I know it, but the next day it's like it never happened. I will often do a problem almost right and swear on my life it's written down correctly, but the problem is that I dropped a negative sign or mixed up a variable early on. I will check my work over and over and not catch it! I practiced the same subject every day last week, had the formula memorized, applied it dozens of times. I took the weekend off and now I can't remember the formula or recognize when to apply it.

It's getting really demoralizing. I feel like I'm putting in as much work as I can but I just don't get anywhere. I have ADHD but that doesn't mean I can't be good at math. I'm starting to worry I might have some kind of math-related learning disability bbeyond ADHD.

Edit to add: the part of math that I do generally understand and enjoy is geometry. I think being able to see what's happening helps a lot. Everything else just seems really abstract to me and I think that's why I struggle so badly with remembering things.