Linked Questions

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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
52 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 ...
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....
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 ...
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(...
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 ...
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