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7
votes
2answers
833 views

Mathematical places to visit [on hold]

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 ...
12
votes
6answers
1k views

What is a field?

I've always wondered about what a field is meant to represent. For example, group automorphisns naturally represent symmetry in many areas. I'm not looking for a solid answer, just an idea.
3
votes
3answers
882 views

Ring theory exercises at the graduate level

Do you know any book or an online source that contains exercises on ring theory? I've solved some exercises of Lang's Algebra and Dummit & Foote's Abstract Algebra but there is a huge gap between ...
21
votes
1answer
741 views

Useful mathematical fora

I want to ask about various mathematical fora and discussion boards available online. I think it might be useful to have such a list here at Math.SE. If I may suggest, it could be useful to keep one ...
15
votes
6answers
8k views

Is there any difference between mapping and function?

I wonder if there is any difference between mapping and a function. Somebody told me that the only difference is that mapping can be from any set to any set, but function must be from $\mathbb R$ to ...
7
votes
3answers
673 views

The pronuncation of “Tychonoff” and “Alaoglu”

I am not quite sure this is the place to ask this sort of question, but I am gonna give a talk on Banach algebra in which I will use theorems named after these two mathematicians whose names I can not ...
5
votes
5answers
1k views

Side-stepping contradiction in the proof of ; ab = 0 then a or b is 0.

Suppose we need to show a field has no zero divisors - that is prove the title - then we head off exactly like the one common argument in the reals (unsurprisingly as they themselves are a field). ...
1
vote
2answers
224 views

If we don't know the actual value of PI then how can we use it in formulations or rely on it for real world calculations? [duplicate]

Possible Duplicate: Do We Need the Digits of $\pi$? Are we at best, estimating things when we use formulas involving pi to describe the area, etc. of things?
2
votes
0answers
111 views

Lectures of many valued logic

I am looking for a good introduction to this topic... something with lots of examples and models would be nice. I am specially interested by the case where the truth values are open sets in a ...
9
votes
3answers
465 views

What is combinatorial homotopy theory?

Edit: After a discussion with t.b. we agreed that this question aims to a different answer from this one, for more information you can read the comment below. Many times I've heard people ...
41
votes
7answers
4k views

What does it take to get a job at a top 50 math program in the U.S.?

I'm a senior undergrad right at a small liberal arts college right now who is applying to math PhD programs in the U.S. I would like to eventually become a professor at a relatively good university ...
63
votes
4answers
8k 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 ...
10
votes
1answer
1k views

Early specialization and career development

I'm in the last year of my undergraduate studies. I recently decided to become a mathematician and I will apply for a two year master degree as a preparation for a PhD. Looking to various master ...
105
votes
5answers
7k views

What do modern-day analysts actually do?

In an abstract algebra class, one learns about groups, rings, and fields, and (perhaps naively) conceives of a modern-day algebraist as someone who studies these sorts of structures. One learns about ...
1
vote
1answer
346 views

Infinite induction “valid”

As you may know, induction works only when we have a statement involving natural numbers. For instance, For every $n$, the intersection of $n$ open sets is open. Now, the corresponding statement for ...
12
votes
6answers
590 views

Read old articles instead books.

I'd like to know if there is a site, or maybe a collection of books, where I can read old articles in mathematics in order to study topics directly from the source, instead reading books in the field. ...
8
votes
2answers
434 views

What should be the next step?

This is a soft/educational question and I'll flag it to be made community wiki. A little bit of background, first. I am in my last undergraduate year, and I took a graduate course in category theory; ...
12
votes
3answers
2k views

Best way to learn Algebraic Geometry?

I've been reading the book Commutative Algebra with a view towards Algebraic Geometry. I was wondering is the best way to learn algebraic geometry through commutative algebra? As the book I'm ...
3
votes
1answer
135 views

summing series using circles inside curves

After watching the infinity elephants video http://www.youtube.com/watch?v=DK5Z709J2eo and seeing how a geometric series could be represented by drawing a circle between a pair of lines, then the ...
2
votes
0answers
452 views

Relationship between Abstract Algebra and Probability Theory

Is there a relationship between abstract algebra and probability theory? I ask this because of the following laws: Axiom of Countable Additivity: If $A_1, A_2, \dots \in \mathcal{B}$ (where ...
3
votes
1answer
148 views

Heuristics suggesting a unit interval is uncountable

This is maybe a soft question, I am not sure yet. Anyway, I am delivering a 8 (+ 4 supervisions) hour course on 'basic set theory' for undergraduates : set notation, bijections, functions, ...
3
votes
3answers
490 views

What are some good examples for suggestive notation?

Motivation: Today I first wondered about and later remembered why the set of all functions from a set $X$ to $Y$ is denoted $Y^X$. They wikipedia page gives the explaination "The latter notation is ...
4
votes
2answers
642 views

How should I teach a high school student about inverse functions?

Today I tried to teach a high school student about inverse functions. I gave him this problem and defined the parts that he didn't understand: Let $f: \mathbb{R} \to \mathbb{R}$, $x \mapsto ...
10
votes
5answers
1k views

The status of high school geometry

Okay, so we've all seen Euclidean geometry in primary and high school. Back then, I really thought of points as indivisible entities in space and lines as 'breadthless lengths'. As far as I could ...
12
votes
2answers
627 views

How rare is it that a theorem with published proof turns out to be wrong?

There is a story I read about tiling the plane with convex pentagons. You can read about it in this article on pages 1 and 2. Summary of the story: A guy showed in his doctorate work all classes of ...
2
votes
2answers
149 views

Testing what probability model to use

Suppose we have some random variable $X$ that ranges over some sample space $S$. We also have two probability models $F$ and $G$. Let $f(x)$ and $g(x)$ be the probability density functions for these ...
1
vote
3answers
280 views

Practical implementation of Mathematics

I am 21 years old now. I have been studying Mathematics in school, college and now in university but I simply feel that I don't practically understand it. For example, I can solve complex questions on ...
26
votes
3answers
920 views

When to skip sections of a book when self-studying?

When you're taking a mathematics class, you usually know exactly what sections of a book you need to know, and you can focus your time on these important sections. However, when studying by myself, ...
13
votes
6answers
751 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
0answers
101 views

What is the convention for using results of theorems left as exercise in the text?

I'm working on an exam, and have a solid proof for one of the problems, but it's reliant on a number of theorems left exercises in the textbook which were not assigned as coursework. What, if any ...
7
votes
4answers
1k views

Citations in Math Papers

I asked a similar question previously, though this is more specific and directed. In the writing of mathematics research papers, when is information cited, such as definitions? I have read that if ...
4
votes
1answer
184 views

Found a simpler proof, now how do I know if it's original?

I've found a simpler proof for some identity/theorem, hypothetically speaking, of course ;) How do I know if it hasn't been done before? For important results it's fairly easy to find. By the way, I ...
18
votes
3answers
2k views

Books to study for Math GRE, self-study, have some time.

I just graduated from a regional university in the US with a minor in mathematics. There is a masters program overseas, for economics, that I want to attend but they require applicants to take the ...
2
votes
0answers
198 views

A classifier based on the Mandelbrot set

I am trying to link these somewhat disparate concepts- a binary classifier and a fractal structure as I think that a binary classifier based on such a structure, in particular Mandelbrot set would be ...
6
votes
1answer
761 views

Is category theory useful in higher level Analysis?

What I mean by higher level before this gets closed is functional analysis, complex analysis and harmonic analysis? I've read looked at the examples in most category theory books and it normally has ...
8
votes
2answers
972 views

Learning roadmap for Class Field Theory and more

I consider to study Class Field Theory and Advance Number Theory by my self this next semester. However, there are many books, lecture notes... I would like to choose the most comprehensive book to ...
4
votes
3answers
211 views

Why $2\sqrt{x} + \sqrt{3}$ can’t be simplified any further?

In How do you simplify the addition of square root, it has been pointed out that it cannot be simplified any further. As obvious as it seems, the following questions seemed need to be addressed: ...
4
votes
2answers
1k views

How to solve this type of Puzzles (Syllogism)?

I have seen lot of questions of below type predictions. I can't figure out the answer. Is there any common method to solve this ? Please consider the following example ...
3
votes
2answers
358 views

Collatz conjecture - example on the opposite situation

In the "Collatz conjecture" we want to find a number that makes the process go on forever never reaching 1. I want to find an example - a problem like "Collatz conjecture" but where you have to find a ...
6
votes
3answers
322 views

Foreign undergraduate study possibilities for a student in Southeastern Europe

In the (non-EU) country I live in, the main problem with undergraduate education is that it's awfully constrained. I have only a minimal choice in choosing my courses, I cannot take graduate courses, ...
20
votes
5answers
2k views

Is it worth it to get better at contest math?

I have never done well in math competitions, and am now past the point at which I can participate in them. I am asking if it is worth it to go back and practice such types of problems until I gain ...
49
votes
7answers
2k views

How to explain to the layperson what mathematics is, why it's important, and why it's interesting [closed]

A mathematician walks into a party. No, this is not the beginning of another joke, nor of a graph theory problem, but rather the beginning of a frequent and often frustrating real-life situation. ...
8
votes
1answer
594 views

Old French papers which haven't been translated into English

So I've been studying French for a few years now and I've decided that translating an old French math paper into English would be a good exercise to further improve my French competency. I would also ...
70
votes
12answers
7k views

Can you give an example of a complex math problem that is easy to solve?

I am working on a project presentation and would like to illustrate that it is often difficult or impossible to estimate how long a task would take. I’d like to make the point by presenting three math ...
14
votes
3answers
1k views

Learning math-oriented French

I'd like to read several papers which I find interesting, but they are all in French. I have no problem with taking a traditional French class or learning it via some other method. However, I realize ...
3
votes
2answers
1k views

Mathematics needed to understand Quantum Topology+Quantum Algebra?

I've recently been given a book called Quantum Topology by Louis H. Kauffman from a friend of mine. I was wondering what branches of mathematics do I need to be able to read this? What branches of ...
0
votes
3answers
305 views

A list of all algebras?

Reading @Qiaochu Yuan's blog the other day, I came across a new term: Poisson algebra. I had never heard of it before and wondered what other algebras are out there that I'm not aware of. Neither ...
10
votes
5answers
721 views

Algebraic topology, etc. for Mac Lane's “Categories for the Working Mathematician”

[NOTE: For reasons that I hope the question below will make clear, I am interested only in answers from those who have read Mac Lane's Categories for the working mathematician [CWM], or at least have ...
2
votes
3answers
420 views

Uniform distribution on $\mathbb Z$ or $\mathbb R$

I was assisting once the course in Probability Theory where students learnt quite quickly that there are ways to assign the uniform distribution to any finite set - or even subsets of $\mathbb R$ of a ...
3
votes
2answers
240 views

Introduction to proofs with a fair amount of hand-holding?

Lately I've gotten a friend of mine interested in mathematics. He has no college-level education to speak of, but is well employed as a software engineer. So I feel he's competent to learn this stuff ...