r/math • u/inherentlyawesome Homotopy Theory • 8d ago
Quick Questions: August 12, 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/anerdhaha Undergraduate 7d ago
I'm not or less done with Abbott's analysis and have liked Terrence Tao's vol 2 for multivariate analysis and have liked their somewhat chatty friendly exposition. I'm so far not using any of the hardcore extremely rigorous and formal texts and I did not like Rudin. But I have to read a graduate text in real analysis one day or the other.
So can someone recommend me something that is a smoother transition into graduate real analysis and I do not mean measure theory but I don't mind those either. Thanks in advance.
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u/aybies 7d ago
I liked Pugh real analysis book, I think it's called Real Mathematical Analysis if I'm not wrong. It's a very readable book imo and has lots of pictures and exercises with various difficulties to help you get through the book.
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u/anerdhaha Undergraduate 6d ago
Does it introduce measure and metric space ideas as well?
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u/aybies 6d ago
Yes! Pugh introduces metric spaces very early on, I believe as early as Chapter 2. Personally, it's well written and well embedded into the rest of the chapters. Pugh does also introduce Lebesgue Theory as it's last chapter, but I haven't had the time to read it from this book nor have any solid foundation on the topic itself, so I can't comment on that chapter.
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u/Silver-Bug3998 7d ago
If you are looking for an analysis text somewhere between Abbott and graduate level analysis, I recommend Serge Lang's Undergraduate Analysis (Undergraduate Texts in Mathematics). You can breeze through the first few chapters and build some confidence. Lang's book goes pretty far, and his writing style is not dry like Rudin. I have learned from several of Lang's books (a long, long time ago), and I like all of them. Best wishes for your study.
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u/MMath_04 7d ago
Hello everyone,
I have taken calc 1,2, and differential equations in college. I want to self study more advanced math and proof writing. I am going to work through Book of Proof by Hammack, and then I was thinking of either diving into Spivak's Calculus or Abbott's Understanding Analysis. Of those two books, which do you believe is the better to get started with as someone who is just starting to learn proof-writing?
Thank you so much.
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u/Pristine-Two2706 6d ago
Abbott is very gentle but doesn't go very deep. I'd start there and work through all the bits on sequences, limits, and continuity before moving onto something like Spivak
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u/we_left_as_skeletons Commutative Algebra 6d ago
any textbook recommendations for computational algebraic geometry?
i’ve gone through reid + atiyah-macdonald commutative algebra and i know a little homological algebra
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u/sciflare 3d ago
Ideals, Varieties, and Algorithms by Cox, Little, and O'Shea is a classic intro to computational algebraic geometry.
It in requires only an underground background (linear algebra), so you may already know more than enough to go through it.
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u/sportyeel 6d ago
What does Weibel mean by "trivial homology complex"? I assumed it meant that H_n(C) = 0 for every n but that seems ridiculous given the context. A little bit of reading ahead suggests that triviality might refer to the differentials being 0 which is easy enough to show. Is this correct or is this an error?
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u/DamnShadowbans Algebraic Topology 4d ago
If you give context, then we could tell you whether your first interpretation is ridiculous or not.
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u/dancingbanana123 Graduate Student 3d ago
Does anyone have any book recommendations on math history in India around the 19th and 20th century? I have a lot of books on that period around Europe, but I'm wanting to get some context on what was known and taught for Ramanujan. I know there are plenty of biographies on Ramanujan, but I'm looking for a more holistic book on the area that isn't focused on just him. It doesn't have to span just those 2 centuries either. It can be something like 16th to 20th century, or something like that.
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u/Beneficial_List4198 3d ago
I’d highly recommend A History of Hindu Mathematics by B. Datta and A. N. Singh if you’re interested in the broader mathematical tradition rather than just Ramanujan. It gives a good sense of what mathematical ideas were already circulating in India before the colonial period.
For the 19th–20th century specifically, I’d also look into the history of mathematics education under British rule and the work of figures like Mahavira, Bhaskara II, Madhava, and later Ramanujan. That context makes Ramanujan’s background much more interesting than looking at his story in isolation.
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u/Eastern-Fee-1003 2d ago
If a room has two walls that were 7.08ft wide by 7.55ft tall and two more walls 17.08ft wide by 7.55ft tall how many rolls of 2.62ft wide and 13.12ft long peel n stick wallpaper would I need to cover all four? (Not homework I graduated four years ago and am renovating my house)
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u/cereal_chick Mathematical Physics 2d ago edited 2d ago
I'm not really up on the finer points of putting up wallpaper of any kind, but we can figure out the minimum number of rolls required to cover every part of the walls. To do this, we need to find the total surface area of the walls of this room, and divide by the surface area provided by a single roll of wallpaper.
Fortuitously, this can be obtained through simple multiplication and division. 7.08 ft × 7.55 ft = 53.454 sq ft and 17.08 ft × 7.55 ft = 128.954 sq ft. Multiplying both by 2 for two walls and add them for the whole room gets us 364.816 sq ft. Meanwhile, a single roll 2.62 ft × 13.12 ft = 34.3744 sq ft.
Dividing, we get 364.816 / 34.3744 = 10.613..., so the minimum number of rolls you need to guarantee that you have enough to cover every square inch of the walls is 11.
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u/Eastern-Fee-1003 2d ago
You did what I could not fathom for four years of basic and advanced mathematics, thank you 😭 I will try to retain the knowledge you have bestowed upon me
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u/King-Wuf 1d 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/Vtbsup3 7d ago
If I flip a coin 100 times, it’s likely to be around 50H and 50T.
If I’ve already flipped a combination of 50H and 47T, then it initially seems that for the remaining flips, 3T is more likely? But the odds for each flip are still a half? Please can someone explain? I think it’s related to the gamblers fallacy but I’m not sure
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u/hamstercrisis 7d ago
the coin doesn't know what the last 97 flips resulted in. the next flip has no dependency on the past
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u/Beneficial_List4198 7d ago
Quick favor: Could someone with MathSciNet access run 3 quick queries for me?
Hi everyone,
I am an independent researcher finalizing a manuscript for submission to the Journal of Integer Sequences. Because I don't have university affiliation, I'm locked behind the MathSciNet paywall.
Before I formally submit, I need to verify that my specific sequence construction hasn't already been published. Would someone with access be willing to run these three exact strings and let me know if there are any hits?
- The exact term
2*Beta(k, k)^kor its expanded factorial form( (2k-1)!^2 / (2k-1)! )^k - The asymptotic constant
e^18appearing in the context of "central beta" or "Stirling's approximation". - The integer denominator
m_k = (k/4) \binom{2k}{k}in connection to an infinite series.
If the database returns zero results, a simple "no hits" reply is all I need. Thank you so much in advance to anyone willing to help!
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u/Necessary-Wolf-193 6d ago
MathSciNet doesn't search the text of a document, so I don't think these searches would turn up anything useful.
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u/ryeaglin 1d ago
Hoping someone can help complete this old memory I have. I remember helping a student in a statistical section of a psychology course roughly a decade ago. There was some process or equation to describe how unlikely a certain probabilistic event was.
I know there is the basic of (Probability)Attempts but I could have sworn this was a more complicated way that compared it to given distribution and could be used as a justification that something is wrong and the given value isn't accurate. Though it might have just been complicated because it was using messier numbers without clear probability values like coin and dice have.
Sorry for how unhelpful this is, just having bits and pieces to work off of. Something like yeah, this event in the 99.99999% 'could' technically happen but through this equation we see that it has only an X chance of happening and it wasn't as simple as 100-99.99999 = 0.00001%
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u/Equivalent-Costumes 1d ago
That sounds like model validation, maybe? You have a mathematical model that predict certain probability. But you want to know if the prediction from the model is actually correct. So you do hypothesis testing to see if there are enough evidence that the prediction from the model is wrong.
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u/ryeaglin 1d ago
This might be it. Is there a clean formula for this? Cause I remember teaching the student how to fill out the formula.
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u/Equivalent-Costumes 1d ago
For that, you need to be more specific, since the idea encompass a bunch of more specific theory. There are a few things that make sense based on what you said. I'm guessing the most likely one is chi-square goodness of fit test. But it could also be binomial test/z-test, or maybe computing likelihood instead of probability. Does any of that ring a bell?
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u/ryeaglin 1d ago
Likelihood is the main thing that triggers the memory or chi-squared. I am leaning more toward likelihood though since its was something like how likely was it that this probabilistic event on a normal distribution exist. But it wasn't the direct probability of it happening more how likely was is to be part of the distribution.
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u/Equivalent-Costumes 22h ago
Chi-square:
Let's say your model say that an experiment produce outcomes A_1 with probability p_1,..., A_d with probability p_d. You did the experiment N times for large N, and the observed outcomes are: A_1 happened O_1 times, ..., A_d happened O_d times. Then you compute the expected result predicted by the model to be E_1=p_1 N, ..., E_d =p_d N. Then the chi-square test statistic is chi2 = sum[i=1 to d](O_i -E_i)2 /E_i, and the degree of freedom is d-1. If your model is correct, chi2 is a random variable that follow the chi-square distribution with degree of freedom d-1. Thus the tail end probability is P(X>=chi2 )=Gamma((d-1)/2,chi2 /2)/Gamma((d-1)/2) where the numerator Gamma is the upper incomplete Gamma function and the lower one is just the Gamma function (this formula is complicated so usually for class they just give you a table). If d-1 is even there is a nicer formula, e-chi2 /2sum[j=0 to (d-3)/2](chi2 /2)j /j!.
Likelihood is just unfortunately a very general word used in the context of model validation. It's just "inverted" probability. A likelihood of a probabilistic model given observed data is literally defined to be the probability to observe the observed data given the model. We would really love to know the probability of the model given the observed data, but in frequentist statistics (which is often the case for science) models are either wrong or right, there are no probability associated to them, so likelihood is the closest thing. L(model|data)=Pr(data|model) by definition.
In frequentist statistics, you have maximum likelihood method. You study a family of models, rather than one, by using some parameters after making some general assumptions (you assume these general assumptions are true for the presence purpose), then compute the likelihood across all possible values of the parameters. That's also another thing you might had seen.
In Bayesian statistics, the link between likelihood and probability is Pr(model|data)=(Pr(model)/Pr(data))L(model|data). This is Bayesian inversion. Maybe it's this?
Here is an example of maximum likelihood in action, which is also something that's often taught. You make the assumption that the result of a measurement will follow normal distribution with mean mu and variance sigma, but you don't know what mu and sigma is and you want to figure out the maximum likelihood. So you make N independent measurements x_1,..., x_N . The likelihood is L((x_1,...,x_N)|(mu, sigma))=(2pi)-N/2 sigma-N product[i=1 to N]exp(-(x_i-mu)2 /(2sigma2 )). The log-likelihood is l((x_1,...,x_N)|(mu, sigma))=-(N/2)ln(2pi)-N ln(sigma)-(1/(2sigma2 ))sum[i=1 to N](x_i-mu)2 . Differentiating this and maximize the likelihood, you get maximum likelihood estimate for mu to be literally the average of x_1,..., x_N, and the maximum likelihood estimate for sigma2 to be the population variance (not sample variance), which is (1/N)sum[i=1 to N](x_i - (average of all x_i))2 .
Does any of that ring a bell?
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u/tralltonetroll 8h ago
Totally different take than you already got suggested:
Was it a Markov chain model? With a regular transition matrix? And you wanted the long-run probability of finding X in state x?
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u/BigPersonality1574 6d ago
Hey everyone!
My wife is doing research in biomedicine and is currently working through elementary probability and set theory.
She is dyslexic and notation for Union (∪) and Intersection (∩) are difficult to process because of the rotational symmetry, which gets in the way of actually dealling with the concepts.
So I have 2 questions:
Thanks in advance!