12
votes
9answers
1k 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 ...
12
votes
2answers
459 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 ...
14
votes
10answers
1k 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
7answers
915 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 ...
39
votes
35answers
6k 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 ...
13
votes
12answers
2k 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 ...
23
votes
3answers
2k 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 $3X^3 + 4Y^3 + 5Z^3 = 0$. ...
41
votes
28answers
4k 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 ...
4
votes
5answers
2k 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 ...
6
votes
1answer
1k views

When are technical assumptions critical? [closed]

Apart from their technical statement and proof, a usual presentation of theorems is by leading up to them with a definite motivation or intuition, for example putting the results in the wider context ...
14
votes
1answer
592 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 ...
23
votes
4answers
2k 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 ...
3
votes
2answers
886 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 ...
32
votes
22answers
6k 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 ...
40
votes
44answers
12k 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 ...
4
votes
2answers
2k 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 ...
10
votes
7answers
1k views

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): ...
43
votes
29answers
9k 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 ...
5
votes
9answers
1k views

Examples of two different descriptions of a set that are not obviously equivalent?

I am teaching a course in enumerative combinatorics this semester and one of my students asked for deeper clarification regarding the difference between a "combinatorial" and a "bijective" proof. ...
16
votes
11answers
4k views

Examples of using physical intuition to solve math problems

For the purposes of this question let a "physical intuition" be an intuition that is derived from your everyday experience of physical reality. Your intuitions about how the spin of a ball affects ...
5
votes
3answers
5k 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 ...
21
votes
2answers
2k views

Examples where the analogy between number theory and geometry fails

The analogy between $O_K$ ($K$ a number field) and affine curves over a field has been very fruitful. It also knows many variations: the field over which the curve is defined may have positive or zero ...
1
vote
1answer
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 ...
17
votes
19answers
2k views

Examples of categorification

What is your favorite example of categorification?
28
votes
17answers
5k views

Wonderful applications of the Vandermonde determinant

This semester I am assisting my mentor teaching a first-year undergraduate course on linear algebra in Peking University, China. And now we have come to the famous Vandermonde determinant, which has ...
8
votes
3answers
876 views

Concavity of $\det^{1/n}$ over $HPD_n$.

One of my beloved theorems in matrix analysis is the fact that the map $H\mapsto (\det H)^{1/n}$, defined over the convex cone $HPD_n$ of Hermitian positive definite matrices, is concave. This is ...
111
votes
67answers
33k 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 ...
86
votes
59answers
15k 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): ...
11
votes
17answers
2k views

What is your favorite isomorphism? [closed]

The other day I was trying to figure out how to explain why isomorphisms are important. I pulled Boyer's A History of Mathematics off the bookshelf and was surprised to find that isomorphism isn't ...
5
votes
9answers
986 views

What category without initial object do you care about?

Recently I have been listening to some constructions that have been designed to accommodate categories without an initial object. The speaker has given some idea of a category or two that he cares ...
11
votes
12answers
2k views

On proving that a certain set is not empty by proving that it is actually large

It happens occasionally that one can prove that a given set is not empty by proving that it is actually large. The word "large" here may refer to different properties. For example, one can prove that ...
5
votes
13answers
1k views

Applications of compactness

Similar to this question: Applications of connectedness I want to collect applications of compactedness. E.g.: compact + discrete => finite, which can be used to prove the finiteness of the ...
8
votes
3answers
1k 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 ...
5
votes
2answers
490 views

Gaining intuition for how submodules behave

I'm studying elementary commutative algebra this semester, largely following Atiyah-MacDonald. I often find myself in a situation where I'm interested in whether some property of an R-module M is ...
15
votes
12answers
2k views

What are some fundamental “sources” for the appearance of pi in mathematics?

I thought it might be fun to ask this question as a way of celebrating Pi Day. One way in which people popularize pi is that they say that even though it's defined in terms of properties of a circle, ...
86
votes
53answers
18k 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 ...
18
votes
8answers
1k views

Cryptomorphisms

I am curious to collect examples of equivalent axiomatizations of mathematical structures. The two examples that I have in mind are Topological Spaces. These can be defined in terms of open sets, ...
2
votes
0answers
239 views

Forgetting extra structure inducing Symmetries

This is a major edit of the original post after receiving helpful comments. It is often the case when one adds additional structure to make a problem more tractable. When one attempts to forget this ...
2
votes
4answers
418 views

Ternary relations that are not binary functions

By far the most prominent elementary relations that are not functions are binary and the most prominent elementary ternary relations are in fact binary functions. "Elementary" shall mean "part of the ...
32
votes
5answers
3k views

What is Yoneda's Lemma a generalization of?

What is Yoneda's Lemma a generalization of? I am looking for examples that were known before category theory entered the stage resp. can be known by students before they start with category theory. ...
49
votes
33answers
8k views

Experimental Mathematics

I would like to ask about examples where experimentation by computers have led to major mathematical advances. Motivation I am aware about a few such cases and I think it will be useful to gather ...
66
votes
41answers
15k views

What are the most attractive Turing undecidable problems in mathematics?

What are the most attractive Turing undecidable problems in mathematics? There are thousands of examples, so please post here only the most attractive, best examples. Some examples already appear on ...
1
vote
1answer
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 ...
7
votes
3answers
898 views

What are some interesting sequences of functions for thinking about types of convergence?

I'm thinking about the basic types of convergence for sequences of functions: convergence in measure, almost uniform convergence, convergence in Lp and point wise almost everywhere convergence. I'm ...
37
votes
28answers
17k views

Applications of the Chinese remainder theorem

As the title suggests I am interested in CRT applications. Wikipedia article on CRT lists some of the well known applications (e.g. used in the RSA algorithm, used to construct an elegant Gödel ...
153
votes
62answers
79k views

Proofs without words

Can you give examples of proofs without words? In particular, can you give examples of proofs without words for non-trivial results? (One could ask if this is of interest to mathematicians, and I ...
13
votes
27answers
2k 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 ...
39
votes
25answers
4k 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 ...
33
votes
12answers
5k views

Combinatorial results without known combinatorial proofs

Stanley likes to keep a list of combinatorial results for which there is no known combinatorial proof. For example, until recently I believe the explicit enumeration of the de Brujin sequences fell ...
41
votes
33answers
3k 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" ...