# All Questions

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### Examples of common false beliefs in mathematics.

The first thing to say is that this is not the same as the question about interesting mathematical mistakes. I am interested about the type of false beliefs that many intelligent people have while ...
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### Polynomial bijection from $\mathbb Q\times\mathbb Q$ to $\mathbb Q$?

Is there any polynomial $f(x,y)\in{\mathbb Q}[x,y]{}\$ such that $f:\mathbb{Q}\times\mathbb{Q} \rightarrow\mathbb{Q}$ is a bijection?
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### Thinking and Explaining

How big a gap is there between how you think about mathematics and what you say to others? Do you say what you're thinking? Please give either personal examples of how your thoughts and words ...
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It can be difficult to learn mathematics on your own from textbooks, and I often wish universities videotaped their mathematics courses and distributed them for free online. Fortunately, some ...
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Lagrange proved that every nonnegative integer is a sum of 4 squares. Gauss proved that every nonnegative integer is a sum of 3 triangular numbers. Is there a 2-variable polynomial $f(x,y) \in ... 7answers 78k views ### Philosophy behind Mochizuki's work on the ABC conjecture Mochizuki has recently announced a proof of the ABC conjecture. It is far too early to judge its correctness, but it builds on many years of work by him. Can someone briefly explain the philosophy ... 11answers 11k views ### Refereeing a Paper [closed] I've refereed at least a dozen papers in my (short) career so far and I still find the process completely baffling. I'm wondering what is actually expected and what people tend to do... Some things ... 62answers 79k views ### Proofs without words Can you give examples of proofs without words? In particular, can you give examples of proofs without words for non-trivial results? (One could ask if this is of interest to mathematicians, and I ... 29answers 13k views ### Which journals publish expository work? I wonder if anyone else has noticed that the market for expository papers in mathematics is very narrow (more so than it used to be, perhaps). Are there any journals which publish expository work, ... 108answers 37k views ### What are some examples of colorful language in serious mathematics papers? [closed] The popular MO question "Famous mathematical quotes" has turned up many examples of witty, insightful, and humorous writing by mathematicians. Yet, with a few exceptions such as Weyl's "angel of ... 79answers 21k views ### Not especially famous, long-open problems which anyone can understand Question: I'm asking for a big list of not especially famous, long open problems that anyone can understand. Community wiki, so one problem per answer, please. Motivation: I plan to use this list ... 7answers 7k views ### Two commuting mappings in the disk Suppose that$f$and$g$are two commuting continuous mappings from the closed unit disk (or, if you prefer, the closed unit ball in$R^n$) to itself. Does there always exist a point$x$such that ... 41answers 50k views ### A single paper everyone should read? [closed] Different people like different things in math, but sometimes you stand in awe before a beautiful and simple, but not universally known, result that you want to share with any of your colleagues. Do ... 64answers 22k views ### Proofs that require fundamentally new ways of thinking [closed] I do not know exactly how to characterize the class of proofs that interests me, so let me give some examples and say why I would be interested in more. Perhaps what the examples have in common is ... 14answers 7k views ### What elementary problems can you solve with schemes? I'm a graduate student who's been learning about schemes this year from the usual sources (e.g. Hartshorne, Eisenbud-Harris, Ravi Vakil's notes). I'm looking for some examples of elementary ... 36answers 24k views ### Why is a topology made up of 'open' sets? [closed] I'm ashamed to admit it, but I don't think I've ever been able to genuinely motivate the definition of a topological space in an undergraduate course. Clearly, the definition distills the essence of ... 31answers 14k views ### What should be learned in a first serious schemes course? I've just finished teaching a year-long "foundations of algebraic geometry" class. It was my third time teaching it, and my notes are gradually converging. I've enjoyed it for a number of reasons ... 8answers 4k views ### Resources for mathematics advising. This question is possibly ill-advised. (If it is not right for this site I will delete it.) I, suddenly, have students. It is very clear to me that there is nothing in my education that has ... 26answers 12k views ### What are some reasonable-sounding statements that are independent of ZFC? Every now and then, somebody will tell me about a question. When I start thinking about it, they say, "actually, it's undecidable in ZFC." For example, suppose A is an abelian group such that every ... 0answers 9k views ### Ultrafilters and automorphisms of the complex field It is well-known that it is consistent with$ZF$that the only automorphisms of the complex field$\mathbb{C}$are the identity map and complex conjugation. For example, we have that ... 67answers 33k views ### Awfully sophisticated proof for simple facts [closed] It is sometimes the case that one can produce proofs of simple facts that are of disproportionate sophistication which, however, do not involve any circularity. For example, (I think) I gave an ... 33answers 27k views ### Widely accepted mathematical results that were later shown wrong? I wonder if there are any examples in the history of mathematics of a mathematical proof that was initially reviewed and widely accepted as valid, only to be disproved a significant amount of time ... 130answers 26k views ### Fundamental Examples It is not unusual that a single example or a very few shape an entire mathematical discipline. Can you give examples for such examples? (One example, or few, per post, please) I'd love to learn about ... 36answers 20k views ### Demonstrating that rigour is important Any pure mathematician will from time to time discuss, or think about, the question of why we care about proofs, or to put the question in a more precise form, why we seem to be so much happier with ... 3answers 5k views ### A Game on Noetherian Rings A friend suggested the following combinatorial game. At any time, the state of the game is a (commutative) Noetherian ring$\neq 0$. On a player's turn, that player chooses a nonzero non-unit element ... 30answers 30k views ### Real-world applications of mathematics, by arxiv subject area? What are the most important applications outside of mathematics of each of the major fields of mathematics? For concreteness, let's divide up mathematics according to arxiv mathematics categories, ... 3answers 4k views ### Analytic tools in algebraic geometry This is not a very precise question, but I hope it will get some good answers. As someone with a background in smooth manifold theory, I have experienced algebraic geometry as a beautiful but foreign ... 61answers 18k views ### Your favorite surprising connections in Mathematics There are certain things in mathematics that have caused me a pleasant surprise -- when some part of mathematics is brought to bear in a fundamental way on another, where the connection between the ... 75answers 30k views ### Best online mathematics videos? I know of two good mathematics videos available online, namely: Sphere inside out (part I and part II) Moebius transformation revealed Do you know of any other good math videos? Share. 7answers 7k views ### Proofs of Bott periodicity K-theory sits in an intersection of a whole bunch of different fields, which has resulted in a huge variety of proof techniques for its basic results. For instance, here's a scattering of proofs of ... 60answers 13k views ### Suggestions for good notation I occasionally come across a new piece of notation so good that it makes life easier by giving a better way to look at something. Some examples: Iverson introduced the notation [X] to mean 1 if X is ... 7answers 6k views ### Does$\mathrm{Aut}(\mathrm{Aut}(…\mathrm{Aut}(G)…))$stabilize? Purely for fun, I was playing around with iteratively applying$\DeclareMathOperator{\Aut}{Aut}\Aut$to a group$G$; that is, studying groups of the form $${\Aut}^n(G):= \Aut(\Aut(...\Aut(G)...))$$ ... 17answers 13k views ### Geometric Interpretation of Trace This afternoon I was speaking with some graduate students in the department and we came to the following quandry; Is there a geometric interpretation of the trace of a matrix? This question ... 15answers 10k views ### When and how is it appropriate for an undergraduate to email a professor out of the blue? This may not be appropriate for MathOverflow, as I haven't seen precedent for this type of question. But the answer is certainly of interest to me, and (I think) would be of interest to many other ... 6answers 24k views ### Where to buy premium white chalk in the U.S., like they have at RIMS? While not a research-level math question, I'm sure this is a question of interest to many research-level mathematicians, whose expertise I seek. At RIMS (in Kyoto) in 2005, they had the best white ... 37answers 58k views ### Too old for advanced mathematics? [closed] Kind of an odd question, perhaps, so I apologize in advance if it is inappropriate for this forum. I've never taken a mathematics course since high school, and didn't complete college. However, ... 14answers 8k views ### Is a free alternative to MathSciNet possible? How could a free (i.e. free content) alternative for MathSciNet and Zentralblatt be created? Comments Some mathematicians have stopped writing reviews for MathSciNet because they feel their ... 1answer 8k views ### What are the shapes of rational functions? I would like to understand and compute the shapes of rational functions, that is, holomorphic maps of the Riemann sphere to itself, or equivalently, ratios of two polynomials, up to Moebius ... 22answers 9k views ### Why do so many textbooks have so much technical detail and so little enlightenment? [closed] I think/hope this is okay for MO. I often find that textbooks provide very little in the way of motivation or context. As a simple example, consider group theory. Every textbook I have seen that ... 28answers 20k views ### Cool problems to impress students with group theory [closed] Since this forum is densely populated with algebraists, I think I'll ask it here. I'm teaching intermediate level algebra this semester and I'd like to entertain my students with some clever ... 41answers 47k views ### Magic trick based on deep mathematics I am interested in magic tricks whose explanation requires deep mathematics. The trick should be one that would actually appeal to a layman. An example is the following: the magician asks Alice to ... 35answers 12k views ### Books you would like to read (if somebody would just write them…) I think that the title is self-explanatory but I'm thinking about mathematical subjects that have not received a full treatment in book form or if they have, they could benefit from a different ... 14answers 11k views ### When should a supervisor be a co-author? What are people's views on this? To be specific: suppose a PhD student has produced a piece of original mathematical research. Suppose that student's supervisor suggested the problem, and gave a few ... 67answers 27k views ### Math puzzles for dinner [closed] You're hanging out with a bunch of other mathematicians - you go out to dinner, you're on the train, you're at a department tea, et cetera. Someone says something like "A group of 100 people at a ... 16answers 22k views ### What's a mathematician to do? [closed] I have to apologize because this is not the normal sort of question for this site, but there have been times in the past where MO was remarkably helpful and kind to undergrads with similar types of ... 12answers 12k views ### Have any long-suspected irrational numbers turned out to be rational? The history of proving numbers irrational is full of interesting stories, from the ancient proofs for$\sqrt{2}$, to Lambert's irrationality proof for$\pi$, to Roger Apéry's surprise demonstration ... 3answers 4k views ### Is R^3 the square of some topological space? The other day, I was idly considering when a topological space has a square root. That is, what spaces are homeomorphic to$X \times X$for some space$X$.$\mathbb{R}$is not such a space: If$X ...
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For a while I have been eying stand-alone eBook readers that use "electronic ink" displays, the most popular ones seem to be the Amazon Kindle readers. My criteria are as follows: It should be able ...