For questions that don't admit a definitive answer. Please do not ask too many of these.
21
votes
2answers
802 views
What can we learn about a group by studying its monoid of subsets?
If $G$ is a group, then $M(G)=2^G$ is has a monoid structure when we define $AB$ to be $\{ab|a\in A,b\in B\}$ and $1_{M(G)}=\{1\}$. How much of the structure of $G$ can be recovered by studying the ...
21
votes
3answers
978 views
A Young Math Enthusiast's Fear
Having been an avid lover of Mathematics, it is my dream to become a mathematician one day. It is purely passion that drives me and I just can't stand the thought of not doing math when I grow older. ...
21
votes
4answers
628 views
Is there anything deep about Fuzzy sets/Fuzzy logic?
I keep coming across the term "fuzzy". I browsed around a bit and read a few articles.. It seems to me that there's absolutely nothing deep or foundational about fuzzy set theory or fuzzy logic. I ...
20
votes
9answers
2k views
Does a Person Need a Mathematics Degree in order to Publish in a Mathematics jounal?
I am a neophyte amateur mathematician. I have been reading a lot about journals and the topic of peer-review in mathematics journals. Does one have to have professional credentials or have a Doctorate ...
20
votes
22answers
6k views
Best Maths books for non-mathematicians
I'm not a real Mathematician, just an enthusiast. I'm often in the situation where I want to learn some interesting Maths through a good book, but not through an actual Maths textbook. I'm also often ...
20
votes
5answers
1k views
Is speed an important quality in a mathematician?
Is solving problems quickly an important trait for a mathematician to have? Is solving textbook/olympiad style problems quickly necessary to succeed in math? To make an analogy, is it better to be a ...
20
votes
15answers
2k views
Useful examples of pathological functions
What are some particularly well-known functions that exhibit pathological behavior at or near at least one value and are particularly useful as examples?
For instance, if $f'(a) = b$, then $f(a)$ ...
20
votes
6answers
953 views
Do online lecture recordings hurt or help math students at university? [closed]
Continuing with my series of soft questions on teaching practice:
My university uses a system whereby all lectures (given via computer slides or hand-writing on a sort of overhead projector called a ...
20
votes
5answers
1k views
Is it possible to be a mathematician without being mathematically talented?
Apparently I'm not mathematically talented enough to be hired as a postdoc. Is there any way to earn a living while continuing to do mathematics I love (commutative algebra)? I don't like teaching or ...
20
votes
3answers
913 views
phd qualifying exams
Where can I find phd qualifying exams questions.Is there any website that keeps a collection of such problems?
I need it for doing some revision of the basic topics.I know of a book but that do not ...
20
votes
5answers
715 views
How is a system of axioms different from a system of beliefs?
Other ways to put it: Is there any faith required in the adoption of a system of axioms? How is a given system of axioms accepted or rejected if not based on blind faith?
(PD: I'm not religious)
20
votes
8answers
731 views
Looking to attain fluency in mathematics, not academic mastery
I'm a business/international relations person and a lot of my job is flying around. I have had a lot of downtime recently, and couldn't find a sustainable hobby to fill in that time.
Until I found ...
20
votes
4answers
1k views
What are the dangers of visual exposition of mathematics?
I've heard several times (such as this one) that it's dangerous to learn/prove/teach mathematics through images. I've also read somewhere that showing mathematics through images helps one's intuition ...
20
votes
6answers
564 views
What is your favorite online graphing tool?
I'm looking for a nice, quick online graphing tool. The ability to link to, or embed the output would be handy, too.
20
votes
3answers
908 views
Why is the Daniell integral not so popular?
The Riemann integral is the most common integral in use and is the first integral I was taught to use. After doing some more advanced analysis it becomes clear that the Riemann integral has some ...
20
votes
8answers
2k views
Which simple puzzles have fooled professional mathematicians?
Although I'm not a professional mathematician by training, I felt I should have easily been able to answer straight away the following puzzle:
Three men go to a shop to buy a TV and the only one ...
20
votes
4answers
501 views
Why do we study prime ideals?
I hope this isn't an inappropriate question here!
I'd like to ask the following (perhaps slightly ill-posed) question: why do we study prime ideals in general (commutative or non-commutative) rings? ...
20
votes
3answers
597 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, ...
20
votes
3answers
2k views
How do people apply the Lebesgue integration theory?
This question has puzzled me for a long time. It may be too vague to ask here. I hope I can narrow down the question well so that one can offer some ideas.
In a lot of calculus textbooks, there is ...
20
votes
1answer
2k views
Grad school with low undergrad GPA
I'm hoping to apply to grad schools this fall. I think I'm a reasonably good candidate, one aspect aside -- I have around a 2.5 undergraduate GPA in-major and failed several math classes in college.
...
20
votes
4answers
376 views
Is $\mathbb{N}$ impossible to pin down?
I don't know if this is appropriate for math.stackexchange, or whether philosophy.stackexchange would have been a better bet, but I'll post it here because the content is somewhat technical.
In ZFC, ...
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 ...
20
votes
0answers
1k views
Undergraduate Research Topics in Analysis [closed]
I am thinking of doing some research (Real Analysis preferably) under a professor this year and I need ideas for interesting topics that might be accessible to an undergraduate.
Does anyone have any ...
19
votes
6answers
674 views
How to learn from proofs?
Recently I finished my 4-year undergraduate studies in mathematics. During the four years, I've met all kinds of proofs. Some of them are friendly: they either show you a basic skill in one field or ...
19
votes
1answer
435 views
How to read a book in mathematics?
How is it that you read a mathematics books?
Do you keep a notebook of definitions? What about theorems?
Do you do all the exercises? Focus on or ignore the proofs?
I have been reading Munkres, ...
19
votes
1answer
663 views
How difficult is it to get an academic job?
Like many others here, I hope to someday become a professor. I've heard that it's very, very difficult and competitive in pure math, and I thought I would come here to ask your opinions.
For those of ...
19
votes
1answer
723 views
Do other non-native english speakers have trouble solving certain kinds of problems?
There are certain problems, I am noting that I am unable to solve because I cannot comprehend what exactly is being asked. The problems themselves might be non wordy, but I try to formulate it in ...
18
votes
8answers
990 views
Real life usage of Benford's Law
I recently discovered Benford's Law. I find it very fascinating. I'm wondering what are some of the real life uses of Benford's law. Specific examples would be great.
18
votes
3answers
875 views
Math without infinity
Does math require a concept of infinity?
For instance if I wanted to take the limit of $f(x)$ as $x \rightarrow \infty$, I could use the substitution $x=1/y$ and take the limit as $y\rightarrow 0^+$.
...
18
votes
4answers
2k views
Is a good GRE score enough for a non-math graduate to be accepted in a decent pure mathematics graduate program?
I have a computer engineering degree , and i have studied several mathematics courses like single variable and multiples variables calculus , complex variables , probability , numerical analysis ... ...
18
votes
6answers
463 views
Struggling with “technique-based” mathematics, can people relate to this? And what, if anything, can be done about it?
I'm a 3rd year undergraduate, majoring in pure mathematics. I've done well in the "proof-based" subjects I've taken, and I think that's because I understand the "rules of the game." That is, predicate ...
18
votes
3answers
978 views
Motivation and methods for self-study
First, a little background: Beginning with calculus in my first semester of college, I fell in love with mathematics. That was the point at which the concepts became interesting to me, and I started ...
18
votes
2answers
159 views
+500
Conjectural closed-form representations of sums, products or integrals
What are some examples of infinite sums, products or definite integrals that have conjectural closed form representations that were confirmed by numerical calculations up to whatever maximum precision ...
17
votes
17answers
1k views
Which mathematicians have influenced you the most?
This question is lifted from Mathoverflow.. I feel it belongs here too.
There are mathematicians whose creativity, insight and taste have the power of driving anyone into a world of beautiful ideas, ...
17
votes
8answers
2k views
Why do introductory real analysis courses teach bottom up?
A big part of introductory real analysis courses is getting intuition for the $\epsilon-\delta$ proofs. For example, these types of proofs come up a lot when studying differentiation, continuity, and ...
17
votes
4answers
686 views
“Casual” mathematical facts with practical consequences
Some mathematical facts -be them approximations or not- can be described as coincidences, without any deeper meaning in themselves, but leading to relevant practical consequences. I was thinking in ...
17
votes
9answers
1k views
Vivid examples of vector spaces?
When teaching abstract vector spaces for the first time, it is handy to have some really weird examples at hand, or even some really weird non-examples that may illustrate the concept. For example, a ...
17
votes
3answers
744 views
Research in algebraic topology
I have started studying algebraic topology with the help of Armstrong(Basic), Massey, and Hatcher. If I plan to do research in algebraic topology in future:
What else should I study after ...
17
votes
6answers
759 views
I'm teaching a college geometry course. What should I cover?
I've been asked to teach "Foundations of Geometry" at the University of South Carolina. Apparently, professors in the past have all done very different things, and I have a lot of choice in the ...
17
votes
5answers
1k views
Soft Question - Intuition of the meaning of homology groups
I'm studying homology groups and I'm looking to try and develop, if possible, a little more intuition about what they actually mean. I've only been studying homology for a short while, so if possible ...
17
votes
2answers
703 views
Grad school & success in the long run
I am an undergraduate mathematics student at some university in Turkey. I am doing well in math classes. Actually, I did not study very hard in my freshmen year and I got not so high grades from ...
17
votes
4answers
550 views
A question about the definition of a neighborhood in topology
Let $X$ be a topological space, and $x \in X$ be a point. There are two prevalent conventions on how to define a neighborhood of $x$:
1) A neighborhood of $x$ is any open subset $W \subset X$ such ...
17
votes
3answers
360 views
How much time is too much (to put into a single problem)?
As a self-studier, I have no due dates and no time pressure. This is partly a good thing since I do have a life outside of my pursuit of mathematics. However, the lack of pressure is also a bad ...
17
votes
3answers
271 views
Why should “graph theory be part of the education of every student of mathematics”?
Until recently, I thought that graph theory is a topic which is well-suited for math olympiads, but which is a very small field of current mathematical research with not so many connections to ...
17
votes
2answers
291 views
Mathematical suggestions for a long commute
I have quite a long commute on my way to my university (about 75 minutes) so I was wondering if there's a math-related podcast or something of the sort you'd recommend. Exercises are appreciated as ...
17
votes
2answers
383 views
Publication Quality Mathematics Diagrams
I am curious what sort of applications people use to make very nice diagrams that often appear in papers and books. I attached an example of the sort of diagram that I am interested in making, ...
17
votes
1answer
185 views
Understanding what $\sqrt{p}$ means for an event of probability $p$
Say I have a random event $E$ with probability $p$. There is a natural interpretation in terms of $E$ for the probability $p^2$: it's the probability that $E$ occurs twice if I perform two independent ...
17
votes
1answer
336 views
How to find exponential objects and subobject classifiers in a given category
In a course I'm learning about Topos theory, there are a lot of exercises which require you to prove explicitly some category is an elementary topos: i.e. to construct exponentials and a subobject ...
16
votes
9answers
906 views
Mathematics and Music
I have heard that, in recent years, many mathematicians as well as music theorists have applied different branches of mathematics to music.
I would like to know about some books/resources relating to ...
16
votes
5answers
1k views
Is 1+1 =2 a theorem?
A theorem is defined to be a mathematical statement that is proven to be true. The statement $1+1=2$ has definitely been proven in the history of mankind (Russel and Whitehead had once proven it in ...
