All Questions
Tagged with examples soft-question
45
questions
16
votes
4
answers
6k
views
Mathematicians learning from applications to other fields
Once upon a time a speaker at the weekly Applied Mathematics Colloquium at MIT (one of two weekly colloquia in the math department (but the other one is not called "pure")) said researchers ...
6
votes
2
answers
406
views
Common/well-known results with natural and/or useful reformulations
$\DeclareMathOperator{\pp}{\mathbb{P}}$My aim here is to have a collection of "natural" not-so-common reformulations/extensions of common/well-known results such that
the reformulation/...
80
votes
22
answers
14k
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 ...
54
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 ...
7
votes
3
answers
635
views
Examples of complicated parametric Jordan curves
For test purposes I need parametric Jordan curves that are complicated in the sense of having many inflection points and ideally no symmetries.
When doing online search I always land at complex ...
- 12.5k
1
vote
1
answer
342
views
Examples of "irregularities" in mathematics, other than prime numbers [closed]
Prime numbers are the prime example (no pun intended) for something that arises apparently without describable patters; we know that infinitely many exist, that gaps between them can be arbitrarily ...
10
votes
1
answer
474
views
Examples of proofs using induction or recursion on a big recursive ordinal
There are many proofs use induction or recursion on $\omega$, or on an arbitary (may be uncountable) ordinal. Are there some good examples of proofs which use a big but computable ordinal?
The ...
- 475
82
votes
11
answers
11k
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,...
0
votes
2
answers
437
views
When was the generalization easier to prove than the specific case? [duplicate]
I distinctly remember from my long-ago undergraduate math that there were some interesting cases where a solution (proof) was sought for some specific thing but it wasn't easy to find - and in a few ...
9
votes
0
answers
601
views
Does anyone use non-sober topological spaces?
Recall that a sober space is a topological space such that every irreducible closed subset is the closure of exactly one point.
Is there any area of mathematics outside of general topology where non-...
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 ...
9
votes
3
answers
2k
views
Need examples of homotopy orbit and fixed points
I am no expert in equivariant homotopy theory. Let's say, I am planing to give a talk on homotopy fixed points and orbits. My audience will be graduate students who are doing algebraic topology or ...
- 2,013
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 ...
5
votes
1
answer
661
views
Examples of "nice" properties of algebraic extensions of $\mathbb{Q}$
I am writing a short survey of some "nice'' properties of algebraic extensions of $\mathbb{Q}$. Let's say a property (P) is nice if
every finite extension of $\mathbb{Q}$ satisfies (P), and
if $K \...
1
vote
1
answer
516
views
Examples of Quot schemes
I'm studying Quot schemes, that I denote with $Quot_{N,X,P}$, with $N \in \mathbb{Z}$, $X \subset \mathbb{P}^d$ and $P \in \mathbb{Q}[t]$. So, I'm looking for explicit examples of Quot schemes. Could ...
- 243
8
votes
2
answers
377
views
Natural $\Pi^1_2$ (or worse) classes of structures?
(To clarify, my interest is mainly lightface, that is, $\Pi^1_2$ instead of $\bf \Pi^1_2$, although it doesn't particularly matter.)
This is just an idle curiosity. In logic, I find myself frequently ...
- 19k
30
votes
14
answers
4k
views
An example of a proof that is explanatory but not beautiful? (or vice versa)
This question has a philosophical bent, but hopefully it will evoke straightforward, mathematical answers that would be appropriate for this list (like my earlier question about beautiful proofs ...
7
votes
7
answers
3k
views
Gelfand representation and functional calculus applications beyond Functional Analysis
I think it is fair to say that the fields of Operator Algebras, Operator Theory, and Banach Algebras rely on Gelfand representation and functional calculus in a crucial way.
I am curious about ...
12
votes
1
answer
681
views
What's the role of $H^{p}(\mathbb{R}^{n})$ in modern (harmonic) analysis?
The classical theory of $H^p$,due to it's heavy dependence on the complex function theory(such as Blaschke products), seemed to have an insurmountable obstacle barrying its extension to several ...
- 1,594
1
vote
0
answers
168
views
Algebraic properties of the semiring of open subsets.
Does anyone know of a useful general topological application of the algebraic properties of the semiring of open subsets of some topological space? Or examples of any such nontrivial properties at all?...
- 3,435
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 ...
19
votes
14
answers
4k
views
Excellent uses of induction and recursion
Can you make an example of a great proof by induction or construction by recursion?
Given that you already have your own idea of what "great" means, here it can also be taken to mean that the chosen ...
5
votes
5
answers
3k
views
Easy and Hard problems in Mathematics [closed]
Modified question:
I would like to know some examples of problems in Mathematics, for pedagogical purposes, which do not involve difficult techiques to solve the problem but with a change of context ...
5
votes
2
answers
1k
views
Is beauty at the high school level even possible? [closed]
This question is a follow up to 74841, and follows from a suggestion by Gian-Carlo Rota that beauty as judged by the educated public differs from that experienced by mathematicians (he gives Euclidean ...
68
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 ...
73
votes
49
answers
26k
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 ...
6
votes
3
answers
430
views
Non-trivial integral forms of algebras
Suppose $\mathcal{A}$ is a $\mathbf{C}$-algebra then an integral form would be a subring $\mathcal{B} \subset \mathcal{A}$ such that the canonical map $\mathcal{B} \otimes_{\mathbf{Z}} \mathbf{C} \...
- 731
7
votes
10
answers
1k
views
Examples of "Unusual" Classifications
When one says "classification" in math, usually one of a handful of examples springs to mind:
-Classification of Finite Simple Groups with 18 infinite families and 26 sporadic examples (assuming one ...
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 ...
24
votes
3
answers
16k
views
Advanced Math Jokes [closed]
I am looking for jokes which involve some serious mathematics. Sometimes, a totally absurd argument is surprisingly convincing and this makes you laugh. I am looking for jokes which make you laugh and ...
1
vote
1
answer
1k
views
Classification Problems [closed]
I was thinking about the famous question in philosophy of mathematics: "When are two proofs the same?" and I was wondering if we could somehow "classify" proofs by establishing some sort of functorial ...
266
votes
67
answers
134k
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 ...
138
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):
...
19
votes
4
answers
4k
views
What are your favorite finite non-commutative rings?
When you are checking a conjecture or working through a proof, it is nice to have a collection of examples on hand.
There are many convenient examples of commutative rings, both finite and infinite, ...
8
votes
3
answers
2k
views
The harmonic (series) beetle: live illustrations of mathematical theorems
In my analysis class I use the following problem to illustrate the divergence
of the harmonic series (consider this as a hint for solving it).
Exercise.
A beetle creeps along a 1-meter infinitely ...
201
votes
72
answers
48k
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 ...
55
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 ...
1
vote
1
answer
2k
views
What are examples of theorems get extensions based on simple observation?
Here are some examples illustrate what I meant:
Bonnet-Myers:Bonnet in 1855 proved n=2 case, Myers in 1941 extended to any dimension using the same idea.
Bishop-Gromov Volume comparison: Bishop knew ...
15
votes
27
answers
3k
views
Justifying a theory by a seemingly unrelated example
Here is a topic in the vein of Describe a topic in one sentence and Fundamental examples : imagine that you are trying to explain and justify a mathematical theory T to a skeptical mathematician ...
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 ...
148
votes
31
answers
68k
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 ...
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" ...
218
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 ...
43
votes
77
answers
10k
views
What is the first interesting theorem in (insert subject here)? [closed]
In most students' introduction to rigorous proof-based mathematics, many of the initial exercises and theorems are just a test of a student's understanding of how to work with the axioms and unpack ...
178
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.