Questions asking for a "big list" of examples, illustrations, etc. Ask only when the topic is compelling, and please do not use this as the only tag for a question.

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19
votes
10answers
861 views

Surprising applications of topology [on hold]

Today in class we got to see how to use the Brouwer Fixed Point theorem for $D^2$ to prove that a $3 \times 3$ matrix $M$ with positive real entries has an eigenvector with a positive eigenvalue. The ...
2
votes
0answers
50 views

What computations would advance math knowledge a lot?

Suppose we where given a super computer that would be capable of computing anything, but only for one day. We could for instance compute many of the Ramsey numbers. What would be some computations ...
-6
votes
0answers
88 views

useful exact sequences [closed]

There are some exact-sequences or long-exact-sequences that are great help in proving to prove some surprising theorem, or have some interesting applications. What's your favorite exact ...
6
votes
1answer
74 views

Theorems discovered without observation

Can you name me a few theorems that were discovered without first observing some special cases? In other words, by brute logic: Starting from the known and logically deducing the unknown? EDIT: As an ...
0
votes
0answers
34 views

Different ways to prove Fundamental Theorem of Algebra

This is just a curosity .I know some proofs of the fact that Every non constant polynomial with complex coefficient has a complex root via using Liouville's theorem in Complex Analysis.Proof goes as ...
1
vote
0answers
40 views

Examples of open problems solved through short proof

Are there good examples of reasonable open problems in mathematics that had an 'obvious' solution via application of a theorem already known/not yet found in mathematics but could have been found with ...
0
votes
0answers
59 views

New proofs of the Fundamental Theorem of Calculus

Apart from the standard one, are there any other proofs of the Fundamental Theorem of Calculus which have been published recently?
2
votes
2answers
299 views

An example of a great explanation or freely accessible article on a math concept

Question: Give an example of a great explanation or freely accessible article on a math concept (suitable at the undergraduate or lower level), and explain why you think it is great. Possible ...
88
votes
41answers
11k views

What's your favorite proof accessible to a general audience?

What math statement with proof do you find most beautiful and elegant, where such is accessible to a general audience, meaning you could state, prove, and explain it to a general audience in ...
4
votes
2answers
103 views

Toy examples for Kan extensions

Background: If $\mathcal{C}$ is a cocomplete category and $f : I \to J$ is a functor between small categories, then $f^* : \mathrm{Hom}(J,\mathcal{C}) \to \mathrm{Hom}(I,\mathcal{C})$ has a left ...
2
votes
3answers
67 views

Examples of orthogonal/orthonormal functions which are not finite degree polynomials?

I've been reading "Fourier Series & Orthogonal Polynomials" by Dunham Jackson. Great introductory read for anyone interested by the way! My question is, what are other examples of Orthogonal ...
1
vote
0answers
36 views

Methods for evaluating polynomials quickly

I am wondering what methods exist for effectively evaluating polynomials (manually or in the head) in a quick, efficient fashion. For example, one of my favorite methods is the "nested form of a ...
23
votes
8answers
369 views

Big list of “guided discovery” books

K. P. Bogart wrote Combinatorics through Guided Discovery, available freely online. In the preface, he writes (emphasis mine): The point of learning from this book is that you are learning how to ...
63
votes
38answers
8k views

A fan, a horn, and a snowflake - unusual math terms [closed]

From time to time, I come across some unusual mathematical terms. I know something about strange attractors. I also know what Witch of Agnesi is. However, what prompted me to write this question is ...
1
vote
0answers
31 views

Applications of Splitting Lemma and Exactness

I'm looking for nice applications of exact sequences, the splitting lemma, and exact functors in algebra and topology (i.e not using the five lemma to get long sequences in homology etc..). For ...
8
votes
4answers
567 views

What are some elementary results (number theory) using theorems that went long-unproven?

Firstly, I do not mind if there are examples from fields other than number theory! This was just the first field, and where I think the richest examples, may come from. Now to elaborate on the title, ...
7
votes
1answer
75 views

For finding limits of functions, when are graphs deceiving?

What are some examples of limits which exist of functions $f:A \to B$ where $A$, $B \subseteq \mathbb{R}$ that require by-hand, "analytical" methods and the value of the limit is seemingly ...
1
vote
2answers
85 views

Elementary Applications of Cayley's Theorem in Group Theory

The Cayley's theorem says that every group $G$ is a subgroup of some symmetric group. More precisely, if $G$ is a group of order $n$, then $G$ is a subgroup of $S_n$. In the course on group theory, ...
2
votes
0answers
65 views

Innocent looking open problems in real analysis

Are there any apparently easy problems or conjectures in basic real analysis (that is, calculus) that are still open? By apparently easy, I mean: so much so, that, if it was for the statement alone, ...
3
votes
2answers
127 views

Connectedness arguments in elementary mathematics?

To begin, let me explain a proof strategy (which I'll call the connectedness principle for want of a better, more canonical term): One way to prove that a mathematical object $O_1$ has some property ...
7
votes
0answers
109 views

What is the most cited mathematical paper?

Just out of curiosity: What is the paper with the largest number of citations in all of mathematics? I think it is Shannon's A Mathematical Theory of ...
-8
votes
6answers
244 views

How many mathematical identities that equal $1$ [closed]

I usually see some identities equal to $1$ , for examples $$\sin ^2(x)+\cos^2(x)=1$$ $$\sec ^2(x)-\tan^2(x)=1$$ $$\csc ^2(x)-\cot^2(x)=1$$ $$\frac{\zeta(2) }{2}+\frac{\zeta (4)}{2^3}+\frac{\zeta ...
66
votes
9answers
4k views

Besides proving new theorems, how can a person contribute to mathematics?

There are at least a few things a person can do to contribute to the mathematics community without necessarily obtaining novel results, for example: Organizing known results into a coherent ...
2
votes
0answers
55 views

Generalizations of de l'Hospital rule

Are there any useful generalizations of de l'Hospital rule? Could you point out some references?
0
votes
1answer
80 views

Common conditions on functions to be morphisms. [closed]

When coming in contact with the concept of morphism one may start to wonder what makes different structured objects of the same kind to be similar in a "morphical" way. At least I did. Below ...
3
votes
0answers
76 views

What will be the consequences if second Hardy-Littlewood conjecture turns out to be true? [migrated]

It is generally believed that the Second Hardy-Littlewood Conjecture is false. But it has not been proved (or disproved) yet. My question is, What would be the consequences if Second ...
1
vote
0answers
43 views

Open problems for which all cases except one have been solved

Keller's conjecture states that in any tiling of Euclidean $n$-space by identical hypercubes there are two cubes that meet face to face. The conjecture has been shown to be true for $n<7$ and ...
3
votes
1answer
46 views

Group structures on Hausdorff space

Could anyone give me some practical (and possibly intuitive) examples of Group structures on Hausdorff spaces? Let us say you had to get freshmen university students interested into fields of maths ...
34
votes
11answers
3k views

What are some theorems that currently only have computer-assisted proofs?

What are some theorems that currently only have computer-assisted proofs? For example, there's the four colour theorem. I am very curious about this and would like to generate a list.
15
votes
1answer
206 views

Learning roadmap request: compiling a “Mathematics Stack Exchange Undergraduate Bibliography” [closed]

[Book recommendation] questions are quite popular on this website, which is, at least for me, one of the best places to get useful and insightful suggestions ...
3
votes
0answers
72 views

Big list of references [divided by categories] that collect commented open problems and conjectures [closed]

The aim of this question is to collect a big list of books or survey papers or websites which collect an up-to-date, comprehensive, well-organized, and possibly commented list of open problems. I ...
2
votes
1answer
100 views

Integration by nonobvious substitutions

The standard technique for evaluating the integral $$\int \sec x \,dx$$ is making the nonobvious substitution $$u = \sec x + \tan x, \qquad du = (\sec x \tan x + \sec^2 x) dx,$$ which transforms the ...
2
votes
4answers
112 views

Reference request: self-contained rigorous introductions to “cool” topics

I am looking for some self-contained (i.e., providing all necessary background information) rigorous introductions to topics perceived as "cool" to propose to (really) advanced high school students ...
7
votes
0answers
182 views

Problems from the Kourovka Notebook that undergraduate students can fully appreciate

The Kourovka Notebook is a collection of open problems in Group Theory. My question is: could you point out some (a "[big-list]" of) problems [by ...
0
votes
0answers
39 views

Books on contemporary set theory [duplicate]

I have gone through Halmos' Naive Set Theory. Now, could you recommend me a good follow-up book for a rigorous treatment of contemporary set theory? (For example, I've been suggested to look at ...
2
votes
1answer
65 views

Popularization of mathematics: inspiring quotes (or interviews) on the fascination of mathematics

I would like to compile a big list of references to particularly inspiring interviews with important mathematicians or famous quotes on the fascination that mathematics holds for ...
2
votes
3answers
178 views

“Methods of Theoretical Physics for Mathematicians”

I read in the Princeton Companion to Mathematics that even pure mathematicians should know some theoretical physics. However, I see that there are many reference books of mathematical methods for ...
8
votes
1answer
164 views

Websites that promote co-operation and social networking among mathematicians

Are there some websites that could be defined as social networks for mathematicians and scientists? What I have in mind is something similar to Academia.edu or ResearchGate, but more specific towards ...
13
votes
1answer
216 views

Collections of undergraduate research projects

I would like to compile a "big list" of undergraduate research projects in the following areas of mathematics: calculus; analysis; abstract algebra; linear algebra; number theory; geometry; ...
2
votes
2answers
176 views

Little, unknown, English or French research journals with good mathematics

In this article by Gian-Carlo Rota, you can read: "I bought a copy of Frederick Riesz' Collected Papers as soon as the big thick heavy oversize volume was published. [...] It was clear that ...
1
vote
0answers
120 views

In how many ways can one establish $\lim_{x\rightarrow 0} \frac{\sin x}{x}=1$? [duplicate]

I was just having fun with $\displaystyle \lim_{x \rightarrow 0} \frac{\sin x}{x}=1$. I would be happy to see how could one show this. Below are two approaches. I like the second more and find it more ...
51
votes
25answers
6k views

Easy example why complex numbers are cool

I am looking for an example explainable to someone only knowing high school mathematics why complex numbers are necessary. The best example would be possible to explain rigourously and also be clearly ...
7
votes
3answers
343 views

On progress in mathematics: some long-open problems and long-standing conjectures

I would like to ask a question here on Math Stack Exchange taking inspiration (and therefore combining) from two well-known threads on MathOverflow: (1) Not especially famous, long-open problems which ...
4
votes
6answers
1k views

What's behind the Banach-Tarski paradox? [closed]

The discovery of the Banach-Tarski paradox was of course a great thing in mathematics but raises the issue of the relation between mathematics and reality. Empirically there are good reasons for faith ...
2
votes
1answer
108 views

Natural progression in a curriculum for self-study of analysis

Would you list what is a natural and effective progression to self-study topics in analysis in order to gain a broad knowledge of the enormous corpus of knowledge that modern analysis involves. As a ...
0
votes
1answer
44 views

Combinatorial techniques, methods, and ideas in (“undergraduate”) real analysis

This question is dual to Probabilistic techniques, methods, and ideas in ("undergraduate") real analysis: I would like to collect some examples of combinatorial arguments to undergraduate ...
3
votes
3answers
145 views

Examples of combinatorial/probabilistic proofs of theorems in linear algebra

Are there any examples of combinatorial/probabilistic proofs of theorems in linear algebra? Motivation: I see here, the inverse is true.
1
vote
2answers
106 views

What is some pure math news website by a publisher? [closed]

Why aren't there be any pure math website by a publisher? I google a lot and resulting only applied math news or math journal that is difficult and inaccessible even to advanced reader I am looking ...
15
votes
7answers
481 views

What are some mathematically productive ways to waste time? [closed]

What are some productive things that can be done (other than directly studying Mathematics) during leisure time that has a side effect to improve oneself at Mathematics? For example, reading ...
6
votes
4answers
179 views

Probabilistic techniques, methods, and ideas in (“undergraduate”) real analysis

As the book Probabilistic Techniques in Analysis by Richard F. Bass shows, nowadays techniques drawn from probability are used to tackle problems in analysis. The mentioned book presents a survey of ...