Questions asking for a "big list" of examples, illustrations, etc. Please do not ask too many of these. Please do not use this as the only tag for a question.

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1
vote
2answers
78 views

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

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 ...
14
votes
7answers
382 views

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

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 ...
21
votes
20answers
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 ...
47
votes
16answers
3k views

Notations that are mnemonic outside of English

Some conventional math notations seem arbitrary to English speakers but were mnemonic to non-English speakers who started them. To give a simple example, Z is the symbol for integers because of the ...
37
votes
19answers
3k 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 ...
4
votes
2answers
93 views

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

As the book Probabilistic Techniques in Analysis by Richard Bass shows, nowadays techniques drawn from probability are used to tackle problems in analysis. The mentioned book presents a survey of ...
6
votes
0answers
73 views

Examples of useful, insightful, and interesting hand-waving

I am really amused by the answers to this question on "Most harmful heuristic" posed on MathOverflow, from which I've benefited a lot. However, it seems to me that some hand-waving may be really ...
8
votes
1answer
136 views

“Novel” proofs of “old” calculus theorems

Every once in a while some mathematicians publish (mostly on the Montly) a new proof of an old (nowadays considered "basic") result in analysis (calculus). This article is an example. I would like ...
8
votes
5answers
147 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 ...
4
votes
2answers
93 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
60 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 ...
0
votes
2answers
98 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 ...
43
votes
14answers
3k views

Largest “leap-to-generality” in math history?

Grothendieck, who is famous inter alia for his capacity/tendency to look for the most general formulation of a problem, introduced a number of new concepts (with topos maybe the most famous ?) that ...
-1
votes
1answer
36 views

All variants of stars and bars / balls and bins problem [on hold]

The Stars and Bars problem or Balls and Bins problem are the the very basic in combinatorics but at the same time are quite helpful for beginners. Can we have list of variants of these problems? Add ...
1
vote
0answers
41 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: ...
10
votes
6answers
602 views

What are some easy-to-remember prime numbers?

This is a question without much mathematical value, but since I don't immediately see an answer on Google I thought I'd ask anyway ... I'm looking for some largeish (> 10,000) easy-to-remember primes, ...
1
vote
1answer
69 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 ...
31
votes
9answers
6k views

Examples of nonabelian groups.

Can anybody provide some examples of finite nonabelian groups which are not symmetric groups or dihedral groups?
6
votes
3answers
3k views

Group theory applications along with a solved example

As I asked in previous question, I am very curious about applying Group theory. Still I have doubts about how I can apply group theory. I know about formal definitions and I can able to solve and ...
4
votes
4answers
215 views

Have any definitions in mathematics been redefined

Based on certain intuitions and motivations we make certain definitions and then proceed to use these concepts in further developing our intuition. For example, we have an intuition that a line has ...
56
votes
4answers
1k views

“Advice to young mathematicians”

I have been suggested to read the Advice to a Young Mathematician section of the Princeton Companion to Mathematics, the short paper Ten Lessons I wish I had been Taught by Gian-Carlo Rota, and the ...
52
votes
16answers
6k views

Interesting “real life” applications of serious theorems

As a student one sometimes encounters exercises which ask you to solve a rather funny "real life problem", e.g. I recall an exercise on the Krein-Milman theorem which was something like: "You have a ...
8
votes
4answers
292 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 ...
61
votes
26answers
4k views

Is there a great mathematical example for a 12-year-old?

I've just been working with my 12-year-old daughter on Cantor's diagonal argument, and countable and uncountable sets. Why? Because the maths department at her school is outrageously good, and set ...
2
votes
3answers
75 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?
32
votes
20answers
2k 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 ...
11
votes
3answers
233 views

Textbooks, lecture notes, and articles from arXiv for undergraduate students

I have found some interesting textbooks and articles on arXiv, such as the following one, that are accessible to an undergraduate student: Course of linear algebra and multidimensional geometry, ...
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 ...
64
votes
24answers
11k views

Mathematicians ahead of their time?

In every field there's always that person who's just years ahead of their time. For instance, Paul Morphy (born 1837) is said to have retired from chess because he found no one to match his technique ...
81
votes
21answers
9k views

Most ambiguous and inconsistent phrases and notations in maths

What are some examples of notations and words in maths which have been overused or abused to the point of them being almost completely ambiguous when presented in new contexts? For instance, a ...
64
votes
21answers
14k views
1
vote
4answers
5k views

Rules for Product and Summation Notation

When we deal with summation notation, there are some useful computational shortcuts, e.g.: $$\sum\limits_{i=1}^{n} 2 + 3i = \sum\limits_{i=1}^{n} 2 + \sum\limits_{i=1}^{n} 3i = 2n + ...
7
votes
1answer
94 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 ...
37
votes
10answers
9k views

Best Algebraic Geometry text book? (other than Hartshorne)

Lifted from Mathoverflow: I think (almost) everyone agrees that Hartshorne's Algebraic Geometry is still the best. Then what might be the 2nd best? It can be a book, preprint, online lecture note, ...
2
votes
0answers
62 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 ...
0
votes
3answers
63 views

What are some textbooks on the same level as Ross's “A First Course in Probability”?

Ideally, I would like to have at least six standard probability texts so that I can compare them to each other. Thank You.
15
votes
9answers
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 ...
166
votes
85answers
14k views

Surprising identities / equations

What are some surprising equations / identities that you have seen, which you would not have expected? This could be complex numbers, trigonometric identities, combinatorial results, algebraic ...
8
votes
2answers
474 views

Video lectures of algebraic geometry (Hartshorne, Shafarevich, … )

I am a commutative algebra student. I wonder if there is some video lectures of algebraic geometry courses available online for free? I'd like the lectures to cover main topics of the books ...
2
votes
1answer
14 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, ...
2
votes
1answer
205 views

Examples of Eilenberg-type Swindles

I am compiling a list of 'swindles' in the style of the Eilenberg-Mazur swindle. I've already got some swindles in K-theory, the Mazur Swindle and the proof of the Cantor–Bernstein–Schroeder theorem. ...
66
votes
11answers
10k views

Results that came out of nowhere.

Most big results in mathematics are built on years and years of groundwork by the author and other mathematicians, such as Wiles' proof of FLT or the classification of finite simple groups. ...
2
votes
2answers
105 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 ...
5
votes
2answers
141 views

How do you compute group cohomology in practice?

If you have a finite group $G$ and a finite $G$-module $K$, and you need to know $H^1(G,K)$ or $H^2(G,K)$, how do you do it? Do you use a computer algebra system? (If so, which one?) Do you use a ...
15
votes
7answers
3k views

List of problem books in undergraduate and graduate mathematics

I would like to know some good problem books in various branches of undergraduate and graduate mathematics like group theory, galois theory, commutative algebra, real analysis, complex analysis, ...
36
votes
7answers
1k views

Original works of great mathematicians

In almost every mathematical text there is a line as This was first proved by Gauss or This formula first appeared in a work of Riemann, but for me it's more like My friend told me once that... For ...
8
votes
3answers
379 views

Instructive examples of elegant, clear, rigorous, terse, but “non-dull” mathematical prose

On the "About" page of the Mathgen project one can read: "More seriously, I think this project says something about the very small and stylized subset of English used in mathematical writing. ...
11
votes
6answers
4k views

Prerequisites/Books for A First Course in Linear Algebra

What mathematical knowledge do I need to begin studying linear algebra? In particular, how much calculus do I need to know? Also, do you have a favorite linear algebra book you can recommend?
6
votes
5answers
95 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 ...
17
votes
6answers
358 views

Open source lecture notes and textbooks

This question is inspired by the popular "Best Sets of Lecture Notes and Articles". Indeed, I would like to collect a "big-list" of open source (that is, with $\LaTeX$ code available) high-quality ...