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2
votes
1answer
57 views

Homotopy Type Theory prerequisites.

I've done some undergraduate level study of algebraic topology (most Hatcher's book) and the smallest amount of type theory in a foundations of mathematics course. Homotopy type theory sounds amazing ...
12
votes
13answers
910 views

What are some interesting sole exceptions or counterexamples?

Many theorems assert that a particular property holds for all objects in a class except those in a given list of exceptions. Examples of rules that admit precisely one exception include: All primes ...
-2
votes
0answers
20 views

Permutations and combinations - fun questions [on hold]

Well I am studying permutations and combinations, and I'm finding it quite an interesting topic. Surprisingly enough, it seems to have practical applications. I was looking for some questions on the ...
4
votes
2answers
146 views
+50

Good source on current 'views and thoughts' on mathematics

Recently I've become interested in the history of the way people think about mathematics. What I mean by this is for example how Godels proof basically put an end to the whole school of thought ...
43
votes
14answers
3k views

Largest “leap-to-generality” in math history?

Grothendieck, who is famous inter alia for his capacity/tendency to look for the most general formulation of a problem, introduced a number of new concepts (with topos maybe the most famous ?) that ...
6
votes
4answers
103 views

Sources for mathematics outside the mathematics world

In this question I would like to ask you about material showing the uses (or occurrences) of mathematics in the everyday world. The aim is to encourage with it a group of young undergraduate ...
1
vote
2answers
38 views

Is there a good short phrase for a point where a function is continuous but not smooth?

Given a point $x_0$ where a function $f$ is $C^0$ but not $C^1$, how could one call this point intuitively? I am not looking for a technically precise term (like a point where $f'$ is ...
344
votes
25answers
66k views

How to study math to really understand it and have a healthy lifestyle with free time?

Here's my problem. I'm studying math and when I really work hard, I think I understand things very good, but that comes at a big cost: in the last few years, I've had practically zero physical ...
0
votes
1answer
55 views

math books of undergraduate/graduate level without formula? [on hold]

Just wondering, is there a math book talks deep into the math ideas (maybe undergraduate or graduate level, so not the pre-algebra content), but comes with no or very few formula?
3
votes
1answer
55 views

Which courses should I take to prepare for PHD in Finance/Econ/OR

Since it's finally the end of the year, I would like to gain some insights about which course should I take that are most helpful to prepare for application to the PHD program in Finance/Financial ...
16
votes
0answers
143 views
+50

Effective Research Notes

Note-taking for research is vital to your success as a mathematician. As I look back at some of my handwritten notes, I realized how poor they were. I had thought to myself, "What happened?" I was ...
-4
votes
0answers
50 views

Give an example of a field K, an extension field F, a subring R of F containing K where R is not a field? [on hold]

Give an example of a field K, an extension field F, a subring R of F containing K where R is not a field? I had $\mathbb{C}$ as a field, $\mathbb{C}(x)$ as a field extension, and $\mathbb{C}[x]$ ...
2
votes
0answers
103 views

Can anyone explain this quote about how mathematicians think?

I found this quote by Stephen Wolfram on page 1177 of his book A New Kind of Science. Yet of the limited set of people exposed to higher mathematics, different ones often seem to think in ...
0
votes
0answers
47 views

What is some reason that there are no book bridge the gap of these three books

I am referring to the (beginner's text- Stochastic Calculus by Mircea Grigoriu and Introduction to Stochastic Calculus by klebaner.) and the advanced texts - stochastic differential equation by ...
9
votes
2answers
110 views

How to select the right books?

As the saying goes, "Give a man a fish, feed him for a day. Teach a man how to fish, feed him for life." I've always had a problem with selecting appropriate books. It could be a problem that I'm a ...
1
vote
0answers
41 views

Physical or geometric meaning of the trace of a matrix

The geometric meaning of the determinant of a matrix as an area or a volume is dealt with in many textbooks. However, I don't know if the trace of a matrix has a geometric meaning too. Is there ...
1
vote
8answers
385 views

Why Not Define $0/0$ To Be $0$?

For every number $x$, $x\times 0=0$, hence $\dfrac{0}{0}$ can be any number! So $\dfrac{0}{0}$ "is knows as indeterminate" [1]. But what if we define it to be $0$? I already have an answer, but ...
3
votes
1answer
40 views

What is the discrete log used for?

Perusing Wikipedia, I stumbled on the discrete logarithm. I looks interesting that we'd be able have a function that could solve $b^k=g$ for integers $b,k,$ and $g$. However, Wikipedia says "No ...
4
votes
1answer
685 views

Complex Variable vs Real Analysis 1

I took Real Analysis 1 last semester, and it was challenging, but not as bad as I thought it would be. I am considering taking Function of a Complex Variable this semester, but I am torn. I don't know ...
1
vote
1answer
65 views

subtle/annoying fallacious proofs [duplicate]

I've been invited to a maths themed Xmas after party. I need to prepare a selection of interesting, and relatively simple fallacious proofs which other guests will try and find the flaw in. I'm trying ...
1
vote
0answers
38 views

Motivation for Putnam (soft question)

This question may be too specific and too vague. But I'm curious about this. How highly are the applicants evaluated in PhD admission if they were ranked above the cutoff of honorable-mention in ...
2
votes
1answer
57 views

Relationship between measure theory and real analysis

Does measure theory generalize real analysis to abstract spaces? For example, you can now talk about convergence even on unordered fields.
3
votes
1answer
44 views

What journals/periodicals are appropriate for serious amateurs to submit findings/research?

Are there journals out there that publish work by serious amateurs (but not at the level of academic researcher). I think that MAA journal American Mathematical Monthly is a good example (I could be ...
4
votes
1answer
87 views

Why are infinite sums so much harder to calculate than the associated infinite integral?

It seems that with continuous functions, we have in calculus an apparatus for "short cutting" an infinite sum. However, when we move to the discrete case, it seems that we don't have the equivalent ...
6
votes
0answers
62 views

Examples of useful, insightful and interesting hand-waving

I am really amused by the answers to this question on "Most harmful heuristic" posed on MathOverflow, from which I've benefited a lot. However, it seems to me that some hand-waving may be really ...
27
votes
7answers
4k views

Genius mathematicians who never published anything

Amongst philosophers, Socrates is an example of a genius with a great influence on human history who never wrote anything. Almost all facts which are known about his revolutionary ideas are written by ...
679
votes
28answers
46k views

Can I use my powers for good?

I hesitate to ask this question, but I read a lot of the career advice from mathOverflow and math.stackexchange, and I couldn't find anything similar. Four years after the PhD, I am pretty sure that ...
19
votes
4answers
419 views

Why there is much interest in the study of $\operatorname{Gal}\left(\overline{\mathbb Q}/\mathbb Q\right)$?

Let's start for a simple quote from wikipedia: "No direct description is known for the absolute Galois group of the rational numbers. In this case, it follows from Belyi's theorem that the ...
6
votes
2answers
107 views

How to explain to a layman why Fermat's Last Theorem involves non-trivial math?

Fermat's Last Theorem states, given$$x^n + y^n = z^n$$ no three integers $x,y,z$ will satisfy the equation given integer value of $n$ greater than two. On the surface this seems like something that ...
3
votes
4answers
212 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 ...
43
votes
17answers
20k views

Good book for self study of a First Course in Real Analysis

Does anyone have a recommendation for a book to use for the self study of real analysis? Several years ago when I completed about half a semester of Real Analysis I, the instructor used "Introduction ...
55
votes
4answers
1k 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 ...
2
votes
1answer
128 views

Why is the axiom of choice not taught from the start to mathematics undergraduates?

I've recently discovered that the following theorems require the axiom of choice to be proven: every surjective function has a right inverse. a real-valued function that is sequentially continuous ...
4
votes
1answer
58 views

Is there a “unifying framework” for harmonic analysis?

Recently, I was exposed to a basic harmonic analysis course. Although the course is almost over, I still can't put my finger on what harmonic analysis is about. I have a vague idea that it is ...
1
vote
2answers
53 views

Probabilistic implications of the existence of non-measurable sets

Measure theory and probability theory are deeply connected through the interpretation of subset measures on the sample space as probabilities of events. A major (and somewhat disturbing) result from ...
43
votes
8answers
7k views

Can someone please explain the Riemann Hypothesis to me… in English?

I've read so much about it but none of it makes a lot of sense. Also, what's so unsolvable about it?
3
votes
2answers
114 views

Is “mixed math” a useful way to learn math?

I was reading a book about how mathematics was taught in Cambridge in the 19th century, and it struck me how much physics was included in the syllabus, and it wasn't optional but everyone had to learn ...
27
votes
9answers
3k 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 ...
5
votes
0answers
20 views

Discrete analogue of Green's theorem

Following formula concerning finite differences is in a way a discrete analogue of the fundamental theorem of calculus: $$\sum_{n=a}^b \Delta f(n) = f(b+1) - f(a) $$ We can think about the Green's ...
13
votes
3answers
590 views

Mathematical structures

Preamble: My previous education was focused either on classical analysis (which was given in quite old traditions, I guess) or on applied Mathematics. Since I was feeling lack of knowledge in 'modern' ...
8
votes
4answers
562 views

A Math book with an inspiring ethos?

I was for some time curious about William Feller's probability tract (first volume); luckily, I could lay my hands on it recently and I find it of super qualities. It provides a complete exposition of ...
92
votes
17answers
12k views

Is 10 closer to infinity than 1?

This may be considered a philosophy but is the number "10" closer to infinity than the number "1"?
4
votes
3answers
5k views

Abstract Algebra book with exercise solutions recommendations.

I am new to studying abstract algebra (and math in general). I've been reading Gilligan and Pinter's books. I am trying to improve my understanding by doing exercises. However none of the books I am ...
3
votes
0answers
40 views

Why is the slope-intercept form of the equation of a line often written $y=mx+b$? Why $m$ instead of $a$?

After a quick google search, I read something about Conway suggesting the $m$ having to do with "modulus" ... This seems odd to me, but perhaps there is some mathematical reason? I've heard of the ...
52
votes
16answers
6k views

Interesting “real life” applications of serious theorems

As a student one sometimes encounters exercises which ask you to solve a rather funny "real life problem", e.g. I recall an exercise on the Krein-Milman theorem which was something like: "You have a ...
-1
votes
0answers
42 views

List of simple, common functions with an incomplete domain or range on $\mathbb{C}$

This may seem like a strange question, but it's an interest of mine and I would appreciate the help of the community in addition to brainstorming on my own. As the question states, I'm looking for ...
59
votes
2answers
1k views
+100

Some users are mind bogglingly skilled at integration. How did they get there?

Looking through old problems, it is not difficult to see that some users are beyond incredible at computing integrals. It only took a couple seconds to dig up an example like this. Especially in a ...
1
vote
2answers
53 views

Self-Learning Geometry

I'm an undergraduate senior wondering where he should start in learning geometry. My university unfortunately offers no such course. Should i begin with riemann geometry or differential geometry and ...
0
votes
1answer
39 views

Probability in a Magical World

We know the frequentist definition of probability-the probability $p$ of an event $E$ is the limiting frequency the event happens when the associated random experiment is repeated large number of ...
1
vote
1answer
31 views

Request for Recommendation of a Handbook of pure Mathematics

I am looking for some books at undergrad level that give a good pedagogical coverage of of topics like topology,group theory,analysis,measure theory and probability with a nice exposition.Even books ...