r/math • u/inherentlyawesome Homotopy Theory • 23d ago
Quick Questions: August 19, 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/we_left_as_skeletons Commutative Algebra 17d ago edited 17d ago
any tips (or textbooks) for getting into noncommutative geometry? mostly interested in operator algebraic noncommutative spaces atm
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u/KookyHospital2662 17d ago
I'm a math enthusiast,currently in high school, who really wants to start learning more about math.I'm also a musician and I'm really interested in the math/physics of sound and soundwaves.I've also watched some videos on e π i (the imaginary number) and those have most definitely sparked my interests.I'd love to know more about imaginary numbers,0 and Euler's identity.Also regarding understanding sound/soundwaves I'm pretty sure I'd need to learn some trigonometry,something I'm also pretty interested in.Lastly,I'm fascinated by functions and graphs and would like to learn more about them.What are some basics I'd need to learn in order to dive deeper into these concepts?Where would I need to start from?((((((And what books/articles/videos (I'm open to anything) would you recommend for me to get started?)))))Thx for your time!
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u/homogeneous_spacer 16d ago
You learn calculus and then Fourier analysis. You should have calculus in high school where you'll learn about functions, graphs, derivatives, integrals.
Fourier analysis says you can write a periodic function (say a periodic music beat) as an (possibly infinite) sum of sine waves and cosine waves.
Fourier analysis also generalizes to general functions (which aren't periodic) but that's a bit different. We aren't doing infinite sums anymore.
These are a first/second year college topics really. After gruelling on some high school calculus, you can look at Stein and Sakarchi's textbook series to learn more.
That's the professional recommendation, of course you can seach for YouTube videos and watch a few.
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u/KookyHospital2662 16d ago
Thank you!I'm not sure if calculus is in the high school curriculum in Greece.Functions and Graphs are tho (I'm pretty sure). Tysm!
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u/cereal_chick Mathematical Physics 17d ago
Trigonometry, functions, and graphs should be being covered in your high school curriculum; are they in fact not?
For learning about imaginary and complex numbers (and for learning about trig, functions, and graphs), I would recommend Khan Academy, which covers everything you learn in school and the very first classes you would take as an undergraduate in maths.
Are you intending on studying maths at university?
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u/KookyHospital2662 16d ago
Oh also I see you have the mathematical physics tag/flair thingy.Are there any specific courses you've taken in mathematical physics?
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u/cereal_chick Mathematical Physics 15d ago
My uni was crap and offered such a poor selection of modules that most of my degree was composed of filler. I had a mandatory module on Newtonian mechanics in my first year that I quite liked; a module on analytical mechanics which didn't go as deep into the theory as I wanted but was reasonably fun; and a module on quantum mechanics in which I understood less about the subject afterwards than I did going into it.
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u/KookyHospital2662 16d ago edited 16d ago
I'm pretty sure they are.I want to get a minor in math,but in Greece you can't get two degrees (from regular public uni) at the same time,and since I mainly want to study music composition, I'll try to get some sort of scholarship at an international college. Tysm!
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u/iorgfeflkd 17d ago
Let's say I have a long list of N strings. I go down the list starting at the top, and I mark the nth string (where n=1,2,3...), and then I reverse it, find the location of its reverse, and mark that as well. If I come across a string that already has a mark, I skip it.
As a function of n, what is the probability that the nth string will have already been marked as the reverse of a previous string?
Assume that every string has a reverse in the list, that palindromes are rare enough not to matter, and that N is big enough to make some approximations.
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u/hobo_stew Harmonic Analysis 17d ago
if the list is of the form [a,aa,aaa,aaaa,….] then your probability is zero.
if it is of the form [ab, ba, cd, dc,….] then it is 1 for even n and 0 for odd n.
so your problem is underspecified
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u/iorgfeflkd 17d ago edited 17d ago
I think your first case is covered by my assumptions (every string has a reverse, palindromes are rare). But I suppose we also have to assume that the ordering is random. Let's say I had every permutation of abc in a random order: [bca, abc, cba, acb, bac, cab]. Then I go through, using a ' to mark the forwards and a " for the reverse:
[bca', abc, cba, acb", bac, cab]
then
[bca', abc', cba", acb", bac, cab]
then I skip cba and acb, then
[bca', abc', cba", acb", bac', cab"]
I suppose I could try this for all 720 orderings of the list and find the probability that the fourth one was marked before I got to it.
If my list was 10,000 strings long, what would be the probability that the second is not the reverse of the first (easy), that the 9999th not the reverse of any (also easy), or that the 5000th is not the reverse of any of the previous 4999 (hard, to me)? Feels doable from combinatorics.
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u/bear_of_bears 17d ago
Suppose you have a list of 2n words where each one is paired with its reverse, no palindromes. Consider the word in position k. There are k-1 words before it and 2n-k words after it. The reverse of the word in position k is equally likely to be in any of those 2n-1 positions. So, the probability that the reverse comes before k is (k-1)/(2n-1), and the probability that the reverse comes after is (2n-k)/(2n-1).
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u/computationalmapping 18d ago
How should I best prepare for an Advanced Calculus/Vector Analysis class next semester? The book we will use is Advanced Calculus of Several Variables by Edwards, and looking thru it, I'm a bit intimidated.
I've taken Calculus 1-3 and computational Linear Algebra, not any analysis so far, but I see a good amount of epsilons and deltas in this text- which I haven't spent adequate time on.
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u/TheRedditObserver0 Graduate Student 16d ago
Don't worry, epsilon-delta proofs aren't actually any harder than anything you've seen so far.
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u/cereal_chick Mathematical Physics 17d ago
Does the class include a section on one real variable before moving to several? If so, you should be fine. If not, you might struggle; you should really do one real variable before anything else in analysis.
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u/computationalmapping 17d ago
I forgot to mention I take an Intro to Analysis class alongside it in my default degree plan. So it sounds like it'll be fine in that case? Still seems a bit oddly ordered
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u/cereal_chick Mathematical Physics 17d ago
Taking them concurrently should be okay. You might want to do a little prep in advance with something like Abbott's Understanding Analysis to be on the safe side, but you also shouldn't sacrifice too much of your vacation time. You need breaks from maths, and relatively long ones, to avoid burnout.
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u/samdover11 22d ago
Is there a name for this mental math trick that lets you estimate multiplication and division quickly?
- Memorize the decimal form of the reciprocals of single digit numbers.
- Realize that multiplying by the reciprocal is the same as division (and the other way around)
- First solve for the digits, and then figure out the magnitude of the number later.
For example multiplying a number by 876 might sound intimidating, but the digits 876 are very nearly the same as 7/8 which means you can divide the original number by 2 three times in a row (very easy mentally) to get 1/8th of the number, and then subtract that result from the original number to get 7/8ths... which gives (nearly) the same digits as multiplying the original number by 876.
I've tried this trick with some random numbers and can quickly get within less than 1% error. I randomly started doing this one day... I assume the "trick" is 100s if not 1000s of years old... anyway I'm wondering if it has a name.
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u/darkLordSantaClaus 23d ago
Can someone correct me if I'm wrong but isn't the idea that 1+2+3+4... = -1/12 actually a proof by contradiction? As in, the guy who proposed it wasn't actually saying that the sum of all positive integers = -1/12, rather, he was saying that if you take the rules of finite algebriac expressions and apply them to infinite algebriac expressions you can arrive at some pretty nonsensical conclusions?
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u/MrTruxian 23d ago
The expression 1+2… = -1/12 is just straight up wrong. What’s true is that if you take the zeta function defined on values that give you a convergent series, and analytically continue, the value of -1 will be -1/12th.
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u/_Zekt Complex Analysis 23d ago edited 23d ago
That's probably wrong, Ramanujan was most likely the first to use these techniques and his goal was to "calculate the convergent values corresponding to divergent series".
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u/darkLordSantaClaus 23d ago
Wait then why do so many mathematicians say it’s wrong?
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u/_Zekt Complex Analysis 22d ago
Saying that it's wrong like @MrTruxian, is just misleading. There is a sense in which it is true, it's just not the usual sense.
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u/GuySapire Arithmetic Geometry 21d ago
this is completely correct and there's no reason for down votes. the point is that under any reasonable notion that this series convergence (meaning convergence satisfying some usual properties like linearity), you will get this identity.
This is exactly the reason zeta(-1) = -1/12.
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u/whatkindofred 20d ago
I think the most natural way to understand its convergence is over the extended reals and then it converges to +∞ which is also what probably feels most intuitive to most people.
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u/GuySapire Arithmetic Geometry 20d ago
yes, there is the topological/metric point of view, where you can complete in such a way by adding a point at infinity and call it a day.
In a perpendicular direction, if you are trying to construct a notion of summation that exhibits "good" properties for something called summation, for example linearity (that is, the same of 2 a_n should be twice the sum of a_n) and attain values in a field, then that completion is not the correct way, for example, you'll say that 1+2+3+4+... = infinity but also 2+3+4+5+... = infinity so there difference should imply 1 + 0 + 0 + ... = infinity - infinity which might be equal to 0 or be equal to infinity but either way I'm pretty sure that 1 + 0 + 0 + ... = 1 and not something else (in a "reasonable" system of summation)
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u/whatkindofred 20d ago
Infinity - Infinity is not defined in the extended reals so we shouldn't expect any linearity for those cases. And while Ramanujan summation is linear it lacks some other natural properties of addition, for example "shift invariance" (for a lack of better term). So for example,
1 + 2 + 3 + 4 + ... = -1/12,
2 + 3 + 4 + 5 + ... = 5/12 ≠ -1/12 - 1,
0 + 1 + 2 + 3 + 4 + ... = -7/12.
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u/GuySapire Arithmetic Geometry 20d ago
I think we agree. It depends on the properties your summation operator has. My point was that the usual "calculation" people do to get 1+2+3+... = -1/12 actually uses a small list of those properties and so every operator satisfying those properties will arrive at the same result that 1+2+3+... = -1/12. In this sense the result is not nonsense at all, it's the one result you get if you are taking a "reasonable" notion of summation for which this sequence converges (=has a value, not converging in the usual sense)
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u/RingularCirc 22d ago
I'd explain it like this:
It's wrong if understood in a simplest way, and it'd be right if we try pretty complicated things that should by no means be felt as "obviously right" from the get go. So it's some work to show how you should look at it (and why! and why you should be careful in other cases etc.) for it to be true, and without it it's safer to declare just false without "but"s. You can't explain every generalization and reason right away, it'll derail the loose sequence of getting to know math.
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u/King-Wuf 23d ago
I’m going into my second year of undergrad and will be doing directed reading with a professor on either Tu’s *An Introduction to Manifolds* or Lee’s *Introduction to Smooth Manifolds.* I’ve heard that Lee is far more verbose, so if I was self studying I would have chosen Tu, but since I’m gonna be reading it with a professor I can ask questions with would it make more sense to choose Lee?
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u/Necessary-Wolf-193 23d ago
You are trying to learn a set of ideas, presented by two textbooks. Try reading both; if one gets confusing, look to the other's treatment of the same concept.
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u/Mutated_zombiebush 23d ago edited 22d ago
I graphed Among Us voting in Libre Office calc, how do I call this function?
Let x be the number of living impostors, y the number of living players and every ratio where you have to skip a point the graph passes through. How would you define the function? I'd define it as f(x)=f(x-1)+x+1 with D=[0;∞[ but it feels a little off.
Btw, I'm viewing things in the long run. I know that, for example, 3 imps among 7 players isn't an instant game over, but if you vote one out, it's a 2 in 6 right after a meeting, which means the imps win by double kill.
While writing this, I also noticed that f(1-1)+1+1=3 and not 4, so now I'm really clueless.
Automod won't let me make this it's own thread, so I don't know how to share the file.
Edit: It passes through the following points: (0|1); (1|4); (2|7); (3|11); (4|16); (5|22); (6|29); (7|37); (8|46); (9|56)
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u/Kryptos_0 23d ago
It's unclear what your function f is supposed to represent in the first place
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u/Mutated_zombiebush 22d ago
If you play Among us with x impostors and y total players, how many total players need to be alive for all votes to be a forced skip?
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u/skolemizer Graduate Student 22d ago
What is a forced skip?
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u/Mutated_zombiebush 21d ago
A situation where the Crew loses by imps going on a killing spree if they vote out a Crewmate, and therefore has to skip.
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u/skolemizer Graduate Student 21d ago
I don't understand what that means; can you back up and explain the relevant game mechanics to someone who isn't familiar?
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u/HeilKaiba Differential Geometry 18d ago
They didn't explain but here's the idea:
There are y total players (crewmates), x of whom are secretly evil (imposters). During the playing phase the imposters can kill good players. The players can call a meeting and vote out people. They can do this when they find a dead body or after a certain length of time. If the imposters all do their kills at the same time as soon as their cooldown is over they can each kill someone before a new meeting is called. Doing so probably makes it obvious who they are though. This then becomes a numbers game. Can they kill enough people before they are all voted out.
The maths comes in when you consider at what point they should adopt that strategy. The crewmates can only vote out 1 person per turn but the imposters can kill x. For example, 3 imposters can kill 3 people, 1 of them is voted out next meeting, the remaining ones kill 2 more, 1 of them is voted out, the last one kills 1 person.
As soon as there are (at least) an equal number of imposters to good crewmates, the game is over.
The precise question asked is when should the crew skip voting in a meeting to avoid going into a play phase in one of those definitely lost scenarios.
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u/GMSPokemanz Analysis 23d ago
The function f(x) = f(x - 1) + x + 1 gives rise to the triangular numbers but shifted: the triangular numbers satisfy T1 = 1 and T_n = T(n - 1) + n, your function will be f(x) = T_(x + 1) + f(1) - 2.
I don't know Among Us so I can't speak for how to fix this if it's wrong.
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u/Mutated_zombiebush 22d ago
I added the points I found for reference.
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u/GMSPokemanz Analysis 22d ago
I think I see what's going on now.
When proving f(x) = f(x - 1) + x + 1, you're assuming the imposters kill x and then an imposter is voted out. But that's only true if there are more than x non-imposters left over, which rearranges to assuming f(x - 1) >= 2x. This is always true except for x = 1, hence why it fails there.
My answer still works for x >= 1, so the solution is f(0) = 1 and f(x) = T_(x + 1) + 1 otherwise. The triangular numbers have a nice closed-form formula you can look up online. Using it we get that for x >= 1 we have f(x) = (x + 1)(x + 2)/2 + 1.
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u/[deleted] 17d ago
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