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325
votes
78answers
147k 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 ...
229
votes
29answers
81k views

Mathematical games interesting to both you and a 5+-year-old child

Background: My daughter is 6 years old now, once I wanted to think on some math (about some Young diagrams), but she wanted to play with me... How to make both of us to do what they want ? I guess ...
761
votes
248answers
202k views

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 ...
85
votes
2answers
112k views

Perfectly centered break of a perfectly aligned pool ball rack

Imagine the beginning of a game of pool, you have 16 balls, 15 of them in a triangle <| and 1 of them being the cue ball off to the left of that triangle. Imagine that the rack (the 15 balls in a ...
222
votes
38answers
93k views

Examples of unexpected mathematical images

I try to generate a lot of examples in my research to get a better feel for what I am doing. Sometimes, I generate a plot, or a figure, that really surprises me, and makes my research take an ...
199
votes
29answers
123k views

Intuitive crutches for higher dimensional thinking

I once heard a joke (not a great one I'll admit...) about higher dimensional thinking that went as follows- An engineer, a physicist, and a mathematician are discussing how to visualise four ...
106
votes
21answers
13k views

Good “casual” advanced math books

I'm curious if there are any good math books out there that take a "casual approach" to higher level topics. I'm very interested in advanced math, but have lost the time as of late to study textbooks ...
102
votes
1answer
29k views

What is the definition of the function T used in Atiyah's attempted proof of the Riemann Hypothesis?

In Michael Atiyah's paper purportedly proving the Riemann hypothesis, he relies heavily on the properties of a certain function $T(s)$, known as the Todd function. My question is, what is the ...
32
votes
8answers
177k views

The factorial of -1, -2, -3,

Well, $n!$ is for integer $n < 0$ not defined — as yet. So the question is: How could a sensible generalization of the factorial for negative integers look like? Clearly a good generalization ...
338
votes
78answers
118k views

Video lectures of mathematics courses available online for free

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 ...
84
votes
24answers
13k views

Noteworthy, but not so famous conjectures resolved recent years

Conjectures play important role in development of mathematics. Mathoverflow gives an interaction platform for mathematicians from various fields, while in general it is not always easy to get in ...
120
votes
45answers
14k views

Old books you would like to have reprinted with high-quality typesetting

There are some questions on mathoverflow such as What out-of-print books would you like to see re-printed? Old books still used with answers that tell us things such as: Mathematicians prefer to ...
271
votes
7answers
127k 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 ...
63
votes
12answers
90k views

If you break a stick at two points chosen uniformly, the probability the three resulting sticks form a triangle is 1/4. Is there a nice proof of this?

There is a standard problem in elementary probability that goes as follows. Consider a stick of length 1. Pick two points uniformly at random on the stick, and break the stick at those points. What ...
169
votes
20answers
26k views

How can a mathematician handle the pressure to discover something new?

Suppose I'm an aspiring mathematician-to-be, who started doing research. Although this is really what I love doing, I found that one disturbing point is that there's always the pressure of discovering ...
178
votes
35answers
113k views

Best Algebraic Geometry text book? (other than Hartshorne)

I think (almost) everyone agrees that Hartshorne's Algebraic Geometry is still the best. Then what might be the 2nd best? It can be a book, preprint, online lecture note, webpage, etc. One suggestion ...
88
votes
77answers
20k views

Tweetable Mathematics

Update: Please restrict your answers to "tweets" that give more than just the statement of the result, and give also the essence (or a useful hint) of the argument/novelty. I am looking for ...
355
votes
15answers
49k views

Why do roots of polynomials tend to have absolute value close to 1?

While playing around with Mathematica I noticed that most polynomials with real coefficients seem to have most complex zeroes very near the unit circle. For instance, if we plot all the roots of a ...
60
votes
18answers
67k views

Reading list for basic differential geometry?

I'd like to ask if people can point me towards good books or notes to learn some basic differential geometry. I work in representation theory mostly and have found that sometimes my background is ...
163
votes
45answers
88k 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 ...
61
votes
12answers
74k views

What practical applications does set theory have?

I am a non-mathematician. I'm reading up on set theory. It's fascinating, but I wonder if it's found any 'real-world' applications yet. For instance, in high school when we were learning the ...
113
votes
2answers
45k views

Estimating the size of solutions of a diophantine equation

A. Is there natural numbers $a,b,c$ such that $\frac{a}{b+c} + \frac{b}{a+c} + \frac{c}{a+b}$ is equal to an odd natural number ? (I do not know any such numbers). B. Suppose that $\frac{a}{b+c} + \...
158
votes
22answers
34k views

The most outrageous (or ridiculous) conjectures in mathematics

The purpose of this question is to collect the most outrageous (or ridiculous) conjectures in mathematics. An outrageous conjecture is qualified ONLY if: 1) It is most likely false (Being hopeless ...
260
votes
42answers
77k views

Widely accepted mathematical results that were later shown to be wrong?

Are there 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 later, possibly ...
142
votes
32answers
52k views

What is convolution intuitively?

If random variable $X$ has a probability distribution of $f(x)$ and random variable $Y$ has a probability distribution $g(x)$ then $(f*g)(x)$, the convolution of $f$ and $g$, is the probability ...
174
votes
30answers
68k 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, e.g....
120
votes
30answers
54k views

What are the most misleading alternate definitions in taught mathematics?

I suppose this question can be interpreted in two ways. It is often the case that two or more equivalent (but not necessarily semantically equivalent) definitions of the same idea/object are used in ...
290
votes
105answers
57k 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 ...
47
votes
6answers
8k views

Why isn't integral defined as the area under the graph of function?

In order to define Lebesgue integral, we have to develop some measure theory. This takes some effort in the classroom, after which we need additional effort of defining Lebesgue integral (which also ...
239
votes
110answers
62k views

What are some examples of colorful language in serious mathematics papers?

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 ...
129
votes
21answers
26k views

Mathematical software wish list

Like many other mathematicians I use mathematical software like SAGE, GAP, Polymake, and of course $\LaTeX$ extensively. When I chat with colleagues about such software tools, very often someone has ...
146
votes
58answers
75k views

Interesting mathematical documentaries

I am looking for mathematical documentaries, both technical and non-technical. They should be "interesting" in that they present either actual mathematics, mathematicians or history of mathematics. I ...
127
votes
26answers
32k views

Has philosophy ever clarified mathematics?

I've recently been reading some standard textbooks on the philosophy of mathematics, and I've become quite frustrated that (surely due to my own limitations) I don't seem to be gleaning any ...
185
votes
11answers
53k 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 ...
54
votes
8answers
33k views

Example of a good Zero Knowledge Proof.

I am working on my zero knowledge proofs and I am looking for a good example of a real world proof of this type. An even better answer would be a Zero Knowledge Proof that shows the statement isn't ...
34
votes
6answers
68k views

Fourier vs Laplace transforms

In solving a linear system, when would I use a Fourier transform versus a Laplace transform? I am not a mathematician, so the little intuition I have tells me that it could be related to the boundary ...
131
votes
45answers
23k views

17 camels trick

The following popular mathematical parable is well known: A father left 17 camels to his three sons and, according to the will, the eldest son should be given a half of all camels, the middle son ...
145
votes
36answers
32k views

Most harmful heuristic?

What's the most harmful heuristic (towards proper mathematics education), you've seen taught/accidentally taught/were taught? When did handwaving inhibit proper learning?
61
votes
12answers
48k views

Why is the gradient normal?

This is a somewhat long discussion so please bear with me. There is a theorem that I have always been curious about from an intuitive standpoint and that has been glossed over in most textbooks I ...
109
votes
17answers
47k views

Periods and commas in mathematical writing

I just realized that I am a barbarian when it comes to writing. But I am not entirely sure, so this might be the right place to ask. When typing display-mode formulae do you guys add a period after ...
253
votes
26answers
45k 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 should ...
70
votes
4answers
17k views

The enigmatic complexity of number theory

One of the most salient aspects of the discipline of number theory is that from a very small number of definitions, entities and axioms one is led to an extraordinary wealth and diversity of theorems, ...
38
votes
18answers
61k views

Suggestions for a good Measure Theory book

I have taken analysis and have looked at different measures, but I am currently looking at realizing a certain problem in a different light and feel that I need a better background in various measures ...
110
votes
61answers
13k views

Nonequivalent definitions in Mathematics

I would like to ask if anyone could share any specific experiences of discovering nonequivalent definitions in their field of mathematical research. By that I mean discovering that in different ...
100
votes
53answers
44k views

Which popular games are the most mathematical?

I consider a game to be mathematical if there is interesting mathematics (to a mathematician) involved in the game's structure, optimal strategies, practical strategies, analysis of the game ...
6
votes
4answers
49k views

Lorentzian vs Gaussian Fitting Functions

This is probably too general a question to ask without some specific context, but I'm going to give it a shot anyway: What are the practical differences between using a Lorentzian function and using ...
147
votes
29answers
27k views

Proposals for polymath projects

Background Polymath projects are a form of open Internet collaboration aimed towards a major mathematical goal, usually to settle a major mathematical problem. This is a concept introduced in 2009 by ...
124
votes
11answers
17k views

Do you know important theorems that remain unknown?

Do you know of any very important theorems that remain unknown? I mean results that could easily make into textbooks or research monographs, but almost nobody knows about them. If you provide an ...
67
votes
23answers
17k views

Which popular games have been studied mathematically?

I'm planning out some research projects I could do with undergraduates, and it struck me that problems analyzing games might be appropriate. As an abstract homotopy theorist, I have no experience with ...
102
votes
11answers
10k views

Examples of notably long or difficult proofs that only improve upon existing results by a small amount

I was recently reading Bui, Conrey and Young's 2011 paper "More than 41% of the zeros of the zeta function are on the critical line", in which they improve the lower bound on the proportion of zeros ...