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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2answers
128 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, ...
3
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1answer
98 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
136 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 ...
9
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0answers
193 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 ...
69
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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
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0answers
62 views

Generalizations of de l'Hospital rule

Are there any useful generalizations of de l'Hospital rule? Could you point out some references? Edit: Something in the spirit of http://arxiv.org/abs/1403.3006, but not necessarily for functions of ...
0
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1answer
82 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 ...
1
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0answers
48 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
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1answer
54 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 ...
35
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12answers
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
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1answer
253 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
82 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
106 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
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4answers
122 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 ...
15
votes
4answers
415 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 referencing them] presented in this book that ...
0
votes
0answers
43 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
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3answers
204 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 ...
12
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1answer
198 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 ...
19
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2answers
476 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
185 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 ...
54
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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
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3answers
362 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
2k 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
134 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 ...
1
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1answer
114 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
173 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
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2answers
122 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
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7answers
513 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
249 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 ...
4
votes
2answers
103 views

Problems whose first solutions had been using Calculus but later was shown to be done by non-Calculus methods

I was wondering about mathematical problems whose first published solutions was obtained by using methods of Calculus but later was shown (or known) to be solvable by using non-Calculus methods. ...
5
votes
1answer
69 views

What are some remarkable and interesting uses of AM-GM Inequality ? Cite and explain with problems.

There are really lot of problems on AM-GM inequality because of its elementary nature and powerful applications. What I want is a collection of questions/problems which look very complex but get ...
20
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19answers
2k views

What are some interesting sole exceptions or counterexamples? [duplicate]

Many theorems assert that a particular property holds for all objects in a class except those in a given list of exceptions. Examples of rules that admit precisely one exception include: All primes ...
1
vote
0answers
60 views

Where can Gaussian Elimination be used?

I have searched for this and came to know about it that it is traditionally used to solve linear equations, finding determinant, rank of matrix, inverse of matrix. There was a problem on codechef: ...
0
votes
2answers
115 views

Why $1$ isn't a prime? [duplicate]

I was wondering the reason behind defining the Prime Numbers in a manner of which $1$ isn't an example. I read in Rotman's A First Course in Abstract Algebra that one reason that $1$ is not called a ...
1
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1answer
83 views

subtle/annoying fallacious proofs [duplicate]

I've been invited to a maths themed Xmas after party. I need to prepare a selection of interesting, and relatively simple fallacious proofs which other guests will try and find the flaw in. I'm trying ...
43
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18answers
4k views

What are some applications of elementary linear algebra outside of math?

I'm TAing linear algebra next quarter, and it strikes me that I only know one example of an application I can present to my students. I'm looking for applications of elementary linear algebra outside ...
9
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3answers
300 views

Examples of useful, insightful, and interesting hand-waving [closed]

It seems to me that some hand-waving (by which I mean some arguments that aim at giving some form of intuition on the problem even at expenses of complete rigour [and not mnemonics for high-schoolers ...
12
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6answers
249 views

Sources for mathematics outside the mathematics world

In this question I would like to ask you about material showing the uses (or occurrences) of mathematics in the everyday world. The aim is to encourage with it a group of young undergraduate ...
0
votes
0answers
41 views

Reference about $p$-homogeneous functions

I'm looking for a book about $p$-homogeneous functions. I am particularly interested in the associated (nonlinear) eigenvalue problems. However, a reference containing most of the known properties of ...
2
votes
3answers
84 views

Are there limits of which we're not able yet to find the value or not even prove non/existence?

I really like working out limits, so I've been wondering: Are there limits we're struggling to evaluate? Are there limits of which we're not succeeding in proving the existence or nonexistence?
18
votes
1answer
417 views

“Novel” proofs of “old” calculus theorems

Every once in a while some mathematicians publish (mostly on the American Mathematical Monthly) a new proof of an old (nowadays considered "basic") result in analysis (calculus). This article is an ...
7
votes
1answer
113 views

Must Have Theorems, Identies, etc… in Your Mathematical Arsenal [closed]

Lately I have been getting into solving problems in some of the math journals I enjoy reading. More and more I find that solvers employ a theorem or identity that makes solving the problem much ...
33
votes
20answers
3k views

Ways to write “50” [closed]

A really good friend of mine is an elementary school math teacher. He is turning 50, and we want to put a mathematical expression that equals 50 on his birthday cake but goes beyond the typical ...
26
votes
11answers
3k views

Great contributions to mathematics by older mathematicians [closed]

It is often said that mathematicians hit their prime in their twenties, and some even say that no great mathematics is created after that age, or that older mathematicians have their best days behind ...
8
votes
4answers
349 views

Unconventional (but instructive) proofs of basic theorems of calculus

Inspired by this questions asked on MathOverflow, I would like to ask if you know some "sophisticated" proofs of the basic theorems in a calculus course (that is, the ones that you can find, for ...
2
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0answers
84 views

Deep questions in number theory not accessible by combinatorial results

Number theory and arithmetic geometry were invented to solve many questions about properties of numbers. What are the some of the foundational results or estimates that are accessible to powerful ...
2
votes
1answer
24 views

Texts on Coxeter groups

I'm looking for an introductory text on Coxeter groups. It can assume undegraduate knowledge of Algebra (Groups up to and including the Sylow theorems in Fraleigh, elementary knowledge of rings, ...
1
vote
2answers
111 views

Examples of categories which naturally include End(O) as object

I want examples of categories $\textbf C$ which naturally include $End_{\textbf C}(O)$ as object for objects $O$ in the category. The set of all endomorphims is always a monoid under the composition ...
17
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10answers
2k views

“Honest” introductory real analysis book

I was asked if I could suggest an "honest" introductory real analysis book, where "honest" means: with every single theorem proved (that is, no "left to the reader" or "you can easily see"); with ...
6
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5answers
244 views

Topic for a lecture intended for High School students [duplicate]

I am not sure if this is the right place to post this, but here is the situation. In about two weeks or so I will be giving a 2-3 hours lecture on some topic in mathematics to freshman and sophomore ...