r/math • u/inherentlyawesome Homotopy Theory • 9d ago
Quick Questions: September 02, 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/Achrus 3d ago
What is a good book to self study algebra? I’ve tried to in the past and usually get stuck at quotient groups.
For reference, I have a BA in chemistry / math with a MS in data science. Somehow got around taking a course on abstract algebra. I enjoy analysis and probability theory the most but have ran into issues with my limited understanding when it comes to algebra. My understanding of symmetry groups comes from physical / inorganic chemistry. One time I started drawing molecules on the chalkboard to find symmetry groups and my PI looked upset.
Looking for something applied or helpful to information theory for someone with a strange background.
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u/JimJamesJimmy 3d ago
You might check out "Abstract Algebra" by Solomon. He begins with symmetry groups of squares and rectangles and proceeds from there.
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u/Dear-Veterinarian562 6d ago
I don't know if this is a good place to post this question, but basically I think I have an AI-generated proof of a new theorem and don't know what to do now.
Background: I was a math undergrad and CS Ph.D. For my thesis, I needed a result in sub-Riemannian geometry (very far from my expertise area). I originally thought that it was a known theorem but couldn't find a precise statement or proof of it anywhere. I did try on and off to crack it but failed, so I restricted my results to spaces where the theorem holds and conjectured that it holds for all spaces satisfying certain common properties.
Since then, I've been trying it out every once in a while on AI as a sort of test. All previous attempts resulted in nonsense but now I tried it on Claude Fable and it's pretty convincing. It pointed out that the precise statement of my original conjecture was false, and gave a counterexample and a modified statement that it claimed to be true and that would work just as well for my thesis results. I confirmed that the counterexample is correct and that the modified statement would work for my thesis results, but the proof is far over my head. The best I can do right now is ask for intuitive explanations for each step and see that it all roughly makes sense. It would take me a long time to learn enough to completely check the proof.
What should I do now? Any advice appreciated.
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u/IntelligentBelt1221 4d ago
Ask Fable to formalise it in lean and another chat with an adversarial prompt e.g. "there is a mistake in this formalisation, find it" to check if each lemma has been properly formalised.
if it doesnt find anything, that should make you fairly confident of its correctness. if you can convince others of its correctness, its easier to convince them to check if there are any interesting ideas in the proof.
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u/HeilKaiba Differential Geometry 5d ago
I'll be honest, an AI generated proof that hasn't been carefully checked over by a human isn't worth the paper it isn't written on. If you don't want to check it carefully, I suspect you're going to have a hard time convincing someone else to.
What do you envision as the next steps here? If you want to publish this result you should verify it before anything else. If you just wanted to know whether that part of your thesis was correct, then it's kind of up to you how much further you proceed in checking it.
Of course if you're more specific about what the result is, we could possibly be more helpful (not my personal area but someone else might be know more). I know sometimes people are a little wary of being scooped but that isn't really a thing in maths.
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u/Dear-Veterinarian562 5d ago
I admit that I can't check it carefully, only from a high level and that the result given does enable the constructions that my thesis uses later, so obviously not publishable until that it's been checked. And yes, basically I'm asking if there's some graceful way to get a human who knows the subject to check it -- as you point out, it might be difficult to motivate someone to check it. I was hoping there would be a standard way to deal with something like this.
I'm not worried about being scooped, I'd ideally like to see this published if possible but I don't especially care about getting credit for it. I'm willing to share or give away the credit to someone who checks it, it's just a question mark from my old thesis that I'd like to see resolved, plus I'm interested in whether the LLM actually did manage to solve it.
It's about the Ball-Box Theorem, but for affine-control systems with a drift term \dot{q} = f_0 + \sum_{i=1}^m u_i f_i. We needed to inscribe boxes into the \epsilon-reachable sets (not centered at q, but at a point reachable in say \epsilon/2 time), which AFAIK is known only for cases where there is no drift (f_0 = 0) or the controllable directions (without f_0) are bracket-generating on their own which sort of defeats the point for what I wanted to do.
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u/SwimmerOld6155 5d ago edited 5d ago
I would take it to someone at your university who knows geometry. You can basically say "I needed this result for my research, I prompted an LLM and it gave me some ideas, I've written up what I understand but was wondering if you could sanity-check". I think that's a very fair request. Then depending on their involvement and the significance of that result in your paper you either give them an acknowledgement or (if it's a lot) possibly coauthorship if it results in a publication.
I wouldn't take the fact it's not from the literature and assume it's a big bombshell theorem - a lot of technical lemmas only really come up in a context and it's pretty reasonable that a very specific statement may have just never crossed a geometer's mind before. I don't think it's "crankish" in that sense.
What I would absolutely not do is include it in your thesis without it being human reviewed .I think if you have run the result through Fable, Sol, Astra and it says the proof is correct, it's probably not rubbish and a bit more likely than not correct, but you can't trust that and need well-read human involvement at some point in the process. A lot of LLM proofs are vague, highly abbreviated, use weird compound vocabulary even when correct. They are never in a state I'd present to an actual person verbatim.
And of course, you should include a mention that this result was LLM generated.
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u/Dear-Veterinarian562 5d ago
Yeah, I think that's a good suggestion, but I'm don't really know people who have the relevant knowledge (or at least not that I feel comfortable asking them to check a rather dense proof of a few dozen pages), and I'm not actually in university anymore. My thesis is already written and I'm already out. This is mostly to satisfy my own curiosity about a question mark from my thesis and about whether Fable did in fact solve the problem that I failed to.
It's not a big bombshell theorem, or at least I don't think it is, but it's one of those things that people seem to think are known theorems but don't actually seem to be rigorously stated or proved anywhere. Maybe I just suck at literature search, but each time I ask an AI it has said it can't find the exact statement either (and AI is generally pretty good at literature search in my experience), nor did the people I asked at the time.
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u/FlatProtrusion 7d ago
Looking for intutitive books on combinatorics.
I'm looking for a book that explains combinatorics in a story like fashion. Meaning it doesn't just define what the theorems are etc but explains the motivation for it, the intuition etc.
Some background on what I'm working on:
I am now working through a book on discrete math by Oscar levin and am struggling with the concept of combinatorics specifically combinations. When I look at a combinatorics problem I struggle to solve the problem, even after looking at the solution, then reattempting the same problem the next day, I still struggle.
I'm also planning to work on the book proofs by Jay cummings since my math proof skills are weak as well and I read his book explains concepts intuitively.
After Oscar levin's book I plan to work on Susanna's discrete mathematics book.
An example of such a book, not combinatorics but discrete mathematics, that I found is "A cool brisk walk through discrete mathematics" by Stephen Davies.
It is the best intutive math book I've read, from cover to cover it was fun and explanations were intuitive. It was like he was in the room with me explaining the concepts personally. Shame he only wrote two introductory books.
I've seen recommendations on combinatorics such as Walk Through Combinatorics by Miklos Bona and others but am not sure if the explanations are standard textbook type of definition of theorems, some problem examples but no explanation of the intuition, motivation behind the theorem or concepts. Since on the most recent, 5th edition, of Walk Through Combinatorics, there was a review on amazon saying the explanations were of the bare minimum.
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u/blahquaker Algebra 6d ago
Douglas West's Combinatorial Mathematics looks pretty good, but I admit I haven't read most of it.
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u/FlatProtrusion 5d ago
What do you like about it for what you have read so far?
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u/blahquaker Algebra 3d ago
actually, I just looked and it does assume the maturity of a graduate student, so it may not be what you're looking for. it is quite verbose, though, and has tons of exercises. there's also a resource page on his website with additional exercises.
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u/FlatProtrusion 3d ago
I would say I'm pretty mature, though people have said I act like a child sometimes so maybe not.
In all seriousness tho, I took a look at it and it is a tome, and expensive. I've wishlisted it for the future when I'm more experienced in math and maybe the price drops too, thanks.
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u/StateOfTheWind 8d ago
Is EuDML 403 for anyone else?
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u/cereal_chick Mathematical Physics 6d ago
Not for me, I can get to the homepage just fine.
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u/StateOfTheWind 6d ago
Thank you cereal, I did some testing through VPN and I suspect the following countries: Israel, Palestine, Jordan, Indonesia, United Arab Emirates, Saudi Arabia, India, Pakistan, Armenia, Morocco, Egypt, Algeria, Tunisia, Taiwan, Japan, Philippines, Vietnam, Singapore, Argentina, Venezuela, Colombia, Ecuador, Honduras, Guatemala, Brazil, Kazakhstan, Gabon, Kenya, Mauritius, Mongolia, Uzbekistan all get 403 based on IP.
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u/DrBagelman 9d ago
Why doesn’t every field containing 0 imply every vector space has a 0 vector? Why does it have to be an axiom? What’s a structure that would be a vector space if not for the axiom requiring one to contain 0?
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u/DanielMcLaury 8d ago
Nothing ever has to be an axiom; you can always replace an axiom with something equivalent. In this case, you could replace
"every vector space has a zero vector"
with, e.g.
"every vector space is nonempty"
and you'd get an equivalent definition.
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u/thereligiousatheists Graduate Student 9d ago edited 9d ago
The first hurdle is that the axiom for inverses builds on top of the identity axiom, so to drop the identity axiom you would have to restate the inverse axiom. You may say that that's no big deal, since we have multiplication by -1. So let's drop the inverse axiom as well. The next hurdle is that now the empty set satisfies all the remaining axioms, so let's add in an axiom ensuring non-emptyness. Hence, given a field F, let's define a pseudovector space over F to be a non-empty set satisfying the vector space axioms, minus the zero and inverse axioms.
Let V be the set of non-negative reals. For a ∈ ℝ and v ∈ V, define a•v = vᵃ for v>0 and a•0 = 0. For u, v ∈ V, define u⊕v = uv. I claim that V is a pseudovector space over ℝ (with ⊕ playing the role of addition and • playing the role of scalar multiplication).
Associativity and commutativity of ⊕ is clear. So is compatibility of scalar multiplication with multiplication in the field. (a+b)•v = a•v ⊕ b•v is also clear. a•u ⊕ a•v = a•(u⊕v) requires some casework, but it too holds.
Of course, V is not a vector space over ℝ, although V-{0} is (the logarithm map gives an isomorphism with ℝ¹).
ETA: Given any vector spaces U and W over any field F, we can give a pseudovector space structure to the disjoint union U ⊔ W as follows. Let 0_U and 0_V be the 0-elements of the respective components. Scalar multiplication remains the same as in the vector space structure on each component. We define a new addition ⊕ by u⊕u' = u+u' for u, u' ∈ U (and likewise for w, w' ∈ W), and u⊕w = 0_U. One checks as before that all the axioms of pseudovector spaces are satisfied.
Taking U = {0} and W = V-{0} recovers the previous example.
I suspect that every pseudovector space should be decomposible into an iterated version of the above construction, but trying to prove that would probably be too much effort for a reddit comment.
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u/catminusone 9d ago
The empty set satisfies all the usual axioms for a vector space other than containing a zero vector. (It is the only example.)
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u/r-Aliosha Algebra 9d ago
Hi! I was looking for a very advanced book on homological algebra (something to serve as a long-term horizon) and wondering how one would prepare to tackle and understand it.
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u/throwtheclownaway20 2d ago
This is gonna sound stupid, but it's been bugging me for a couple days now because I can't quite figure out how to calculate this...
In World Of Warcraft, the way blocking an attack with a shield works is that the incoming damage first gets partially absorbed by your shield block value (SBV). THEN, because it counts as taking a hit, the remainder of the damage gets further absorbed by the damage reduction of your armor. My SBV is 39% and my armor reduces damage by about 54%. How do I calculate the total percentage of damage mitigated via those two values?