7
votes
1answer
114 views

Handwaving gone wrong

My motivation for this question is twofold: On one hand, I'm studying algebraic topology, where - at least in the book written by Hatcher - there is quite a lot of handwaving (e.g. maps are continous ...
1
vote
1answer
50 views

Some examples of local and nonlocal properties

Today I learned that continuity at a point is a local property. Concretely, if $f: \mathbb R \to \mathbb R$ is continuous on $[-K,K]$ for all $K \in \mathbb R$ then $f$ is continuous on $ \mathbb ...
4
votes
3answers
63 views

Examples of “eventually reaches y under iteration” other than the Collatz problem

The Collatz conjecture states that iteratively applying the map $$f(n) = \begin{cases} n/2 &\text{if } n \equiv 0 \pmod{2}\\ 3n+1 & \text{if } n\equiv 1 \pmod{2} .\end{cases}$$ to any ...
3
votes
1answer
224 views

Example of non-noetherian algebras which are tensor products of noetherian algebras

We suppose all rings are commutative with unity. I am looking for examples of a tensor product $B\otimes_A C$ which is not noetherian, where $A$ is a noetherian ring and $B, C$ are noetherian ...
7
votes
1answer
108 views

What is your favorite group? [closed]

I would like to know about your favorite group(s). Since groups do appear everywhere in mathematics and there are plenty of them, which ones have drawn your attention the most or surprised you? Please ...
11
votes
1answer
341 views

What are some examples of hard theorems in category theory?

I'm currently learning some category theory, but so far I've used it only as a handy way to talk about various related concepts in algebra and topology with some nice, easy-to-prove lemmas like "left ...
3
votes
3answers
166 views

What are some examples of “exotic” algebraic structures? [closed]

I guess that I'm quite familiar with the basic "everyday algebraic structures" such as groups, rings, modules and algebras and Lie algebras. Of course, I also heard of magmas, semi-groups and monoids, ...
10
votes
0answers
259 views

Papers with unorthodox writing style

I'm not sure if this is the right forum for this question, in any case probably CW is appropriate? I've been looking around the mathblogosphere for the past few weeks and ran into mathgen. It's ...
14
votes
4answers
795 views

Can we get un-obvious results by defining sophisticated topologies?

What I originally found so interesting about general topology was how general a type of thing a topology is, and how the terminology open, closed, compact, continuous, convergence et cetera means ...
4
votes
1answer
204 views

List of proofs of non-trivial theorems which were unnoticed to be wrong for at least a few years

For example, the Weber's proof of Kronecker–Weber theorem. I would like to know such proofs. It seems to be important for me to remember that a widely accepted proof might be wrong.
6
votes
3answers
165 views

Use of infinity as an “idealistic approximation”

There have been several recent posts about the work of N. J. Wildberger, a finitist who seems to think that mathematics should only focus on things that have some sort of "real world" connection, ...
157
votes
21answers
15k views

Nice examples of groups which are not obviously groups

I am searching for some groups, where it is not so obvious that they are groups. In the lectures script there are only examples like $\mathbb{Z}$ under addition and other things like that. I ...
2
votes
0answers
51 views

Are there many spaces which have a regular $G_\delta$-diagonal but is not submetrizable?

Are there many spaces which have a regular $G_\delta$-diagonal but is not submetrizable? Submetrizable = if we can choose a coarser topology on the space $X$ and thus make it a metrizable space. ...
3
votes
1answer
143 views

Worst category with first isomorphism?

I am no expert in category theory, but from VIII of Algebra: Chapter 0 I learnt that In an abelian category every $A\xrightarrow{\phi}B$ can be decomposed into \begin{equation}A\twoheadrightarrow ...
3
votes
1answer
228 views

Convergence Counterexamples

I'm trying to compile a list of counterexamples for convergence implications (or rather, the lack of). I have an incomplete list and I hope to get it all together in one piece. I'm currently working ...
29
votes
9answers
5k views

Examples of nonabelian groups.

Can anybody provide some examples of finite nonabelian groups which are not symmetric groups or dihedral groups?
26
votes
1answer
826 views

Counterexample Math Books

I have been able to find several counterexample books in some math areas. For example: $\bullet$ Counterexamples in Analysis, Bernard R. Gelbaum, John M. H. Olmsted $\bullet$ Counterexamples in ...
10
votes
4answers
200 views

Examples of Monads and their Algebras

I'd like to get some examples of monads; specifically, I'd love a big list of different monads and a description of what their algebras are. Alternatively, online resources and especially exercices ...
148
votes
31answers
8k views

Can't argue with success? Looking for “bad math” that “gets away with it”

I'm looking for cases of invalid math operations producing (in spite of it all) correct results (aka "every math teacher's nightmare"). One example would be "cancelling" the 6s in $$\frac{64}{16}.$$ ...
11
votes
4answers
454 views

Counterexamples in complex analysis

In contrast to other topics in analysis such as functional analysis with its vast amount of counterexamples to intuitively correct looking statements (see here for an example), everything in complex ...
2
votes
0answers
92 views

Ways to look at categories [closed]

The axioms of category theory arise in many contexts structures and maps between them structures and interpretations between them theories and interpretations between them theories and proofs ...
5
votes
4answers
429 views

Applications of quadratic forms

It seems that a lot of great mathematicians spent quite a while of their time studying quadratic forms over $\mathbb{Z},\mathbb{Q},\mathbb{Q_p}$ etc. and there is indeed a vast and detailed theory of ...
21
votes
17answers
1k views

Accidents of small $n$

In studying mathematics, I sometimes come across examples of general facts that hold for all $n$ greater than some small number. One that comes to mind is the Abel–Ruffini theorem, which states that ...
10
votes
3answers
263 views

Deducing results in linear algebra from results in commutative algebra

Here are two examples of results which can be deduced from commutative algebra: Any $n\times n$ complex matrix is conjugate to a Jordan canonical matrix (can be proven using the structure theorem ...
30
votes
17answers
6k views

Examples of mathematical induction

What are the best examples of mathematical induction available at the secondary-school level---totally elementary---that do not involve expressions of the form $\bullet+\cdots\cdots\cdots+\bullet$ ...
267
votes
31answers
16k views

Examples of apparent patterns that eventually fail

Often, when I try to describe mathematics to the layman, I find myself struggling to convince them of the importance and consequence of 'proof'. I receive responses like: "surely if the Collatz ...
2
votes
5answers
4k views

List applications of sets & relations in science/business/tech that a highschooler can understand

What are some applications of sets & relations in science/business/tech that a highschooler can understand? To kindle a young mind, what examples can be given?
31
votes
5answers
1k views

False beliefs about Lebesgue measure on $\mathbb{R}$

I'm trying to develop intuition about Lebesgue measure on $\mathbb{R}$ and I'd like to build a list of false beliefs about it, for example: every set is measurable, every set of measure zero is ...
11
votes
4answers
1k views

Famous uses of the inclusion-exclusion principle?

The standard textbook example of using the inclusion-exclusion principle is for solving the problem of derangement counting; using inclusion-exclusion (and some basic analysis) it can be shown that ...
5
votes
2answers
325 views

Famous Finite Sets [closed]

What are the most famous (or most beautiful, IYO) finite sets in mathematics? I'm especially looking for 'large' sets that contain more than $2^{10} \approx 1000$ but fewer than $2^{20} \approx ...
41
votes
12answers
2k views

Examples of results failing in higher dimensions

A number of economists do not appreciate rigor in their usage of mathematics and I find it very discouraging. One of the examples of rigor-lacking approach are proofs done via graphs or pictures ...
2
votes
0answers
105 views

List of large classes of functors providing morphisms

I recently posted this question on mathoverflow and it was closed as being too localized. I am hoping to more precisely say what I mean here. I recently learned, through my Topology coursework, that ...
1
vote
0answers
110 views

Potential computational questions that could be asked about p-adic numbers and Galois Theory

I have an exam on P-Adic integers (and a bit on Galois Theory) that my professor said would be very computational, but he never does any examples of the theorems he proves in class. He said the exam ...
11
votes
6answers
2k views

Examples of non symmetric distances

It is well known that the symmetric property is $d(x,y)=d(y,x)$ is not necessary in the definition of distance if the triangle inequality is carefully stated. On the other hand there are examples of ...
37
votes
3answers
2k views

What is the spectral theorem for compact self-adjoint operators on a Hilbert space actually for?

Please excuse the naive question. I have had two classes now in which this theorem was taught and proven, but I have only ever seen a single (indirect?) application involving the quantum harmonic ...
174
votes
30answers
12k views

A challenge by R. P. Feynman: give counter-intuitive theorems that can be translated into everyday language

The following is a quote from Surely you're joking, Mr. Feynman . The question is: are there any interesting theorems that you think would be a good example to tell Richard Feynman, as an answer to ...