Linked Questions
18 questions linked to/from Fundamental Examples
52
votes
11
answers
6k
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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....
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 ...
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 ...
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 ...
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 ...
67
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 ...
97
votes
50
answers
41k
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 ...
62
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 ...
13
votes
16
answers
2k
views
Individual mathematical objects whose study amounts to a (sub)discipline? [closed]
Certain mathematical objects have a theory so rich that their study
alone arguably constitutes a distinct (sub)discipline. My own list
would begin with
1) the absolute Galois group of the rationals;
...
45
votes
3
answers
4k
views
What was the relative importance of FLT vs. higher reciprocity laws in Kummer's invention of algebraic number theory?
This question is inspired in part by this answer of Bill Dubuque, in which he remarks that the fairly common belief that Kummer was motivated by FLT to develop his theory of cyclotomic number fields ...
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 ...
127
votes
63
answers
27k
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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 ...
52
votes
37
answers
83k
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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 ...
202
votes
72
answers
49k
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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 ...
181
votes
60
answers
42k
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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 ...