Questions tagged [examples]

For questions requesting examples of a certain structure or phenomenon

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401 votes
84 answers
185k 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 ...
291 votes
34 answers
50k 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 ...
271 votes
67 answers
136k 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 ...
220 votes
140 answers
48k 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 ...
202 votes
72 answers
49k views

What are your favorite instructional counterexamples?

Related: question #879, Most interesting mathematics mistake. But the intent of this question is more pedagogical. In many branches of mathematics, it seems to me that a good counterexample can be ...
177 votes
80 answers
65k 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.
152 votes
52 answers
23k views

Experimental mathematics leading to major advances

I would like to ask about examples where experimentation by computers has led to major mathematical advances. A new look Now as the question is five years old and there are certainly more examples of ...
148 votes
31 answers
69k 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 ...
139 votes
59 answers
31k views

Jokes in the sense of Littlewood: examples? [closed]

First, let me make it clear that I do not mean jokes of the "abelian grape" variety. I take my cue from the following passage in A Mathematician's Miscellany by J.E. Littlewood (Methuen 1953, p. 79): ...
135 votes
43 answers
37k views

What are the most attractive Turing undecidable problems in mathematics?

What are the most attractive Turing undecidable problems in mathematics? There are thousands of examples, so please post here only the most attractive, best examples. Some examples already appear on ...
114 votes
32 answers
20k views

What notions are used but not clearly defined in modern mathematics?

"Everyone knows what a curve is, until he has studied enough mathematics to become confused through the countless number of possible exceptions." Felix Klein What notions are used but not ...
111 votes
6 answers
10k views

Counterexamples in algebraic topology?

In this thread Books you would like to read (if somebody would just write them...), I expressed my desire for a book with the title "(Counter)examples in Algebraic Topology". My reason for doing so ...
103 votes
17 answers
15k views

Theorems that are essentially impossible to guess by empirical observation

There are many mathematical statements that, despite being supported by a massive amount of data, are currently unproven. A well-known example is the Goldbach conjecture, which has been shown to hold ...
97 votes
46 answers
18k views

Examples of theorems with proofs that have dramatically improved over time

I am looking for examples of theorems that may have originally had a clunky, or rather technical, or in some way non-illuminating proof, but that eventually came to have a proof that people consider ...
95 votes
8 answers
12k views

Mistakes in mathematics, false illusions about conjectures

Since long time ago I have been thinking in two problems that I have not been able to solve. It seems that one of them was recently solved. I have been thinking a lot about the motivation and its ...
91 votes
2 answers
7k views

$A$ is isomorphic to $A \oplus \mathbb{Z}^2$, but not to $A \oplus \mathbb{Z}$

Are there abelian groups $A$ with $A \cong A \oplus \mathbb{Z}^2$ and $A \not\cong A \oplus \mathbb{Z}$?
Martin Brandenburg's user avatar
86 votes
6 answers
16k views

What are the most elegant proofs that you have learned from MO?

One of the things that MO does best is provide clear, concise answers to specific mathematical questions. I have picked up ideas from areas of mathematics I normally wouldn't touch, simply because ...
84 votes
11 answers
12k views

What are examples of (collections of) papers which "close" a field?

There is sometimes talk of fields of mathematics being "closed", "ended", or "completed" by a paper or collection of papers. It seems as though this could happen in two ways: A total characterisation,...
84 votes
34 answers
9k views

books well-motivated with explicit examples

It is ultimately a matter of personal taste, but I prefer to see a long explicit example, before jumping into the usual definition-theorem path (hopefully I am not the only one here). My problem is ...
82 votes
17 answers
11k views

Examples of algorithms requiring deep mathematics to prove correctness

I am looking for examples of algorithms for which the proof of correctness requires deep mathematics ( far beyond what is covered in a normal computer science course). I hope this is not too broad.
81 votes
30 answers
68k views

Applications of the Chinese remainder theorem

As the title suggests I am interested in CRT applications. Wikipedia article on CRT lists some of the well known applications (e.g. used in the RSA algorithm, used to construct an elegant Gödel ...
80 votes
22 answers
15k views

How would you have answered Richard Feynman's challenge?

Reading the autobiography of Richard Feynman, I struck upon the following paragraphs, in which Feynman recall when, as a student of the Princeton physics department, he used to challenge the students ...
80 votes
23 answers
18k views

Algebraic geometry examples

What are some surprising or memorable examples in algebraic geometry, suitable for a course I'll be teaching on chapters 1-2 of Hartshorne (varieties, introductory schemes)? I'd prefer examples that ...
76 votes
34 answers
7k views

Dimension leaps

Many mathematical areas have a notion of "dimension", either rigorously or naively, and different dimensions can exhibit wildly different behaviour. Often, the behaviour is similar for "nearby" ...
74 votes
51 answers
27k views

An example of a beautiful proof that would be accessible at the high school level?

The background of my question comes from an observation that what we teach in schools does not always reflect what we practice. Beauty is part of what drives mathematicians, but we rarely talk about ...
74 votes
1 answer
5k views

$R$ is isomorphic to $R[X,Y]$, but not to $R[X]$

Is there a commutative ring $R$ with $R \cong R[X,Y]$ and $R \not\cong R[X]$? This is a ring-theoretic analog of my previous question about abelian groups: In fact, in any algebraic category we may ...
Martin Brandenburg's user avatar
73 votes
30 answers
9k views

What are some examples of ingenious, unexpected constructions?

Many famous problems in mathematics can be phrased as the quest for a specific construction. Often such constructions were sought after for centuries or even millennia and later proved impossible by ...
73 votes
10 answers
10k views

Riemannian surfaces with an explicit distance function?

I'm looking for explicit examples of Riemannian surfaces (two-dimensional Riemannian manifolds $(M,g)$) for which the distance function d(x,y) can be given explicitly in terms of local coordinates of ...
Terry Tao's user avatar
  • 108k
72 votes
13 answers
10k views

The use of computers leading to major mathematical advances II

I would like to ask about recent examples, mainly after 2015, where experimentation by computers or other use of computers has led to major mathematical advances. This is a continuation of a question ...
70 votes
28 answers
7k views

Examples where it's useful to know that a mathematical object belongs to some family of objects

For an expository piece I'm writing, it would be useful to have good examples of the following phenomenon: (1) ${\cal X}$ is a parameterized family of somethings. (Varieties, schemes, manifolds, ...
69 votes
2 answers
8k views

Galoisian sets of prime numbers

The question is about characterising the sets $S(K)$ of primes which split completely in a given galoisian extension $K|\mathbb{Q}$. Do recent results such as Serre's modularity conjecture (as proved ...
Chandan Singh Dalawat's user avatar
67 votes
28 answers
12k views

Examples of seemingly elementary problems that are hard to solve?

I'm looking for a list of problems such that a) any undergraduate student who took multivariable calculus and linear algebra can understand the statements, (Edit: the definition of understanding here ...
67 votes
10 answers
12k views

Non-homeomorphic spaces that have continuous bijections between them

What are nice examples of topological spaces $X$ and $Y$ such that $X$ and $Y$ are not homeomorphic but there do exist continuous bijections $f: X \to Y$ and $g: Y \to X$?
63 votes
7 answers
8k views

Theorems demoted back to conjectures

Many mathematicians know the Four Color Theorem and its history: there were two alleged proofs in 1879 and 1880 both of which stood unchallenged for 11 years before flaws were discovered. I am ...
62 votes
22 answers
18k views

What's a groupoid? What's a good example of a groupoid? [closed]

Or more specifically, why do people get so excited about them? And what's your favorite easy example of one, which illustrates why I should care (and is not a group)?
62 votes
6 answers
5k views

Simplest examples of nonisomorphic complex algebraic varieties with isomorphic analytifications

If they are not proper, two complex algebraic varieties can be nonisomorphic yet have isomorphic analytifications. I've heard informal examples (often involving moduli spaces), but am not sure of the ...
Ravi Vakil's user avatar
  • 3,837
61 votes
7 answers
16k views

Is there a measure zero set which isn't meagre?

A subset of ℝ is meagre if it is a countable union of nowhere dense subsets (a set is nowhere dense if every open interval contains an open subinterval that misses the set). Any countable set ...
Anton Geraschenko's user avatar
56 votes
28 answers
11k views

Nontrivial question about Fibonacci numbers?

I'm looking for a nontrivial, but not super difficult question concerning Fibonacci numbers. It should be at a level suitable for an undergraduate course. Here is a (not so good) example of the sort ...
56 votes
16 answers
14k views

Examples of using physical intuition to solve math problems

For the purposes of this question let a "physical intuition" be an intuition that is derived from your everyday experience of physical reality. Your intuitions about how the spin of a ball affects ...
56 votes
21 answers
17k views

Wonderful applications of the Vandermonde determinant

This semester I am assisting my mentor teaching a first-year undergraduate course on linear algebra in Peking University, China. And now we have come to the famous Vandermonde determinant, which has ...
56 votes
14 answers
6k views

Does any method of summing divergent series work on the harmonic series?

It's sort of folklore (as exemplified by this old post at The Everything Seminar) that none of the common techniques for summing divergent series work to give a meaningful value to the harmonic series,...
Qiaochu Yuan's user avatar
55 votes
30 answers
7k views

What are examples of good toy models in mathematics?

This post is community wiki. A comment on another question reminded me of this old post of Terence Tao's about toy models. I really like the idea of using toy models of a difficult object to ...
53 votes
15 answers
5k views

Request for examples: verifying vs understanding proofs

My colleague and I are researchers in philosophy of mathematical practice and are working on developing an account of mathematical understanding. We have often seen it remarked that there is an ...
50 votes
15 answers
11k views

Explicit computations using the Haar measure

This question is somewhat related to my previous one on Grassmanians. The few times I've encountered the Haar measure in the course of my mathematical education, it's always been used in a very ...
Thierry Zell's user avatar
  • 4,536
50 votes
12 answers
7k views

Combinatorial results without known combinatorial proofs

Stanley likes to keep a list of combinatorial results for which there is no known combinatorial proof. For example, until recently I believe the explicit enumeration of the de Brujin sequences fell ...
50 votes
4 answers
5k views

Difficult examples for Frankl's union-closed conjecture

Frankl's well-known union-closed conjecture states that if F is a finite family of sets that is closed under taking unions (that is, if A and B belong to the family then so does $A\cup B$), then there ...
gowers's user avatar
  • 28.7k
49 votes
30 answers
7k views

Taking a theorem as a definition and proving the original definition as a theorem

Gian-Carlo Rota's famous 1991 essay, "The pernicious influence of mathematics upon philosophy" contains the following passage: Perform the following thought experiment. Suppose that you are ...
49 votes
6 answers
6k views

What is Yoneda's Lemma a generalization of?

What is Yoneda's Lemma a generalization of? I am looking for examples that were known before category theory entered the stage resp. can be known by students before they start with category theory. ...
49 votes
5 answers
4k views

are there natural examples of classical mechanics that happens on a symplectic manifold that isn't a cotangent bundle?

I'm curious about just how far the abstraction to a symplectic formalism can be justified by appeal to actual physical examples. There's good motivation, for example, for working over an arbitrary ...
symplectomorphic's user avatar
48 votes
5 answers
15k views

Algebraically closed fields of positive characteristic

I'm taking introductory algebraic geometry this term, so a lot of the theorems we see in class start with "Let k be an algebraically closed field." One of the things that's annoyed me is that as far ...
Harrison Brown's user avatar

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