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Results tagged with soft-question
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user 3272
Questions that are about research in mathematics, or about the job of a research mathematician, without being mathematical problems or statements in the strictest sense. Do not use this tag for easy or supposedly easy mathematical questions.
42
votes
Accepted
Are there situations when regarding isomorphic objects as identical leads to mistakes?
Inside of the complex numbers there are lots of examples of distinct fields which are isomorphic. For instance, there are three subfields of the form ${\mathbf Q}(\alpha)$ where $\alpha^3 = 2$: take …
- 50.6k
10
votes
Accepted
"Epicycles" (Ptolemy style) in math theory?
Euler found values of the Riemann zeta-function by artful manipulations of divergent series, e.g., interpreting a function that's $(-1)^{n/2}$ at even $n > 0$ and $0$ at odd $n > 0$ as $\cos(\pi n/2)$ …
10
votes
What is the point of reading classics over modern treatments?
A question very similar to yours was asked quite a few years ago on MO here. Take a look at the answers there.
9
votes
Golden ratio in contemporary mathematics
In every real quadratic field $K$, the unit group of its ring of integers $\mathcal O_K$ is known to have the form $\pm u^\mathbf Z$ for a unique number $u > 1$, which is called the fundamental unit o …
22
votes
Origins of names of algebraic structures
Ring came from Zahlring, which was Hilbert's term for what we would essentially call a ring of algebraic integers. Dedekind earlier used the term ordnung (= order, taken from the Linnean classificati …
21
votes
Accepted
Sylow Subgroups
Victor, you should check out Sylow's paper. It's in Math. Annalen 5 (1872), 584--594. I am looking at it as I write this. He states Cauchy's theorem in the first sentence and then says "This import …
- 50.6k
16
votes
Are there any "related rates" calculus problems that don't feel contrived?
Here are two examples I think are interesting. (Update: I have added a third example at the end, more intricate than the other two.)
A ladder that is leaning against a wall starts slipping down. If t …
17
votes
Accepted
The interrelationship problem of modern mathematics – How to deal with it in first year grad...
Of course you should show students, taking into account their backgrounds, that the material they are learning in one course is relevant elsewhere. It makes it clearer to the students that topics th …
9
votes
Applications of functional analysis beyond analysis(towards algebra, geometry, number theory...
In number theory there are $p$-adic Banach spaces and $p$-adic Banach algebras (e.g., the Tate algebras), and more generally there is the whole subject of $p$-adic functional analysis. Applications of …
11
votes
Magic trick based on deep mathematics
Here is a card trick from Edwin Connell's Elements of Abstract and Linear Algebra, page 18 (it can be found online). I always do this trick to my undergraduate number theory class in the first minute …
23
votes
Accepted
Complex Analysis applications toward Number Theory
This question is like asking how abstract algebra is useful in number theory: lots of it is used in certain areas of the subject so there's no tidy answer. You probably won't be using Morera's theore …
- 50.6k
13
votes
Counterexamples in algebra?
If $f$ and $g$ are relatively prime in ${\mathbf Q}[X]$ then the mapping ${\mathbf Q}[X]/(fg) \rightarrow {\mathbf Q}[X]/(f) \times {\mathbf Q}[X]/(g)$ given by $h \bmod fg \mapsto (h \bmod f, h \bmod …
7
votes
Do empirical studies have a place in contemporary mathematics research?
The Birch and Swinnerton-Dyer conjecture was formulated on the basis of substantial computer calculations. They looked at the growth of products $\prod_{p \leq x} N_p(E)/p$ for large $x$ and various …
21
votes
Widely accepted mathematical results that were later shown to be wrong?
Any rational function field over a finite field has genus $0$ and class number $1$, where the class number of a function field over a finite field is the number of degree-zero elements of the divisor …
26
votes
Recent Applications of Mathematics
The new area of discrete differential geometry is solving problems in computer graphics, such as creating more lifelike hair in animation. I became aware of this from an article in the New York Times …