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.
23
votes
28answers
2k views
Classical texts that should not be missing from any shelf [closed]
It seems to me as if many modern texts are rather streamlined. They are designed not to expect too much from the reader but they often miss the depth of respective classical literature.
The purpose ...
20
votes
1answer
567 views
Expository articles on Analysis and Probability theory
When I come across a notion from algebra or number theory which I don't know I usually check Keith Conrad's page to see if he has written something about it. Key features of his articles are a very ...
-3
votes
1answer
143 views
Examples of interesting sequences [closed]
Any good series or sequences you have?
For example if you sum the reciprocal of primes this diverges.
As Q is de-numerable, almost any Cauchy sequence we pick will not converge in Q.
Stuff like ...
3
votes
1answer
296 views
Fundamental groups of certain 3-manifolds
I'm starting my master's thesis on geometry/topology & group theory.
I'd like to know examples of fundamental groups of 3-manifolds having geometric structure of the following types:
...
5
votes
2answers
370 views
Mathematical places to visit
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
1answer
289 views
Properties of generalized limits aka nets
I want to find some article or a book which contains all general properties of nets. Of course some of them similar to properties of sequences with almost the same proofs, but I don't fill the edge, ...
27
votes
4answers
2k views
Books that every student “needs” to go through
I'm still a student, but the same books keep getting named by my tutors (Rudin, Royden).
I've read Baby Rudin and begun Royden though I'm unsure if there are other books that I "should" be working on ...
27
votes
5answers
1k views
False beliefs about Lebesgue measure on $\mathbb{R}$
I'm trying to develop intuition about Lebesgue measure on $\mathbb{R}$ and I'd like to build a list of false beliefs about it, for example: every set is measurable, every set of measure zero is ...
13
votes
6answers
568 views
In what fields would you like to see applications of mathematics?
There are very few disciplines which mathematics has not penetrated. As a pupil finds such gem in the calculus problem of theory of rumors, he wonders if such field has application in vaudevillian ...
1
vote
2answers
129 views
Interesting problems concerning factorization and simplifictions of expressions
I feel that deep within the knowledge of many users here dwell great problems regarding clever tricks for simplifying problems, and factoring algebraic expressions. I would be very much interested in ...
12
votes
1answer
230 views
Understanding sub-atomic particles, for mathematicians
I have a masters degree in pure mathematics and I'm working towards my dream of a PhD, but I know very very little about sub-atomic particles. I would like to find some good popular science books or ...
8
votes
4answers
957 views
Famous uses of the inclusion-exclusion principle?
The standard textbook example of using the inclusion-exclusion principle is for solving the problem of derangement counting; using inclusion-exclusion (and some basic analysis) it can be shown that ...
1
vote
0answers
134 views
Problems and conjectures that have positive practical consequences for society, once solved
This question made me think a bit about how mathematics can be used in such a way that society benefits from it.
I think there are quite a lot of good answers to the aforementioned question. Still, ...
0
votes
2answers
1k views
Best books on Linear Algebra [duplicate]
Possible Duplicate:
Prerequisites/Books for Linear Algebra
I've studied from David Poole's Linear Algebra: A Modern Introduction However, it's not very complete. I want to study subjects as ...
18
votes
10answers
2k views
“Immediate” Applications of Differential Geometry
My professor asked us to find and make a list of things/facts from real life which have a differential geometry interpretation or justification. One example is this older question of mine. Another ...
4
votes
1answer
166 views
2-dimensional $\ell$-adic representations [closed]
In an assignment, I have to give an example of a 2-dimensional $\ell$-adic representation of the absolute Galois group of $\mathbb{Q}$, bu I am faced with the problem that I do not a lot of these. Or ...
17
votes
2answers
680 views
Geometric proof for inequality
While on AOPS, I saw this interesting problem. I was wondering how many different approaches could be used to tackle the problem.
In other words I am looking for interesting and unique ways to solve ...
2
votes
6answers
497 views
What are the most common errors in math exams: when is asked to study the function f(x)=…?
I'm new here and I would like to know what teachers have saw in their experience about errors in students exams;
I'm interested to know what are the most common errors in exams about "calculus", more ...
3
votes
2answers
594 views
Books for high school students starting on college math
I am a student beginning high school from India. I have recently developed a taste for physics and mathematics. I am doing Lagrangian and Quantum Mechanics in Physics. But my mathematics is not too ...
8
votes
2answers
537 views
Who are some forgotten mathematicians? [closed]
In Thomas' Calculus, he presents ''Nicole Oresme's Theorem'':
$$
\sum_{n=1}^\infty {n\over 2^{n-1}}=4.
$$
My first reaction was "who is this person?''.
As it turns out, he was a Frenchman from the ...
0
votes
0answers
408 views
Working through Math 55 problem sets as self-study
I am not a professional mathematician, but have learnt Engineering Mathematics in college and worked through parts of maths textbooks myself. The latter include the first few chapters of include Real ...
2
votes
0answers
185 views
Euler's problem and Proofs from the book [closed]
I have received the book "Proofs from the books" 4th edition by Springer, and I found 3 problems on Number Theory relate to L.Euler. They are "Representation number as sum of two squares", "Quadratic ...
13
votes
3answers
1k views
The main attacks on the Riemann Hypothesis?
Attempts to prove the Riemann Hypothesis
So I'm compiling a list of all the attacks and current approaches to Riemann Hypothesis. Can anyone provide me sources (or give their thoughts on possible ...
12
votes
7answers
1k views
What are some good iPhone/iPod Touch/iPad Apps for mathematicians?
There are lots of good apps for teaching mathematics to children but I would like to learn about apps for undergraduate/graduate/research levels.
Helper questions
Any algebra system (like ...
5
votes
2answers
273 views
Famous Finite Sets [closed]
What are the most famous (or most beautiful, IYO) finite sets in mathematics? I'm especially looking for 'large' sets that contain more than $2^{10} \approx 1000$ but fewer than $2^{20} \approx ...
14
votes
5answers
1k views
What are some common proof strategies in mathematics?
I want to start out by saying that I am new at proof based mathematics. I am used to seeing patterns and using them to solve similar problems. However, I have found this is not a very good way to ...
21
votes
10answers
5k 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 ...
23
votes
6answers
1k views
Do We Need the Digits of $\pi$?
I was reading today that someone found $\pi$ to the ten trillionth digit. Whenever I read that $\pi$ has been calculated to more digits, I ask myself whether this is useful. I know that there are ...
18
votes
10answers
1k views
Seeking a layman's guide to Measure Theory
I would like to teach myself measure theory. Unfortunately most of the books that I've come across are very difficult and are quick to get into Lemmas and proofs. Can someone please recommend a ...
32
votes
4answers
3k views
Lesser-known integration tricks
I am currently studying for the GRE math subject test, which heavily tests calculus. I've reviewed most of the basic calculus techniques (integration by parts, trig substitutions, etc.) I am now ...
4
votes
2answers
146 views
Expository texts on advanced subjects
I'm reading The Princeton companion to Mathematics and Basic Notions of Algebra by Shafarevich. Both of them are really pleasant reading, the first one treats the topics from a more elementary point ...
12
votes
2answers
806 views
Contemporary mathematician one should know about
While reading The Princeton companion to Mathematics Timothy Gowers choose not to list alive mathematicians in the last part but I think it's important to know about them. Recently I've read about ...
0
votes
1answer
44 views
Books on graphical models
I looked at Koller & Friedman's Probabilistic Graphical Models, but their use of non-standard notation is prompting me to see if there is anything else out there.
I'd like to find an introductory ...
6
votes
3answers
1k 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 ...
3
votes
0answers
42 views
Basis-dependent results on matrices
It is common fashion to try to formulate results about matrices in a basis-free way, using linear algebra. What are some good examples of situations where this is impossible? I illustrate what I have ...
2
votes
2answers
103 views
Topologies on spaces of mappings
Given two topological spaces $X, Y$, the only example I know of a topology on the space $\mathcal C(X,Y)$ of continuous mappings from $X$ to $Y$ is the compact-open topology. However I presume that ...
10
votes
3answers
966 views
Real-world uses of Algebraic Structures
I am a Computer science student, and in discrete mathematics, I am learning about algebraic structures. In that I am having concepts like Group,semi-Groups etc...
Previously I studied Graphs. I ...
4
votes
1answer
181 views
Request for “Nice” Papers in i) Differential Geometry ii)Analysis iii)Topology, to Go Over
Everyone:
I am a little frustrated with the progress in my bottom-up learning process, and I think I might get better results by doing some more top-down, i.e., by reading and trying to make sense of ...
3
votes
5answers
507 views
Primer on complex analysis and Riemann surfaces for undergraduate physics / theoretical physics majors
Ref: The Road to Reality: a complete guide to the laws of the universe, (Vintage, 2005) by Roger Penrose [Chap. 7: Complex-number calculus and Chap. 8: Riemann surfaces and complex mappings]
I'm ...
8
votes
2answers
216 views
What experimental-mathematical problem would you try to solve if you had a supercomputer?
At the present moment, what open mathematical problem do you seriously think you could solve if you had a very powerful computer at your disposition?
I mean something like the Four-Color Problem, i.e. ...
39
votes
19answers
2k views
What is your favorite application of the Pigeonhole Principle?
The pigeonhole principle states that if $n$ items are put into $m$ "pigeonholes" with $n > m$, then at least one pigeonhole must contain more than one item.
I'd like to see your favorite ...
1
vote
3answers
195 views
Recent practical advances
I love interesting and deep mathematical results, but on the other hand I cannot object when someone says that most likely all these complicating abstract theorem will not make a change to human kind ...
5
votes
2answers
720 views
Do there exist groups whose elements of finite order do not form a subgroup? [duplicate]
Possible Duplicate:
Examples and further results about the order of the product of two elements in a group
I was browsing around, and came across the little exercise that elements of finite ...
3
votes
0answers
520 views
Common tricks to compute series and integrals (common substitutions)
Tomorrow, I will have a test about Calculus 1 and 2 and up to now, I was solving exercises over and over again but I have the impression that I don't learn a lot from doing this. I believe that when ...
11
votes
2answers
351 views
Books on topology and geometry of Grassmannians
Grassmannians are a pretty useful subject in numerous fields of mathematics (and physics). In fact, it was the first non-trivial higher-dimensional example that was given in an introductory projective ...
13
votes
2answers
390 views
open conjectures in real analysis targeting real valued functions of a single real variable
I am hoping that this question (if in acceptable form) be community wiki.
Are there any open conjectures in real analysis primarily targeting real valued functions of a single real variable ? (it may ...
9
votes
5answers
459 views
Software to draw links or knots
I am looking for software that can aid me in drawing knots and links. There are of course (examples) knotplotters all over the web, but they can only draw specific knots.
What I am looking for is the ...
6
votes
0answers
283 views
Positive definite function zoo
A positive definite function $\varphi: G \rightarrow \mathbb{C}$ on a group $G$ is a function that arises as a coefficient of a unitary representation of $G$.
For a definition and discussion of ...
1
vote
0answers
104 views
Origins of mathematical terms? [closed]
I'm interested in the development and naming of mathematical terms that we probably take for granted. Why is integral called the integral? Who first used pathological in the mathematical sense? Please ...
-1
votes
1answer
331 views
Self-Contained Treatments of Stokes's Theorem for Manifolds [closed]
I am seeking to compile a list of textbooks that provide self-contained treatments of Stokes's Theorem in the language of differential forms and manifolds. By "self-contained", I mean the statements ...
