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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3
votes
8answers
297 views

simple theorems that non-mathematicians can understand and appreciate.

For example, I stated this fact/theorem at a dinner to some friends and they were pretty impressed. Given any sequence of n integers, positive or negative, not necessarily all different, some ...
4
votes
3answers
100 views

''Reading'' polynomials at the first glance

I'm reading Proofs from the Book, and I ran into following theorem: Suppose all roots of polynomial $x^n + a_{n-1}x^{n-1} + \dots + a_0$ are real. Then the roots are contained in the interval: $$ - ...
8
votes
3answers
339 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
0answers
90 views

What is a good source of problem-solving type problems?

I am not looking for contest problems where there is a clever trick or a standard approach, I am looking for more creative and open-ended problems such as this , and I am not looking for questions ...
1
vote
0answers
413 views

What are real life applications of Diophantine equations?

Are there any real life applications of linear Diophantine equations? I am looking for examples which will motivate students.
3
votes
4answers
163 views

Enlightening Mathematical Models

What is your favourite Mathematical Model? What features make it intuitive or elegant? This question is largely inspired by an example and a desire to find other's like it. Suppose we have two ...
8
votes
3answers
226 views

Examples of $\mathcal{O}_X$-modules that are not quasi-coherent sheaves

Let $X = \operatorname{Spec} k[x]_{(x)}$ which consists of two elements, the generic point $\zeta$ corresponding to the zero ideal and the closed point $(x)$. Define an $\mathcal{O}_X$-module ...
41
votes
14answers
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 ...
49
votes
16answers
3k views

Rigour in mathematics

Mathematics is very rigorous and everything must be proven properly even things that may seem true and obvious. Can you give me examples of conjectures/theories that seemed true but through rigorous ...
4
votes
1answer
188 views

List of proofs of non-trivial theorems which were unnoticed to be wrong for at least a few years

For example, the Weber's proof of Kronecker–Weber theorem. I would like to know such proofs. It seems to be important for me to remember that a widely accepted proof might be wrong.
7
votes
7answers
175 views

Mathematical Games suitable for undergraduates

I am looking for mathematical games for an undergraduate `maths club' for interested students. I am thinking of things like topological tic-tac-toe and singularity chess. I have some funding for this ...
18
votes
3answers
622 views

How to Garner Mathematical Intuition

Motivated by Why Is Intuition so Important to Mathematicians but Missing from Mathematics Education? $^{1}$ by Leona Burton, I would like to learn about specific ideas or strategies to attain ...
19
votes
4answers
555 views

What newer mathematics fields helped to solve or solved problems from older fields of mathematics?

I usually have a more or less formed template of conversation to talk with people about mathematics, It's importance, methods, history, etc. I've been for some time interested in newer fields of ...
2
votes
3answers
180 views

Pop Math Book You Would Love to See Written

I don't usually like these types of big list questions, but I think this is actually fairly important as far as education, informing the public about what we do, and getting people excited about math ...
4
votes
4answers
131 views

Resources to help an 8yo struggling with math

Friends of mine asked me for suggestion for one of their children (age 8) who had bad scores at the local Star test (the family is based in California). Both parents work, so they have also limited ...
10
votes
3answers
301 views

categorical generalizations of familiar objects

A couple of days ago I've learned that you can define trace in a very abstract setting. Namely, suppose $F\colon A\to B$ is a functor between two categories. Suppose $E,G\colon B\to A$ are two ...
0
votes
5answers
428 views

Unprovability results in ZFC

I am looking for real-world examples of unprovable statements in ZFC. So, not contrived logical formulae but statements that are of importance for ordinary mathematicians. Could you please point to ...
7
votes
1answer
127 views

Applications of Serre duality

I am reading about Serre duality theory in algebraic geometry from Hartshorne, and am wondering what kinds of applications it has. It seems that most applications go through some version of the ...
5
votes
0answers
87 views

Classes of groups known to be realizable (IGP)

A finite group $G$ of order $n$ is said to be realizable (over $\mathbb{Q}$) if there exists a Galois extension $L/\mathbb{Q}$ such that $\mathrm{Gal}(L/\mathbb{Q})=G$. I'm curious what classes of ...
-2
votes
1answer
104 views

Article for a maths magazine [closed]

Any idea for an unique proof or theorem or any kind of mathematical philosophy to be presented in a maths magazine?
6
votes
1answer
120 views

Are there treelike representations of axioms, theorems, lemmas and corollaries?

I was watching Bill Shillito Lectures on Higher Mathematics, in the second episode he says the basic stuff about axioms and theorems - that axioms are unproved statements in which we build theorems ...
5
votes
2answers
330 views

Abstract Algebra/ Linear Algebra classic problems [closed]

I am studying for an Algebra/Linear Algebra exam coming up in August.In preparation for my exam I have worked on a lot of problems from Dummit and Foote and Hoffman and Kunze' books. I would like to ...
3
votes
1answer
872 views

Recommended maths book for beginner to study in computer science

I am going to study computer science next year. I am afraid I can't handle the mathematics in the university because I only know some basic mathematics, such as set theory, simple probability, simple ...
1
vote
1answer
261 views

Practice Problem Books

The Analysis I/II/III (Differentiation and Continuity/Sequence and Series/Integration) published by AMS. The first one is this. It's a problem-solution book. I found it excellent because of the ...
1
vote
1answer
110 views

What are the major properties of indexed unions and intersections?

I'm pretty confident reasoning about expressions involving the following operations. binary unions binary intersections complementation However, I am less confident in my ability to reason about ...
1
vote
1answer
267 views

what is the best book to study contour integration?

what is the best book or website to study contour integration ? I find in some question answer using contour integration but I can't understand how they do that so is there any help ?
3
votes
5answers
323 views

Geometry books with beautiful diagrams

What are some geometry books with particularly beautiful diagrams? Old or new. Could be on 'standard' material or specialised on one particular topic. Something for the connoisseur of mathematical ...
10
votes
4answers
163 views

Example of integral pairs or triples ($I$, $J$, $K$…)

A somewhat common trick when evaluating integrals is to add a second (or third) integral to help integrate the first. As an example if one where to compute $$ I = \int \sin \log x \, ...
13
votes
3answers
369 views

Interesting finite-cardinality theorems

What are the most interesting results in mathematics that say there are only finitely many of something? Examples: If it's ever shown that there are only finitely many twin primes, that would fit ...
3
votes
1answer
69 views

List of first-order foundations of mathematics

I would like to get an overview over what different first-order systems suitable for the formalization of "classical" mathematics are currently known. So far I only know the variants of ZFC, ...
13
votes
5answers
544 views

Favourite applications of the Nakayama Lemma

Inspired by a recent question on the nilradical of an absolutely flat ring, what are some of your favourite applications of the Nakayama Lemma? It would be good if you outlined a proof for the result ...
4
votes
1answer
168 views

List of explicit enumerations of rational numbers

A well-known mathematical fact is that the rational numbers are countable, i.e. there is a bijective function $$f:\mathbb{N}\rightarrow \mathbb{Q}$$ I am interesting in making a list of all explicit ...
7
votes
6answers
858 views

Finite vs infinite dimensional vector spaces

What familiar and intuitive properties of finite dimensional vector spaces fails in infinite dimensions? For example: In infinite dimensions there are non-continuous linear maps. In infinite ...
1
vote
2answers
249 views

Which ring homomorphisms preserve/reflect what?

Exams are coming up and I'm getting kind of desperate. So more now than ever, whatever help you're able to provide is much appreciated. In the abstract algebra exam I'm currently preparing for, ...
4
votes
0answers
100 views

What are some great graduate textbooks with solutions in the back to the problems?

I can think of Aubin's A Course in Differential Geometry, as well as Knapp's books. Any other great ones you know of? Especially in the GSM series from AMS (blue and yellow covers).
25
votes
8answers
2k views

“It looks straightforward, but actually it isn't”

In a previous topic, I asked about proof of statements which are simple but incorrect. Here, I ask about statements which seems, at a first glance, straightforward, but if we try to write a proof, ...
7
votes
1answer
216 views

Question for mathematicians who started before the computer era: what constants did you have memorized, in what form, and why?

A former department chair at BYU, Wayne Barrett, would always amaze grad students by his vast knowledge of mathematical constants, like the radical form of $\cos(2\pi/5)$. I've never memorized ...
7
votes
5answers
437 views

Most astonishing applications of compactness theorem outside logic

The compactness theorem has a lot of applications to logic and model theory. I'm looking for applications. I'm looking for theorems in other areas of mathematics which seem at first sight to have ...
5
votes
2answers
202 views

Astonishing and innocent results with the axiom of choice

The product of nonempty sets is nonempty. I am fascinated that such a simple and seemingly intuitive statement can lead to rather astonishing results such as the Banach-Tarski paradox or the solution ...
7
votes
3answers
140 views

Realizing groups as symmetry groups

We're supposed to think of (non-Abelian) groups as groups of symmetries of some object. Sometimes it isn't obvious what this object is. For example, the fundamental group of a topological space acts ...
2
votes
1answer
154 views

Accessible Applications of Graph Ramsey Theory

I am giving a short lecture series on graph Ramsey theory to a group of gifted high school seniors. The brief outline is to start with the "six people at a dinner party" question, transition into the ...
41
votes
19answers
4k views

Theorems' names that don't credit the right people

The point of this question is to compile a list of theorems that don't give credit to right people in the sense that the name(s) of the mathematician(s) who first proved the theorem doesn't (do not) ...
51
votes
20answers
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 ...
6
votes
2answers
178 views

What happens after the cardinality $\mathfrak{c}$?

While having measure theory this year the following came in my mind: When we go from finite objects to infinite we "lose" a lot of properties. For example the summation isn't well defined ...
1
vote
1answer
118 views

A question on mean value inequality

It is known that mean value inequality is very useful. It is: For any $0 \le a_i (i=1,2,\dots,n)$, $$ a_1 a_2\dots a_n\le (\frac{a_1+a_2+\dots + a_n}{n})^n \tag1 $$ My question is: how many ...
22
votes
5answers
413 views

Famous Problems Where We Only Know the Elementary

Define a graph with vertex set $\mathbb{R}^2$ and connect two vertices if they are unit distance apart. The famous Hadwiger-Nelson problem is to determine the chromatic number $\chi$ of this graph. ...
51
votes
9answers
3k views

Surprisingly elementary and direct proofs

What are some examples of theorems, whose first proof was quite hard and sophisticated, perhaps using some other deep theorems of some theory, before years later surprisingly a quite elementary, ...
12
votes
4answers
892 views

Examples of nonlinear ordinary differential equations with elementary solutions.

I am looking for nice examples of nonlinear ordinary differential equations that have simple solutions in terms of elementary functions. (But are not trivial to find, like, for example, with ...
17
votes
5answers
414 views

Examples of properties that hold almost everywhere, but that explicit examples unknown.

In measure theory one makes rigorous the concept of something holding "almost everywhere" or "almost surely", meaning the set on which the property fails has measure zero. I think it is very ...
4
votes
0answers
184 views

Good examples of proofs in mathematics exemplary of creative reasoning [closed]

Just what the title says. I'm not looking for any proofs that require specialized knowledge past the very fundamentals of real analysis. I'm looking for proofs for important results (don't have to be ...