For questions whose answers can't be objectively evaluated as correct or incorrect, but which are still are relevant to this site. Please be specific about what you are after.

learn more… | top users | synonyms

30
votes
2answers
919 views
+100

Intuitive reasoning why are quintics unsolvable

I know that quintics in general are unsolvable, whereas lower-degree equations are solvable and the formal explanation is very hard. I would like to have an intuitive reasoning of why it is so, ...
0
votes
1answer
7 views

online notes on symmetric spaces

Can anyone suggest some good online lecture notes on symmetric spaces? I am interested in reading from Helgason, which is a very tough book to read. So I am searching for some places where the ...
488
votes
145answers
30k views

What was the first bit of mathematics that made you realize that math is beautiful? (For children's book)

I'm a children's book writer and illustrator, and I want to to create a book for young readers that exposes the beauty of Mathematics. I recently read Paul Lockhart's essay "The Mathematician's ...
1
vote
1answer
20 views

What does an ideal generated by a subset look like?

I am acquainted with principal ideals relatively well but I am struggling to understand the more general definition of ideals generated by an arbitrary subset of a ring. I suppose my question comes ...
5
votes
1answer
70 views

Studying Deformation Theory of Schemes

Versal Property Local Deformation Space Mini-versal deformation space I came across these words while studying these papers a) Desingularization of moduli varities for vector bundles on curves, ...
4
votes
3answers
259 views

How to check my answer in combinatorics problems

Combinatorics problems (combinations and permutations) are an absolutely maddening subject for me. I can seem to work my way to the answer, provided I already know the correct answer. However, I can ...
6
votes
12answers
999 views
+50

Irrational numbers in reality

I have a square stone slab 1 metre by 1 metre, by the Pythagorean identity the diagonal from one corner to another is given by $\sqrt 2$. However $\sqrt 2$ is an irrational number, could someone ...
11
votes
1answer
173 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 ...
3
votes
3answers
2k views

Useful trigonometry tricks/shortcuts

I'm curious as to any "tricks" or shortcuts that could help make verifying/solving trigonometric identities easier, for example one is: $$a\cos\theta+b\sin\theta = \sqrt{a^2+b^2}\,\cos(\theta-\phi)$$ ...
16
votes
6answers
5k views

Calculus self taught? Books?

I recently graduated with a degree in bachelor of science with a focus interactive and multimedia design. I had to opportunity to take 1 C++ course and 1 HTML course. I was also only required to take ...
1
vote
0answers
34 views

Generalizing in Mathematics

I was reading the book "Fermat's Last Theorem" by Simon Singh when it hit me that this theorem is so contrived, andyet it lead to several important breakthroughs in mathematics and especially the ...
65
votes
5answers
2k 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 ...
1
vote
2answers
33 views

Significance and physical meaning of diagonalization of linear maps and bilinear forms, eigenvalues and eigenvectors

In linear algebra, I have studied the diagonalization of a linear map and of a bilinear form; and also the concepts of eigenvalues and eigenvectors. However, the importance of diagonalizing a linear ...
1
vote
0answers
17 views

Is it more useful to study lots of methods/theorems or work with details?

Suppose one has to learn a new subject on book that contains hundreds of pages and hundreds of problems. Is it more useful to learn book by reading it and skipping those parts I don't understand and ...
49
votes
17answers
16k views

Real life applications of Topology

The other day I and my friend were having an argument. He was saying that there is no real life application of Topology at all whatsoever. I want to disprove him, so posting the question here What ...
1
vote
1answer
42 views

Why do some authors use terms “non-ascending” and “non-descending” instead of ascending and descending?

In my math book, everywhere the author has used "non-ascending" instead of descending and "non-descending" instead of ascending. I was wondering if there is some special meaning or use associated with ...
2
votes
0answers
43 views

What abstract structures allows us to describe “nets that converge toward each other”?

Topological spaces are equipped with a bare minimum of structure to allow for a formalization of the statement "the net $a$ converges to the point $x$." Actually this isn't strictly true, but its true ...
11
votes
8answers
249 views

What are the theorems in mathematics, proved using completely different ideas?

I know this question can have many answers. But I would like to know about theorems which can give completely different proofs. For example: I read from the book "Proof from the Book," there ...
0
votes
2answers
26 views

Books on Lebesgue Integration

I am having Measure Theory as a subject in my course.It is having these as topics: 1.Lebesgue measure on the line 2.Measurable functions 3.Lebesgue integral 4.Convergence almost everywhere ...
0
votes
1answer
57 views

Understanding the condition of $\diamondsuit^+$

I try to understand $\diamondsuit^+$, a variant of the diamond principle. It says that there is a sequence $\langle \mathcal{A}_\alpha\rangle_{\alpha<\omega_1}$ of collection of subsets which ...
7
votes
1answer
152 views

How important is Differential Geometry for Number Theory?

The title pretty much says it. To elaborate slightly, I am, of course, aware of the huge role played by Algebraic Geometry in Number Theory but I'm not so sure about Differential Geometry. I would be ...
14
votes
5answers
330 views
+100

Lang's Linear Algebra: what's next?

I've completed the study of Lang's Linear Algebra ($3^\text{rd}$ edition). To put it simply, I have enjoyed the subject and I would like to know "what's next". In other words, I would like to know ...
0
votes
0answers
26 views

Different methods used to show the existence of integer solutions

Let $A_{n},B_{n},C_{n}$ be three sequences of positive integers. I want to know the different methods used to show the existence of integer solutions $x$ and $y$ for the equation: ...
0
votes
3answers
126 views

The obivious “why”-questions [on hold]

Whenever someone is introduced to a mathematical concept but doesn't stuck to it long enough there are always this "why"-questions. Why is $8+2=2+8$ etc. The answer to that is obvious I think for the ...
20
votes
10answers
903 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 ...
0
votes
4answers
99 views

Why non-real means only the square root of negative?

Once in 1150 AD, an Indian mathematician Bhaskara wrote in his work Bijaganita (algebra) that, There is no square root of a negative quantity, for it is not a square However later on in 1545 an ...
2
votes
2answers
63 views

Request for apps for Mathematical Drawing [duplicate]

I have been for long looking for some software apps which can help me draw various mathematical and geometrical figures and drawings.Can someone please tell me something about these which will run on ...
0
votes
2answers
72 views

Universe as a finite 3-manifold without boundary

My question is soft and imprecise, as I know very little differential topology. Nevertheless, I hope it makes some $\epsilon>0$ of sense. Assume the Universe is a 3-manifold without boundary, ...
88
votes
41answers
11k views

What's your favorite proof accessible to a general audience? [on hold]

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 ...
1
vote
0answers
18 views

Etymology of the term coherent in sheaf theory

Why has been introducted the term coherent in sheaf theory? What's its intuitive significance? In italian we translate it with the term "coerente", in the sense of compactness.
2
votes
0answers
23 views

Red-Black tree - “Insert-Delete” [on hold]

I am looking at red-black trees. Unfortunately in my lecture notes, the operations "Insert" and "Delete" are not well explained. Could you explain to me steps that we have to do for these two ...
3
votes
0answers
50 views

Combinatorics project ideas for high school students

It's that time again! Last year I asked for high school project ideas in the area of algebraic geometry, this year it's combinatorics (you can include graph theory and combinatorial game theory if you ...
0
votes
0answers
48 views

Taking Putnam as a freshman.

Currently, in 11th grade, I've always thong about participating exams like the Putnam. I have however, sent the problems, and they seem, to be grueling hard!! I have access to problem solving ...
1
vote
1answer
34 views

How should I think when combining multiple inequalities?

When reading/writing papers, I have always find it not obvious when two or more inequalities are combined. For example, taken from my current research $$\text{Pr}(X \le ab) \le -a (1-p)^{-N} (1 - ...
2
votes
1answer
78 views

Trigonometry book which develops geometric intuition.

I'm looking for a trigonometry text that helps develop a lot of geometric intuition and goes deep into the subject. Also some geometry problems which actually require thinking about would be in order. ...
2
votes
2answers
66 views

British “S-Level” Mathematics Books

The British S-level exams (not to be confused with A-levels or O-levels) were said to be challenging exams that were used to select who got a place at the University of Oxford or Cambridge. Is anyone ...
7
votes
1answer
274 views

The Purpose of Master Thesis

I am posting this question in the aftermath of the earlier posting in this link. Here are what I would like to know more about master and PhD thesis: (1) I understand that schools' math departments ...
3
votes
1answer
98 views

How important is the own talent for research of your PhD supervisor?

Currently I am in the process of finding a PhD. Some potential supervisors are more didactical than others, some are nicer and warmer than others, and some are more famous mathematicians than others. ...
33
votes
18answers
6k views

Examples of mathematical induction

What are the best examples of mathematical induction available at the secondary-school level---totally elementary---that do not involve expressions of the form $\bullet+\cdots\cdots\cdots+\bullet$ ...
2
votes
2answers
301 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 ...
0
votes
0answers
20 views
-1
votes
1answer
64 views

Best schools for commutative algebra [on hold]

I will be applying to graduate programs this fall and I was curious which schools have the best commutative algebra groups. I know berkeley and michigan are up there, but what are others?
1
vote
0answers
20 views

Different notation for Jacobi symbol

Is there a different, sort of established, notation for the Legendre / Jacobi / Kronecker symbol $\left(\frac{a}{b}\right)$? If yes, where is it used (in which texts)? I'm asking, because I ...
1
vote
0answers
31 views

Is Problem Solving Strategies by Engel sufficient?

Is a book like, Problem Solving Strategies by Arthur Engel sufficient for the Putnam Exam or should I consult something else? I asked a similar question asking for recommendation, no one discussed ...
0
votes
1answer
66 views

Book recommendation for Putnam/Olympiads

I have been concentrating on olympiad questions, and PUTNAM exams, Putnam is my main focus. Can you suggest a book from one of these: Problem Solving Strategies By Arthur Engel Putnam and Beyond by ...
12
votes
5answers
389 views

Beautiful Mathematical Images [on hold]

My Maths department is re-branding itself, and we've been asked to find suitable images for the departmental sign. Do you have a favourite mathematical image that could be used for the background of ...
7
votes
2answers
834 views

Mathematical places to visit [closed]

There are certain buildings and places on this planet where mathematicians can find delight because of the history, the art, the architecture, and for other reasons. For example, the Alhambra with ...
1
vote
0answers
24 views

Understanding the Jacobian past calculus

What's taught in calculus: In the calculus of multiple variables I learned that the Jacobian $$\textbf ...
0
votes
0answers
16 views

Role of functional equations in current panorama of pure mathematics

It seems that currently functional equations are greatly explored as a research field. I would like to know what is the importance and role of such a field in the panorama of the current development ...
0
votes
1answer
18 views

Motivation for the binary entropy function

What is the motivation for the definition of the binary entropy function $H(x) = -p\log_2(p) - (1-p)\log_2(1-p)$? I understand that we want the entropy to be zero at $p = 0$ and $p = 1$ (no ...