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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9
votes
2answers
485 views

Surprising applications of cohomology

The concept of cohomology is one of the most subtle and powerful in modern mathematics. While its application to topology and integrability is immediate (it was probably how cohomology was born in the ...
0
votes
1answer
112 views

Examples of non-trivial closed subschemes of a complete non-projective non-singular variety

Let $k$ be an algebraically closed field. A variety over $k$ is a separated integral scheme of finite type over $k$. Let $V$ be a complete non-projective non-singular variety over $k$. Let $Z$ be a ...
5
votes
1answer
94 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 ...
2
votes
3answers
133 views

University-level books focusing on intuition?

I help some students with difficulties in Mathematics and Physics (especially math, physics, and engineering majors). While in high school they usually don't study, or are not interested, etc., in ...
2
votes
0answers
30 views

List of crucial results deserving more attention for first course in Real Analysis

Can it help to form a list of crucial results for basic courses that are concealed as exercises or neglected? I don't know of other resources for this, as I wrote here. I am happy for this to be ...
0
votes
2answers
47 views

What is list of common integral that have no closed form?

What is list of common integral that have no closed form? It's diffucult for me to google it for some reason.
5
votes
1answer
95 views

Why Did You Specialize in X?

For those of you who are researchers or graduate students, why did you choose to specialize in the field of mathematics X (as opposed to some other field Y)? Is it because you think X is important, ...
6
votes
3answers
192 views

Definitions which should be propositions/theorems

I am asking for a list of concepts which some sources present as definitions whereas other sources pose them as propositions/theorems. For example, most abstract algebra books will define a group ...
82
votes
30answers
16k views

Examples of mathematical results discovered “late”

What are examples of mathematical results that were discovered surprisingly late in history? Maybe the result is a straightforward corollary of an established theorem, or maybe it's just so simple ...
4
votes
2answers
95 views

What are different notations used by mathematicians and physicists?

One can find many cases that mathematicians and physicists use different notations for the same concepts. Here is a few cases I find. Inner product of vectors: Mathematicians use $(a,b)$ or ...
1
vote
1answer
143 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 ...
17
votes
8answers
436 views

Proofs of AM-GM inequality

The arithmetic - geometric mean inequality states that $$\frac{x_1+ \ldots + x_n}{n} \geq \sqrt[n]{x_1 \cdots x_n}$$ I'm looking for some original proofs of this inequality. I can find the usual ...
3
votes
1answer
139 views

Hard problems book in linear algebra

Could you suggest me a book where I can find hard problems in Linear Algebra for an undergraduate student? Thanks in advance.
2
votes
0answers
215 views

List of Common or Useful Limits of Sequences and Series

There are many sequences or series which come up frequently, and it's good to have a directory of the most commonly used or useful ones. I'll start out with some. Proofs are not required. ...
3
votes
1answer
220 views

Example of non-noetherian algebras which are tensor products of noetherian algebras

We suppose all rings are commutative with unity. I am looking for examples of a tensor product $B\otimes_A C$ which is not noetherian, where $A$ is a noetherian ring and $B, C$ are noetherian ...
76
votes
18answers
7k views

How do you describe your mathematical research in layman's terms?

"You do research in mathematics! Can you explain your research to me?" If you're a research mathematician, and you have any contact with people outside of the mathematics community, I'm sure ...
7
votes
0answers
172 views

Most famous competition problems? [closed]

When I've attended math competition discussions, I've often heard people remark "oh, this is a famous problem" or say that it's similar to one. Most of them I've actually never heard of before. ...
6
votes
2answers
129 views

Ways to induce a topology on power set?

In this question, two potential topologies were proposed for the power set of a set $X$ with a topology $\mathcal T$: one comprised of all sets of subsets of $X$ whose union was $\mathcal T$-open, one ...
0
votes
1answer
67 views

Creative easy combinatorics problems. [closed]

I would like cool problems of the following style: how many marbles need to be taken out of a jar to guarantee we have one of each color? I need some cool problems for some classes I want to give to ...
6
votes
3answers
208 views

What are some properties that imply that a group must be the trivial group?

In the problem posed in this question of mine we want to show that a particular group is both perfect and solvable, and therefore trivial, and this turns out to be useful in proving the result. What ...
15
votes
3answers
289 views

which exact integration techniques belong in a first year calculus/analysis course?

At our university we are now discussing changes to the course contents and there is some heated discussion regarding integration in the first year calculus courses. Currently, the techniques of exact ...
7
votes
1answer
106 views

What is your favorite group? [closed]

I would like to know about your favorite group(s). Since groups do appear everywhere in mathematics and there are plenty of them, which ones have drawn your attention the most or surprised you? Please ...
2
votes
2answers
163 views

When are analytical solutions preferred over numerical solutions in practical problems?

In most engineering or applied math papers that I read, the authors seem to obtain solutions to say, a system of differential equations, using numerical methods, rather than analytical techniques. ...
0
votes
0answers
93 views

What are some great Bachelor's Project subjects in the field of Mathematical Optimization Theory?

Currently, I ought to pick a subject for my third year mathematics bachelor's thesis. I would like to research something in the field of mathematical optimization theory. I have a background in basic ...
0
votes
1answer
133 views

Most “beautiful” presentations of the basic proofs for vector spaces?

I am familiar with the standard proofs presented in textbooks for stuff like linear independence/dependence, the dimensions of common vector spaces, any basis for a vector space V must be linearly ...
2
votes
1answer
284 views

Lecture Notes in Real Analysis

I understand that this question was partially addressed here but I would like to have a question dedicated to just real analysis. I am looking for both elementary real analysis (advanced calculus type ...
4
votes
2answers
114 views

Theorems that have proofs from the outside of the original field of math

I would like to know more examples of theorems, which "belong to one field of math", but their proofs are from the "outside of the field". I am mostly interested in proofs that are not too long ...
6
votes
2answers
127 views

Sources of Elementary Number Theory Problems

I am looking for sources of interesting and challenging problems that would suitably accompany an honors level introductory number theory course. What are some good sources for interesting elementary ...
17
votes
3answers
283 views

Results in graph theory proved using other areas of math, and vice versa

I'm curious about learning graph theory, as it seems to pop up in some unexpected places. In order to get a partial feel for the subject, I was wondering if anyone could point me to some survey ...
7
votes
1answer
227 views

“All math is useful eventually”

We have all heard the argument : a lot of mathematics that was thought to be useless, abstract constructions with no links to the real world ended up being of use, like some arithmetic is useful in ...
2
votes
2answers
64 views

when is a ring a free module over a subring?

Let $S \subset R$ be rings, $S$ not necessarily an ideal of $R$, and $S \neq R$. Is there anything that can be said about when $R$ is free as an $S$-module?
2
votes
1answer
133 views

Statements equivalent to the Axiom of Choice

The Axiom of Choice reads: The product of a collection of non-empty sets is non-empty. As you know well, this axiom is equivalent to many other statements. A few examples (probably the most ...
10
votes
1answer
324 views

What are some examples of hard theorems in category theory?

I'm currently learning some category theory, but so far I've used it only as a handy way to talk about various related concepts in algebra and topology with some nice, easy-to-prove lemmas like "left ...
17
votes
1answer
512 views

“Bad” Mathematics in Movies

There's a website and a companion book to it about bad physics in movies, called "Insultingly Stupid Movie Physics". Similar issues may exist about mathematics: What are the differences between ...
2
votes
3answers
64 views

“Asymmetric” results in maths analogous to “Parity violation” of the weak force?

Disclaimer: I'm not a physicist and I don't claim to be one so if I have any mistakes I’ll be glad to be corrected. One feature of the standard model of particle physics is that the weak force is not ...
30
votes
14answers
776 views

Interesting but short math papers?

Is it ok to start a list of interesting, but short mathematical papers, e.g. papers that are in the neighborhood of 1-3 pages? I like to read them here and there throughout the day to learn a new ...
3
votes
3answers
159 views

What are some examples of “exotic” algebraic structures? [closed]

I guess that I'm quite familiar with the basic "everyday algebraic structures" such as groups, rings, modules and algebras and Lie algebras. Of course, I also heard of magmas, semi-groups and monoids, ...
1
vote
0answers
34 views

limit formalisms

Let $f:\mathbb R \to \mathbb R$ be a function and $a\in \mathbb R$ a point. The Cauchy definition of the limit $\lim _{x\to a}f(x)=L$ is well-known. For pedagogical reasons I'm interesting in a ...
2
votes
1answer
114 views

Book about elementary geometry , triangles, circles … [duplicate]

Currently, I'm studying a little about geometry and I was trying to find out some good book about it on internet, however I didn't find anything that I thought nice to me or what I really expected to ...
2
votes
3answers
399 views

Book with lots of geometry theorems

I want to study geometry and was looking for some book that has lots of theorems and covers almost all Euclidean geometry that is needed for High School and Maths Olympiads. Thanks.
18
votes
1answer
492 views

The most active fields of mathematics?

Which fields of mathematics are the most active at this time -- by number of papers published, grant money, people working in them or by any other measure? Any trends in this regard?
1
vote
0answers
61 views

How calculators compute. [duplicate]

I would like to teach a class on the "magic" behind the calculator, so I would like to generate a list of "algorithms" for how a calculator computes the things we want it to. I will get the ball ...
3
votes
3answers
256 views

Examples where derivatives are used (outside of math classes)

I want to know what is the use of derivatives in our daily life. I have searched it on google but i haven't find any accurate answer. I think it is mostly used in Maths but I want to know its use in ...
26
votes
5answers
439 views

Other interesting consequences of $d=163$?

Question: Any other interesting consequences of $d=163$ having class number $h(-d)=1$ aside from the list below? Let $\tau = \tfrac{1+\sqrt{-163}}{2}$. We have (see notes at end of list), ...
3
votes
1answer
213 views

Best book ever on Galois theory (and differential galois theory) [closed]

Which is the single best book for Galois theory (that includes differential Galois theory) that everyone who loves pure Mathematics should read?
2
votes
1answer
58 views

Find multiple integrals $I_{\max}(k,n)$ and $I_{\min}(k,n)$ in various ways

$I_{\max}(k,n)=\underbrace{\int\limits_0^1\int\limits_0^1\dots\int\limits_0^1}_k\left(\max\limits_{1\le i\le k}x_i\right)^n\,dx_1dx_2\dots dx_k$ ...
9
votes
4answers
227 views

Math games for car journeys

On long car journeys with kids we are all familiar with "I spy" or "Twenty questions". What math related games can one play on a car journey instead that are fun and offer similar variety?
5
votes
1answer
83 views

Nice examples of finite things which are not obviously finite

This question is in the spirit of the question "Nice examples of groups which are not obviously groups". There are many impressive finiteness results in mathematics. For example: The finiteness of ...
0
votes
1answer
62 views

Real examples of non-positive definite integrals

We need to get some examples of non-positive definite integrals in real practice (mathematical physics, engineering, mathematical statistics and etc).
3
votes
1answer
251 views

Good book for logic self-study

I know a similar question has already been asked, but can anyone suggest a good book on mathematical logic that includes answers to exercises? I am looking for something that is conducive to ...