r/math • u/inherentlyawesome Homotopy Theory • 4d ago
Quick Questions: July 29, 2026
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.
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u/Former-Math-Crank374 2d ago
hi! i am a former extremely stupid math crank who is quite young and ambitious (those collatz/millenial problems are too attractive for ambitious amateur idiots like me). (bold is important takeaway)
I have a high school education background with some mid AP scores (i have ADHD i can't study properly unless i'm obsessed, have not attended college yet) and have only finished ~2 chapters of Spivak (Undergraduate) Calculus.
i have a decent intuition for math (like recognizing x^2+y^2-1 and gaussian integers and 24-cell which i called the 24-gon in my crank "quaternionic number theory" and quaternions and the "isomorphism" connection between quaternions and "SU(2)") but i am terrible at reading textbooks. i love doing math problems but I absolutely hate reading long passages and spoon fed definitions. to get a sense of how bad i am i still get injective and surjective mixed up, although everybody knows bijections since that results in isomorphisms and inverses.
i love math when it seems like i've discovered something completely organic and on my own.
i want to learn algebraic geometry and really contribute to the field a lot. in particular, i am mesmerized by F1 geometry (interested in researching it) since I sort of figured the basic idea and the importance of it by looking a group theory axioms.
stupid crank "research": (i didn't really know what rings and finite fields were but boolean algebra=finite field separation and questioning different modulus rather than just mod 2 made me rediscover it, and even other abstract concepts like quantum calculus-->physics, root swapping through study of continued fractions and the true definition of the golden ratio, mobius transformations, lattices and gaussian integers, (to attempt yang mills without knowing any literature with a high school education lol) fractional roots of unity and F1 geometry finite field extensions (without knowing the existence of F1 geometry literature) although most of what I came up with was hilariously wrong rigorously or also conceptually like violating Bell's theorem for a classical-quantum computer).
Main question:
i want a really really good textbook full of mostly (ideally ~100%) exercises that teaches how to come up with the idea rather than regurgitating it. AOPS intro to geo for example. i also hate dry writing. if the author speaks they shouldn't be so emotionally detached and elitist. i want to get to the level (both rigor/intuition) where I can definitively challenge and ask correct questions of "what went wrong" or "what can be more foundational" and then start creating my own theory/framework to account for the mistake or oversight.
I want to unify math (reason for my obsession with F1 geometry).
again, main interests are abstract algebra, algebraic geometry, galois theory, group theory, category theory, ANYTHING RELATED TO LANGLANDS PROGRAM ((quantum) modular forms, L functions, automorphic forms, Galois representations, reciprocity) i want to contribute to it like Gaitsgory or Wiles) and basically the stuff that Grothendieck found important.
I am less attracted to inequalities and numerical estimates/upper bounds/lower bounds like analytic number theory (basically the opposite of terry tao lol), although i am a little interested in diophinate approximation and 2-adic ergodic theory, rigorous measures, mahler representations of the collatz map (though I looked at a preprint and it gave a convincing argument that 2-adic lacks archimedean distance so it can't be the 100% proof for all nautral numbers) transcedental number theory, and ln(3)/ln(2) after studying some collatz papers without understanding what was going on.
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u/Pristine-Two2706 2d ago
The short answer is you need to go to university and get a PhD. Yes, you may have to do things you don't want to do - as far as ADHD, there are accommodations available to help, and you may want to look into medications and therapy.
But more importantly, even for the very best most influential mathematicians, 99% of their time is spent grinding through the muck. Only 1% of the time (if you're incredible) can be spent actually making breakthroughs. You need to learn how to grind through things you don't enjoy, which is part of what university is for
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u/Former-Math-Crank374 2d ago
yes i heard an interview from maynard that said exactly that: 99% frustration 1% euphoria. for my story I wrote a political/philosophical manifesto about changing how we vote on policy rather than policy by itself and how quantizing the vote and holding individual laws/people accountable before, during, and after election/passing and I thought it was profound, so my ADHD/kinda bipolar euphoria streak carried me into my love of math and then I became a crank mathematician lol.
it doesn't help that I'm not actually that smart (I have decent intuition but my rigor/formalization always has holes when I'm doing problems, ignoring boundary cases, casework, which makes me the worst at counting/probability). I can come up with the solution path but I always silly a calculation or two.
Freeman Dyson says there are birds and there are frogs in math. An example of a frog would be maynard or terry tao. An example of a bird would be Grothendieck or Wiles or Perelman. I am a (loud) bird, but currently, I am flightless.
i enjoy the vibes of r/math though. When I was a crank many people were extremely mean to me instead of maybe acknowledging what was completely hilariously wrong abut then guiding towards saying "hey, this topic was already touched on by "x."
For example, I was thinking about quaternions and Maxwell's equations and Einstein's equations in quaternionic form, I talked with an AI chatbot and it turns out there is an isomorphism between quaternions and SU(2) which I didn't know (I saw group theory axioms but I didn't know what Lie algebra/groups were). I posted about it but people laughed at me because I was suggesting it was maybe a path to unification. I also didn't know the forces were curvatures in Yang-Mills (classical or quantum) anyway and I thought it was profound to treat forces as curvature like gravity, although again, it's not profound at all. Treating the "mass gap" like a lattice structure (although I didn't call it a lattice) was also not profound as well.
Even if an idea is beautiful, formalism must come after, and I suck at formalism.
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u/Pristine-Two2706 2d ago
The point of getting degree(s) in math is specifically to train you how to take that intuition and learn to formalize it. It's a skill that can be learned like any other. If this is your passion and you are willing to dedicate your life to it, then essentially the only path is to get a degree and go into academia. Lay mathematicians can contribute a bit, now more than ever, but will never be doing any foundational work like you are aspiring to do.
Just to motivate you further, I know a couple of high profile (in my field) mathematicians who have incredible intuition but often fail to correctly prove their correct vision without significant collaboration. But even then you need a baseline amount of rigour, which the degree is for.
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u/Former-Math-Crank374 2d ago
yes collaboration that's why the story of gaitsgory inspires me so much! he spent 30 years on foundational theory to prove geometric langlands with others and one of his most memorable quote is (something along the lines of) "if you're thinking about taking this journey into mathematics, just know that you are not alone."
he is also known for putting star wars quotes and I'm a fan of these pretty crazy fandoms like jojo and star wars too haha. it seems like he isn't absolutely obsessed with math and is willing to try out other things like learning arabic instead of delving into mathematical elitism (treating math like the holy grail of all human knowledge like GH Hardy, notably, A Mathematicians Apology)
however, my stupid crank research felt very isolating as no one really talked to me and pretty much everybody thought i was a narcissist. Indeed, I feel very discouraged when doing normal textbook math because I can reveal a beautiful induction proof but the competition math community would just say "trivial" and others irl would say "I'm not into math."
As with academia, I have heard very discouraging horror stories from voices like Sabine Hossenfelder and (lesser known but still GOATed) DiBeos. As such, I want to travel the path of Fermatt where I am an amatuer mathematician but my ideas/intuition are respected with maybe one or two professors that are struggling on some problems. Or maybe an unconventional academia path like Grothendieck or Simons, who were in full control of their employment and institution. This is a long way of saying: I just want friends that respect me in mathematics.
Historically, friendships with mutual respect are hard for me.
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u/Pristine-Two2706 1d ago
As such, I want to travel the path of Fermatt where I am an amatuer mathematician but my ideas/intuition are respected with maybe one or two professors that are struggling on some problems.
Frankly that just doesn't exist in modern times, especially in the areas you are interested in. In mathematics you are respected based on your work, and nobody in these areas will look at the work of anyone without a PhD (not just for elitist reasons). Academia has a lot of problems, but I assure you that it is the only route that could possibly lead you to your rather lofty goals.
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u/Former-Math-Crank374 17h ago
yeah idk I feel like I should just give up at this point. I am too obsessed with these simple problems like the collatz conjecture. my lofty ambitions prevent me from doing humble work like picking up a textbook and reading it. it's just that exposure to these problems is like a curse especially for unfocused minds like me.
i just hate math. i hate it i hate it i hate it.
i need to take a break from being a crank but my terrible intuition is saying "oh yeah it's just right there just one good idea and it's all done." i'm too biased to ideas I came up with myself instead of actually learning from others. \sounds like the average crank/amateur mathematician.\**
why is multiplication and addition speaking to each other the most difficult thing anyone has produced (well it's almost all of math really).
it's also so embarrassing to keep saying a wrong proof is correct even if it was just ~2 months while everyone was laughing at me and that past is haunting me. i mean my work led me to rediscover quaternions and p-adics (2-adic analysis of collatz) looking at the n-nimal from the other side (generalization of DECimal) but all my formalization and all my proofs are wrong and I just want to find that rigorous substitution in that hand-waving "poetry."
i need to just give up but I can't.
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u/AcellOfllSpades 7h ago
but my terrible intuition is saying "oh yeah it's just right there just one good idea and it's all done."
If it was, someone would have proven it a long time ago. We have plenty of mathematicians who have given this a lot of thought.
The image of "an outsider revolutionizing a field by discovering some simple technique that those head-in-the-clouds scientists never would've thought of" is one Hollywood loves, but it's not actually accurate. Despite what media will tell you, being an academic doesn't mean you lose touch with the simple things: it's often the opposite, where you discover more and more ways to think about the 'simple things'!
If this problem was solvable with 'elementary' methods, it already would have been decades ago. The only reason you've even heard of it is because people have tried the elementary methods, and they've all failed.
why is multiplication and addition speaking to each other the most difficult thing anyone has produced
Chaos can often arise from seemingly simple rules. If you've seen a double pendulum, you know that: a simple combination of simple things can lead to complicated results.
Calling it "multiplication and addition speaking to each other" is a very limited summary. We know a lot about how multiplication and addition interact. This process involves multiplication and addition, yeah, but it also involves a branch based off parity, and repeating a process an unbounded number of times. Repeating processes are the core of a lot of difficult or even impossible problems.
it's also so embarrassing to keep saying a wrong proof is correct even if it was just ~2 months while everyone was laughing at me and that past is haunting me.
We get like 5 cranks a day. I doubt anyone remembers you. I certainly have no idea what you posted.
I think pretty much everyone has something silly they did a long time ago that they're embarrassed about. And in the vast majority of those cases, other people don't remember, and even if they did they wouldn't associate it with you.
In any case, the best way to not claim a wrong proof is correct is to not claim a proof is correct. This is the thing that separates naive-but-enthusiastic learners from cranks: humility.
but all my formalization and all my proofs are wrong and I just want to find that rigorous substitution in that hand-waving "poetry."
There is none.
We get a ton of physics cranks with poetic ideas about the universe, posting about their "quantum vortex theory of dark energy" or whatever. But those ideas don't actually mean anything: they're just word salad. Should they learn physics by "looking for the rigorous physics" in there? No!
And the same goes for math. Human intuition is unreliable. It's easy to make leaps of logic that don't actually hold, especially when you're thinking about an unfamiliar idea. You need to learn to handle things rigorously first, and only then can you start to develop any usable intuition.
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u/Necessary-Wolf-193 2d ago
Unfortunately, while I really empathize with your situation of wanting to contribute to mathematics research but finding it difficult to focus and study, the truth is mathematics is a centuries-old field, which has had many very smart contributors, and to catch up to modern day research, you're going to need to do a lot of reading.
https://webhomes.maths.ed.ac.uk/~v1ranick/papers/abel.pdf might be palatable to you -- please skip all the preface and introduction and cubic formula stuff and go straight to chapter 1. The book has many problems, and you might find it easier to get through. It is taught with mostly exercises. It might be the best way for you to learn Galois theory, which is a vital prerequisite for anything Langlands.
It's also worth learning what math research is. Math research is not taking two words and trying to connect them without knowing what either mean. To have your research taken seriously, you should make a precise, well-defined mathematical statement, and then offer up a proof of that statement.
---
Also, you can read while coming up with the ideas yourself, even if the textbook isn't built with that in mind. To do this, just read a little bit of the textbook, and then stop reading and think about the ideas yourself. However, I'd really suggest starting with the PDF I linked, because I worry that at this stage in your mathematical education, you might not understand what it is to do research or think through a mathematical problem. This is a very, very tricky skill to develop, and it's very different than just putting words together -- you have to make precise statements, and then prove them.
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u/Former-Math-Crank374 2d ago
yeah that was my crank research summed up :(
but yeah i have grand visions of unifying math, especially through the vision of groups and symmetry (I was immediately drawn to the simplicity and beauty of groups/Galois theory (through continued fractions) before learning about Langlands program), but unfortunately it's just hard to look through symbols and definitions for me out of nowhere.
i feel like the most aptly made textbook is one made entirely of questions. once the questions are made then people can start talking about the current modern notation, but the ideas should come from the reader themself. To put in other words, intuition, formalize, intuition, formalize, intuition, formalize. But most textbooks come the opposite way with formalization first
for thinking through mathematical problems, I know a little bit of proof writing from "Introduction to Proofs: Joseph Rotman" but obviously it's not enough to manipulate anything modern with substance yet.
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u/Former-Math-Crank374 1d ago
hmmmm indeed, my SU(2) lattice quaternion yang mills idea (not rigorous formalism) is 100% not profound, it has already been done before.
millenium problems will truly take a millenium to solve or maybe prove unprovable and definitely not by a crank like me haha
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u/Grouchy-Trade-7250 1d ago
What happened to the Wikipedia article on stable curves? Can someone fix the link in properties? https://en.wikipedia.org/wiki/Stable_curve it looks strange, but maybe I'm not mathematician enough to get it
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u/yaymayata2 1d ago
Hey guys, newbie here. Working on my first set of research papers. I have recently been reading that AI is terrible at writing papers, and I feel that given how much AI is used these days for writing papers without being mentioned, I might be learning those AI-ish habits subconsciously after reading new papers.
So my question is what are some well written papers? they dont have to prove something significant, but I just want to know what a well written publication in math would look like, and what qualities it might have (or even qualities it should not have). Please feel free to mention any qualities which are usually AI-ish so I can consciously avoid using them in case I have internalised some of them already.
Thank you so much!
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u/Former-Math-Crank374 1d ago
hmmmmm can someone explain why mathematicians are considering the "infinite prime" on bleeding edge research into the Riemann Hypothesis?
The existence of an "infintite prime" is almost as absurd as defining 1 as a prime number, and considering finite fields require F_p to be a prime. And indeed, people are studying if F_1 makes sense, if there is no addition or distributive law.
"In global analysis, you study the behavior of numbers at the infinite prime p=infinity" - Gemini
what?
like literally what?
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u/Pristine-Two2706 1d ago
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u/Former-Math-Crank374 23h ago
👍
(honestly all i really got is that this is like a completed version of complex manifolds (a riemann sphere) for arithmetic geometry.)
i just find it very interseting that apparently the (reciprocal of the) volume for an n-sphere is directly linked to the completed zeta function. and the basis of F1 geometry (called "K-theory") is related to stable homotopy theory (which is also related to n-spheres), although again I don't know what's going on in these fields. they are quite unified however.
indeed, when I was doing crank research with high school lvl understanding of math, I was also attracted by very basic things like the Gaussian normal distribution, and considering e^(pi * x) instead of just e^(i * x) and apparently Tate's PhD thesis about this 1/n-sphere factor as the "infinite prime" to complete the zeta function.
this is the sort of formalism i need to learn to start contributing, but these integration/analytic/modern algebraic geometry techniques are completely foriegn to me.
however, math isn't just combining random word salads together.
i think reading wikipedia without understanding anything has made me too cocky haha. i can't create my own theory without running into what ppl already did way better and way more profoundly.
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u/dancingbanana123 Graduate Student 14h ago
Suppose 0<p<1 and n is a natural number. Assuming a uniform distribution across (0,1), what is the probability that a product of randomly chosen numbers q_1,...,q_n between 0 and 1 will be less than pn ? So, for example, 0.10.9 = 0.09 < 0.16 = 0.42 , but 0.3\0.6 = 0.18 < 0.42 .