Linked Questions
18 questions linked to/from Fundamental Examples
83
votes
28
answers
7k
views
What could be some potentially useful mathematical databases?
This is a soft question but it's not meant as a big-list question. I have recently been asked whether I want to provide feedback at the pre-beta stage on a forthcoming website that will provide a ...
181
votes
60
answers
42k
views
Examples of eventual counterexamples
Define an "eventual counterexample" to be
$P(a) = T $ for $a < n$
$P(n) = F$
$n$ is sufficiently large for $P(a) = T\ \ \forall a \in \mathbb{N}$ to be a 'reasonable' conjecture to ...
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 ...
52
votes
11
answers
6k
views
What is an important mathematical question?
$\DeclareMathOperator\GL{GL}$Many times I have heard people say sentences like X is an important question/ X is a natural question. I find this very surprising because to me it's all a matter of taste....
75
votes
13
answers
8k
views
Counterexamples in PDE
Let us compile a list of counterexamples in PDE, similar in spirit to the books Counterexamples in topology and Counterexamples in analysis. Eventually I plan to type up the examples with their ...
127
votes
63
answers
27k
views
Counterexamples in algebra?
This is certainly related to "What are your favorite instructional counterexamples?", but I thought I would ask a more focused question. We've all seen Counterexamples in analysis and ...
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 ...
68
votes
30
answers
36k
views
A book you would like to write
Writing a book from the beginning to the end is (so I heard) a very hard process. Planning a book is easier. This question is dual in a sense to the question "Books you would like to read (if somebody ...
61
votes
41
answers
22k
views
Most intricate and most beautiful structures in mathematics
In the December 2010 issue of Scientific American, an article "A Geometric Theory of
Everything" by A. G. Lisi and J. O. Weatherall states "... what is arguably the most
intricate structure known to ...
97
votes
50
answers
42k
views
Theorems that are 'obvious' but hard to prove
There are several well-known mathematical statements that are 'obvious' but false (such as the negation of the Banach--Tarski theorem). There are plenty more that are 'obvious' and true. One would ...
8
votes
2
answers
1k
views
Examples of analytic functions to motivate a first course in complex variables
[Changed title as a plea to re-open the question.]
If one is to motivate a course in complex variables, what specific analytic (holomorphic/meromorphic) function of one variable would you cite as an ...
47
votes
9
answers
6k
views
What examples of distributions should I keep in mind?
I'm learning a bit about the theory of distributions. What examples of distributions will help me develop good intuition?
Definitions: Let $U$ be an open subset of $\mathbb{R}^n$. Write $C_c^\infty(...
53
votes
37
answers
83k
views
What is your favorite "strange" function? [closed]
There are many "strange" functions to choose from and the deeper you get involved with math the more you encounter. I consciously don't mention any for reasons of bias. I am just curious what you ...
36
votes
17
answers
6k
views
Canonical examples of algebraic structures
Please list some examples of common examples of algebraic structures. I was thinking answers of the following form.
"When I read about a [insert structure here], I immediately think of [example]."
...
51
votes
25
answers
11k
views
Theorems for nothing (and the proofs for free) [closed]
Some theorems give far more than you feel they ought to: a weak hypothesis is enough to prove a strong result. Of course, there's almost always a lot of machinery hidden below the waterline. Such ...