All Questions
95
questions
1
vote
0
answers
203
views
Results that hold for the complex numbers but not for algebraically closed fields of characteristic zero
When a result is stated for the field of complex numbers it can usually be extended to a result for an algebraically closed field of characteristic zero. I would like to see a list of results that ...
- 147
0
votes
0
answers
167
views
A zoo of derivations
Recall that given a $k$-algebra $A$, a derivation on $A$ is a $k$-linear morphism $d:A\to A$ such that $$d(ab)=d(a)b+ad(b).$$
The use of derivations is of paramount importance in mathematics. I think ...
3
votes
1
answer
496
views
Novel examples, proofs or results in mathematics from arithmetic billiards
The goal of the post is get a repository of mathematical results, proofs or examples by users of the site, arising from arithmetic billiards in number theory, analysis, geometry,….
Wikipedia has an ...
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/...
103
votes
17
answers
15k
views
Theorems that are essentially impossible to guess by empirical observation
There are many mathematical statements that, despite being supported by a massive amount of data, are currently unproven. A well-known example is the Goldbach conjecture, which has been shown to hold ...
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 ...
49
votes
30
answers
7k
views
Taking a theorem as a definition and proving the original definition as a theorem
Gian-Carlo Rota's famous 1991 essay, "The pernicious influence of mathematics upon philosophy" contains the following passage:
Perform the following thought experiment. Suppose that you are ...
72
votes
13
answers
10k
views
The use of computers leading to major mathematical advances II
I would like to ask about recent examples, mainly after 2015, where experimentation by computers or other use of computers has led to major mathematical advances.
This is a continuation of a question ...
27
votes
1
answer
825
views
"Non-categorical" examples of $(\infty, \infty)$-categories
This title probably seems strange, so let me explain.
Out of the several different ways of modeling $(\infty, n)$-categories, complicial
sets and comical sets allow $n = \infty$,
providing ...
43
votes
10
answers
5k
views
What are some examples of proving that a thing exists by proving that the set of such things has positive measure?
Suppose we want to prove that among some collection of things, at least one
of them has some desirable property. Sometimes the easiest strategy is to
equip the collection of all things with a measure, ...
0
votes
1
answer
308
views
Examples of additive categories [closed]
I already this question here but I didn't get any satisfactory answer, so I will try in MO now.
There are a lot of interesting and creative examples of categories, such as for example, the category ...
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 ...
13
votes
2
answers
1k
views
Contrasting theorems in classical logic and constructivism
Is it possible there are examples of where classical logic proves a theorem that provably is false within constructivism? Is so what are some examples?
What are some examples of most contrasting ...
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 ...
4
votes
1
answer
325
views
Very canonical constructions
You have two categories $C_1$ and $C_2$. We call a map of the classes $\mathrm{Ob}(C_1)\rightarrow \mathrm{Ob}(C_2)$ a construction. Sometimes you can find a functor $C_1\rightarrow C_2$ inducing this ...
10
votes
0
answers
196
views
Examples of automorphic representations to keep in mind
I have recently started studying the automorphic science and find it somewhat hard to form intuition. Can we have a list of examples of automorphic representations that you usually use to test a new ...
user140615
33
votes
8
answers
3k
views
Big list of comonads
The concept of a monad is very well established, and there are very many examples of monads pertaining almost all areas of mathematics.
The dual concept, a comonad, is less popular.
What are ...
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 ...
4
votes
0
answers
158
views
What are some interesting examples of cooperative games that can be naturally generalised to a stochastic version of it?
In classical, deterministic cooperative game theory, there are $N$ players that can form $2^{N}$ coalitions. Each of these coalitions is assigned a value by means of the characteristic function $v ( \...
- 4,226
18
votes
7
answers
2k
views
Examples of residually-finite groups
One of the main reasons I only supervised one PhD student is that I find it hard to find an appropriate topic for a PhD project. A good approach, in my view, is to have on the one hand a list of ...
7
votes
1
answer
457
views
Example of a smooth family of projective surfaces with non-vanishing integrals of Todd classes
Motivation:
Let $\pi\colon S \rightarrow B$ be smooth projective morphism of relative dimension 2 over a smooth projective scheme $B$. If the stucture sheaves of the fibres do not have higher ...
- 76
2
votes
0
answers
179
views
Right adjoint completions
Forgive me if this question is not well thought out. I don't know how else to ask it.
The nlab page on completion gives some examples of completions which are left adjoints. These completions are "...
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 ...
82
votes
17
answers
11k
views
Examples of algorithms requiring deep mathematics to prove correctness
I am looking for examples of algorithms for which the proof of correctness requires deep mathematics ( far beyond what is covered in a normal computer science course).
I hope this is not too broad.
10
votes
1
answer
611
views
Problems which use S₄ → S₃
I need examples of problems which use, directly or indirectly, the homomorphism $S_4\to S_3$ in the solution (its kernel is $\mathbb{Z}_2\oplus\mathbb{Z}_2$).
Obvious candidates:
Lagrange resolvent (...
- 42.4k
16
votes
5
answers
1k
views
What are examples when the equality of some invariants is good enough in algebraic topology?
As far as my understanding goes, most of the tools of algebraic topology (homotopy groups, homology groups, cup product, cohomology operations, Hopf invariant, signature, characteristic classes, knot ...
3
votes
2
answers
293
views
Examples of TVS with no non-trivial open convex subsets
I give here the classical example of the space $E = L^p([0,1])$ which has no open convex subsets apart from $\emptyset$ and $E$. Consequently, there is no non-trivial continuous linear form on $E$.
...
- 1,355
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 ...
30
votes
11
answers
5k
views
What are your favorite concrete examples of limits or colimits that you would compute during lunch?
(The title was initially "What are your favorite concrete examples that you would compute on the table during lunch to convince a working mathematician that the notions of limits and colimits are not ...
14
votes
2
answers
671
views
Occurrences of D. H. Lehmer's 10-th degree polynomial
Salem numbers and Lehmer's minimum height problem are venerated not only in number theory and diophantine analysis, where they are considered naturally interesting for their own sake, but also in ...
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 ...
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 ...
37
votes
12
answers
3k
views
Interesting conjectures "discovered" by computers and proved by humans?
There are notable examples of computers "proving" results discovered by mathematicians, what about the opposite:
Are there interesting conjectures "discovered" by computers and proved by humans?
...
33
votes
3
answers
3k
views
Arithmetic geometry examples
(This is inspired by Algebraic geometry examples.)
I want to collect here (counter)examples in arithmetic geometry.
Curves violating the Hasse principle: The Selmer curve $3X^3 + 4Y^3 + 5Z^3 = 0$. ...
73
votes
30
answers
9k
views
What are some examples of ingenious, unexpected constructions?
Many famous problems in mathematics can be phrased as the quest for a specific construction. Often such constructions were sought after for centuries or even millennia and later proved impossible by ...
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 ...
18
votes
1
answer
1k
views
What are "good" examples of string manifolds?
Based on this mathoverlow question, I would like to have a similar list for the case of string manifolds. An $n$-dim. Riemannian manifold $M$ is said to be string, if the classifying map of its bundle ...
46
votes
4
answers
9k
views
What are "good" examples of spin manifolds?
I'm trying to get a grasp on what it means for a manifold to be spin. My question is, roughly:
What are some "good" (in the sense of illustrating the concept) examples of manifolds which are spin (...
0
votes
5
answers
3k
views
Examples of finite dimensional non simple non abelian Lie algebras
Hello, I have recently started reading about Lie algebras. However all the examples I have encountered so far are simple and semisimple Lie algebras. Thus I would love to see ...
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 ...
8
votes
6
answers
1k
views
Spaces of filters
This question arose more from curiosity than from an actual problem. There are situations when you embed some space $X$ in a set of filters on $X$, which inherits properties of $X$ or has even better ...
7
votes
10
answers
1k
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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 ...
9
votes
6
answers
4k
views
Examples of naturally occurring Quadratic forms or quadrics.
I am always fascinated when a quadratic form (or a quadric) arises naturally. I have
some elementary examples, but most of all, I want to learn more examples. I hope this question isn't considered too ...
16
votes
12
answers
5k
views
Examples of $G_\delta$ sets
Recall that a subset $A$ of a metric space $X$ is a $G_\delta$ subset if it can be written as a countable intersection of open sets. This notion is related to the Baire category theorem. Here are ...
- 18.4k
12
votes
8
answers
3k
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Applications of the notion of of Gromov-Hausdorff distance
I am looking for applications of the notion of Gromov-Hausdorff convergence to prove theorems that a priori have nothing to do with it. Examples that I am aware of (thanks to wikipedia and google):
...
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 ...